Painlevé analysis, Lie symmetries, and integrability of coupled nonlinear oscillators of polynomial type

PAINLEVE ANALYSIS, LIE SYMMETRIES, AND INTEGRABILITY OF COUPLED NONLINEAR OSCILLATORS OF POLYNOMIAL TYPE

M. LAKSHMANAN Centre for Nonlinear Dynamics, Department of Physics, Bharathidasan University, Tiruchirapalli-620 024, India and R. SAHADEVAN Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Madras-600 005, India

NORTH-HOLLAND

PHYSICS REPORTS (Review Section of Physics Letters) 224, Nos. I & 2 (1993) 1—93. North-Holland

PHYSICS REPORTS

Painlevé analysis, Lie symmetries, and integrability ofcoupled nonlinear oscillators of polynomial type M. Lakshmanan~and R. Sahadevan”~ ~ Centre /clr Nonlinear Dynamics, Department of Physics, Bharaihidasan University, Tiruchirapalli-620 024, India b) Ramanujan Institute for Advanced Study in Mathematics, University ofMadras, Madras-600 005, India Received August 1992; editor: D.K. Campbell Contents: I. Introduction 1.1. Preliminary considerations 1.2. Organization of the review 2. Classification of singular points 2.1. Fixed and movable singularities 2.2. Linear odes and fixed singularities 2.3. Nonlinear odes and movable singularities 3. Historical development of the Painlevé approach and integrability of odes 3.1. First-order nonlinear odes and Kovalevskaya’s rigid-body problem 3.2. Second-order nonlinear odes and Painlevé transcendental equations 3.3. Painlevé property of odes: strong and weak P-properties 4. Painlevé method of singular-point analysis for odes 4.1. Leading-order behaviour 4.2. Resonances 4.3. Evaluation of arbitrary constants 4.4. Further remarks 5. Painlevé analysis and integrability of two coupled nonlinear oscillators 5.1. Quartic anharmonic oscillators 5.2. Henon—Heiles system

3 3 5 6 6 7 7 8 8 9 10 11 12 13 15 17 19 19 25

5.3. Sextic anharmonic oscillators 5.4. Polynomial-potential nonlinear oscillators 6. Painlevé analysis and integrability of three coupled anharmonic oscillators 6.1. Quartic anharmonic oscillators 6.2. Sextic nonlinear oscillators 7. Application of Painlevé analysis to N coupled higher-dimensional anharmonic oscillators 7.1. N coupled quartic anharmonic oscillators 7.2. N coupled sextic anharmonic oscillators 8. Invariance analysis, Lie symmetries, and integrability of Hamiltonian systems 8.1. Invariance conditions, determination of infinitesimals and first integrals of motion 8.2. Applications 8.3. Three coupled quartic and cubic nonlinear oscillators 8.4. Integration of symmetries 9. Nonintegrability aspects of the Henon—Heiles and the system of two coupled quartic anharmonic oscillators 9.1. Henon—Heiles system 9.2. Two coupled quartic anharmonic oscillators 10. Results and discussions References

28 29 33 33 39 45 45 58 61 62 65 68 70

84 84 88 91 91

Abstract: In recent investigations on nonlinear dynamics, the singularity structure analysis pioneered by Kovalevskaya, Painlevé and contempories, which stresses the meromorphic nature ofthe solutions of the equations of motion in the complex-time plane, is found to play an increasingly important role. Particularly, soliton equations have been found to be associated with the so-called Painlevé property, which implies that the solutions are free from movable critical points/manifolds. Finite-dimensional integrable dynamical systems have also been found to possess such a property. In this review, after briefly presenting the historical developments and various features of the Painlevé (P) method, we demonstrate how it provides an effective tool in the analysis of nonlinear dynamical systems, starting from simple examples. We apply this method to several important coupled nonlinear oscillators governed by generic Hamiltonians of polynomial type with two, three and arbitrary (N) degrees of freedom and classify all the P-cases. Sufficient numbers of involutive integrals of motion for each of the P-cases are constructed by employing other direct methods. In particular, we examine the question ofintegrability from the viewpoint ofsymmetries, explicitly demonstrate the existence ofnontrivial extended Lie symmetries for the P-cases, and obtain the required integrals of motion by direct integration of symmetries. Furthermore, we briefly explain how the singularity structure analysis can be used to understand some of the intrinsic properties of nonintegrability and chaos with special reference to the two-coupled quartic anharmonic oscillators and Henon—Heiles systems. ‘Present address: Department of Mathematics, La Trobe University, 3083 Bundocra, Australia.

0370-1573/93/S24.00

©

1993

Elsevier Science Publishers BY. All rights reserved.

1. Introduction 1.1. Preliminary considerations In the description of natural phenomena, the evolution of typical dynamical systems is often described by nonlinear ordinary and partial differential equations [1,2]. Characteristically, these nonlinear dynamical systems exhibit regular as well as chaotic trajectories in phase space, depending on the number of dependent variables involved, the nature and the range of the external forces and the parameters involved and the energy of the system [3, 4]. The identification, characterization and classification of regular and chaotic regimes of dynamical systems are in general hampered by the absence of systematic and well defined analytical techniques to handle them [5]. It is in fact one of the important problems in nonlinear dynamics to identify when a given system displays regular motion. In other words, under what conditions the given nonlinear dynamical system, be it Hamiltonian or non-Hamiltonian, becomes completely integrable and when it is nonintegrable exhibiting irregular or chaotic motion. The obvious and fundamental questions which arise in this regard are what is meant by integrability and when does it occur. The answer to the former question is somewhat vague as the concept of integrability is itself in a sense not well defined and there seems to be no unique definition for it as yet [6]. The latter is even more difficult to answer, as no well defined criteria seem to exist to identify integrable cases. Integrability can be considered as a mathematical property that can be successfully used to obtain more predictive power and quantitative information to understand the dynamics of the system globally [7,81.~Recent investigations [1, 2, 9—11], which are in a sense a revival of the efforts of the mathematicians of the last century, show that the integrability nature of dynamical systems can be methodically investigated using the following two broad notions. The first one uses essentially the literal meaning: integrable integrated with the required number of integration constants; nonintegrable proven not to be integrable. This loose definition of integrability can be related to the existence of single-valued, analytic solutions, a concept originally advocated by Fuchs [12], Kovalevskaya [13, 14], Painlevé [15] and others [16, 17] for differential equations, thereby leading to the notion of “integrability in the complex plane”. The second notion, particularly applicable to Hamiltonian systems, is to look for a sufficient number of single-valued, analytic, involutive integrals of motion: N integrals for a Hamiltonian system with N degrees of freedom, so that the associated Hamilton’s equations of motion can in principle be integrated by quadratures in the sense of Liouville [5]. For example, the existence of such involutive integrals of motion can be verified systematically through a generalised Lie symmetry analysis [18] as discussed later in this review. The combination of the two aforementioned notions has met with remarkable success in recent years in predicting the integrable cases of dynamical systems. The primary motivation of this review is to demonstrate explicitly how these two notions combine to provide effective analytical techniques to determine the integrability nature of nonlinear dynamical systems in general, and in —

—

3

4

M. Lakshmanan and R. Sahadevan. Coupled nonlinear oscillators

particular to the case of coupled nonlinear oscillators of polynomial type governed by a generic Hamiltonian. The basic idea of the singular-point structure analysis, or popularly now known as Painlevé (P) analysis, is to identify and characterize the nature of the singularities (poles, branch points, both algebraic and logarithmic types, and essential singularities) admitted by the general solution in the complex plane of the independent variable, and to find conditions under which the solution is meromorphic or related to meromorphism. Correspondingly, an ordinary differential equation (ode) is said to possess the P-property only if its general solution is free from movable critical points [17] in a sense to be specified later. If a given nonlinear dynamical system governed by a system of odes satisfies the P-property, then it is expected to be integrable and the corresponding solution can be given in terms of a suitable Laurent series expansion in the neighbourhood of a movable singular point. Kovalevskaya’s successful investigation on the integrability property of the rigid-body problem is the first and standing example of this concept. In recent times several authors [19—31] have applied the P-analysis successfully to a variety of dynamical problems, both Hamiltonian and non-Hamiltonian, and identified a considerable number of new integrable systems. In spite of these successful investigations, the direct connection between the P-property and the integrability of nonlinear dynamical systems has not yet been rigorously established in a strictly mathematical sense. However, the works of Adler and van Moerbeke [32, 33], Yoshida [34], and Ercolani and Siggia [35] have to some extent paved the way for understanding the connection between P-property and integrability. Thus whenever a nonlinear dynamical system characterized by odes passes the P-test, its integrability has been supported by some other means; for example, by constructing the required number of independent integrals of motion. In the case of Hamiltonian systems with two degrees of freedom, Darboux [36] may have been the first to give a systematic method of constructing the second integral of motion, the first integral of motion being the Hamiltonian. This direct method has been modified recently by Hall [37] and the results obtained up to now have been reviewed by Hietarinta [38]. Chandrasekhar [39] has successfully exploited the idea of Darboux to construct the time-independent second and third integrals of motion for Hamiltonian systems with three degrees of freedom to some extent which will be discussed in this review later. The question of integrability can also be related to the existence of nontrivial symmetries [40,41], a concept originally advocated by Lie a century ago for differential equations. The fundamental idea of Lie’s theory is to find the one-parameter infinitesimal symmetries leaving the equations of motion invariant. These symmetries are in general of two types, (i) Noether and (ii) non-Noether; the former ones leave the Lagrangian invariant while the latter ones do not. In the former case, by virtue of Noether’s theorem [42], we may construct the associated integrals of motion using these infinitesimal symmetries and thereby ensuring complete integrability. In fact, we find that nontrivial generalized Lie symmetries exist only for those parametric values for which the P-property holds. We will also discuss this approach and its applicability to coupled nonlinear oscillators in detail in this review. Thus the present review will be concerned with a detailed exposition of the Painlevé property and its applicability to a class of coupled nonlinear oscillators of polynomial type with two, three, and N degrees of freedom. In an earlier work, Ramani, Grammaticos and Bountis [65] have discussed the general aspects of the Painlevé property and singularity analysis applicable to odes and pdes, whereas the present work makes an in-depth analysis for a specific class of coupled nonlinear oscillator systems.

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

5

1.2. Organization of the review To be self-contained, a brief outline of the nature of the singularities admitted by odes, both linear and nonlinear, is given in section 2. Section 3 contains a short account of the historical development of the P-analysis. In section 4, we sketch the various features of P-analysis and also demonstrate how this approach provides an effective tool in analysing the integrability nature of nonlinear odes with some typical examples. A systematic investigation of the P-property of certain two-coupled nonlinear oscillators with polynomial interactions governed by the generic Hamiltonian H2=~(p~+p~)+V(x,y),

(1.1)

is presented in section 5 for the following potentials. (a) Two coupled quartic nonlinear oscillators [19,43], 2 + By2 + ~xx4+ fly4 + öx2y2. V(x, y)

=

(1.2)

Ax

(b) Henon—Heiles systems [20,44], V(x, y)

=

~(Ax2 + By2) + ~x2y

—

~fly3.

(1.3)

(c) Two coupled sextic anharmonic oscillators [45] V(x,y)= Ax2 +By2 + cx6 +fly6 + ö

4y2 + 5 1x

2y4.

(1.4)

2x

(d) Two coupled nonlinear oscillators with polynomial potential [45] [N/2)

V(x,y)= ~

(1.5)

~NkXY,

k=O

where A, B, ~, ~Nk~ fi, c5, 5i and ‘~2are parameters and the symbol [ ] denotes the greatest integer. The complete integrability of all the identified P-cases is demonstrated by explicitly constructing the time-independent second integrals of motion. In section 6, we discuss the application of the P-analysis to the system of three-coupled nonlinear oscillators defined by the Hamiltonian H 3

=

~(p~ + p~+ p~)+ V(x, y, z),

(1.6)

by considering V(x, y, z) to be a quartic or sextic polynomial [43,45,46], 2 +By2 + Cz2 +ccx4+fly4+yz4 -l-5x2y2 +fly2z2 +wx2z2, V(x,y,z)= Ax V(x, y, z) = Ax2 + By2 + Cz2 + cxx6 + fly6 + ‘yz6 + ö 4y2 + 5 2y4 + 5 4z2 1x 2x 3x + c5 2z4 + 5

4x

4z2 + ~ 2z4 + ~ 2y2z2 6y 7x

5y

,

(1.7)

(1.8)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

6

where A, B, C, cc, fi, y, (5,ö~,~2, ö7~c and w are parameters. Here again, we explicitly evaluate the time-independent involutive second and third integrals of motion for all the predicted P-cases of each of the systems (1.7) and (1.8) in order to prove their complete integrability. Section 7 deals with the nature of the P-property of N coupled nonlinear oscillators defined by the Hamiltonian V(x1),

HN=~~pX2+

i=1,2,...,N,

(1.9)

where V(x1) is either a quartic or a sextic polynomial [24,45], V(x1)

=

2x~,~

~ A,x~+ ~ cc~x~ +~

=

(1.10)

fiji,

fl1~xj (I j)

V(x

2+~cc 1)=~A1xj

6+~ 1xj

i.j=1 ij)

fl

(1.11)

2x~x~,

1~x~x~+~ ~ i,j,k=1

yjjkXj

(ij,&k)

and as before the A~,cc

1, f3,~and ~ijk are parameters. The P-analysis shows that the systems (1.10) and (1.11) possess the P-property for a set of 4N 4 and N + 1 distinct parametric choices, respectively. The complete integrability is established by explicitly constructing N time-independent integrals I~,j= 1, 2, ,N obeying the Poisson bracket [Ii, Ik]pB = 0,j, k = 1, 2, , N for each of the P-cases. In section 8, we consider the question of integrability from the viewpoint of the Lie symmetry. After a brief introduction to the theory of Lie invariance applicable to Lagrangian systems, we consider the application for the specific cases of two and three coupled nonlinear oscillators. We also demonstrate how the required involutive integrals of motion can be obtained by a direct integration of the symmetries. Section 9 is concerned with the nonintegrability aspects of Henon—Heiles and two coupled quartic nonlinear oscillators. Finally, in section 10, we give a brief discussion of the various results obtained for the coupled nonlinear oscillators and indicate some further recent trends in this area. —

...

...

2. Classification of singular points 2.1. Fixed and movable singularities [48,49] Let us consider odes in the complex plane of the independent variable. The general solution of odes may cease to be analytic at certain points, called singularities. Poles, branch points (both algebraic and logarithmic) and essential singularities are all such points. Since the solutions are functions of the constants of integration, these singular points may depend on the constants (ultimately on the initial conditions of the problem). In turn these singular points may be placed anywhere in the complex plane and in such a case they may be called movable singular points. On the other hand, if the singularities do not depend upon the integration constants, they are fixed singular points. Among the singular points, the branch points and essential singularities are usually referred to as critical points.

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

7

2.2. Linear odes andfixed singularities Consider an nth order linear ode, d~w/df+ p1(z)d”’w/dz”1 + where the functions p~,i = 1, 2,

...

,

+ p~(z)w=0

(2.1)

,

n, are analytic at the ordinary point z

0 e C1. Obviously, eq. (2.1) admits n linearly independent analytic solutions in the neighbourhood of z0 and is given by w(z) = ~ c~w1(z),

(2.2)

where c~are integration constants. The singularities of the solution (2.2) must be located at the singularities of the coefficients p~,i = 1, 2, , n, which are all fixed. Thus one may conclude that the singularities admitted by linear odes are fixed and independent of integration constants. ...

2.3. Nonlinear odes and movable singularities In contrast to the linear case, the solutions of nonlinear odes can admit movable singularities as well as fixed singular points. We may illustrate this nature with the following examples: 2w z—~-—w2=0, ddz

dw —+w2=0, dz

dw ——expw=0, dz

dw 2 —+w(logw) =0. dz (2.3a, b, c, d)

Equation (2.3a) has the solution w(z) = 2/z with a local Laurent series expansion (2/z) x [z~(z z 2 + ~z 0) 0(z z0)~’+ ~ + ...], where z0 is the integration constant, and so z = 0 is a fixed singularity independent of the integration constant z0. The general solution of (2.3b) is w(z) = (z z0) 1, where z0 is the integration constant, which also locates the singularity, namely a movable pole. In the case of eq. (2.3c) the general solution takes the form w(z) = log(z z0) and hence admits a movable logarithmic branch point. It can be straightforwardly checked that the general solution of (2.3d) is w(z) = exp(z z0)~ and therefore it has a movable essential singularity. As noted earlier, if the critical points are independent of the initial conditions, they are fixed. Then the fundamental existence theorem guarantees that the general solution of an nth-order ode is completely and uniquely specified by the knowledge of the values which w(z) and its n 1 derivatives assume at the noncritical point z0 [47,48]. On the other hand, if a movable critical point is admitted, then there will be considerable difficulties in the problem of analytic continuation of solutions [49, 50]; the latter requires a specific, unique path, but the movable critical points prevent one from choosing such paths due to the multiple number of choices, and so the uniqueness theorem may not hold. This in turn may lead to the nonintegrable nature of the corresponding odes. Possibly motivated by the above considerations and by the development of the theory of complex variables, a problem of intense interest arose in the theory of differential equations in the second half of 19th century. It concerned the classification of odes according to the singularities, movable critical or not, admitted by the solutions. —

—~

—

—

—

—

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

8

3. Historical development of the Painlevé approach and integrability of odes 3.1. First-order nonlinear odes and Kovalevskaya’s rigid-body problem [8,51] By considering a first order ode of the form dw/dz

F(w, z),

=

(3.1)

where F is rational in wand analytic in z, Fuchs [12] concluded that out of all the forms ofF in(3.1) the only equation which is free from movable critical points is the generalized Riccati equation dw/dz

p0(z) + p1(z)w + p2(z)w2

=

(3.2)

,

where the p~,i = 0, 1, 2, are analytic. The fact that eq. (3.2) is free from movable critical points also leads to the conclusion that ifw 1, w2 and w3 are three particular solutions, the general solution may be obtained by a superposition principle, (w

—

w1)/(w

—

w2)

A(w3

=

—

w1)/(w3

—

w2),

(3.3)

where A is a constant. Possibly familiar with the above results and the then current Jacobi’s work on elliptic functions, Kovalevskaya was led to analyse the problem of a heavy rigid body under the influence of gravity [8, 13, 14] in terms of the singularity structure exhibited by the general solution. The equations of motion are of the form dI/dt

=

I A Q + mgr0

A

e,

de/dt

=

e

A

Q,

(3.4)

where the angular velocity Q and the angular momentum I are given by Q = ~ Q~e1,

1= AQ1e1 + BQ2e2 + CQ3e3

(3.5)

,

with respect to a moving trihedral e1, i = 1, 2, 3 fixed on the body. Then the vertical unit vector e and the centre of mass r0 are given by e = cce1 + fie2 + ye3

,

r0

=

x0e1 + y0e2 + z0e3

(3.6)

,

where cc, fi and y refer to the direction cosines which define the orientation of the heavy rigid body. Equations (3.4) can be rewritten in component form as AdQ1/dt

—

C)Q2Q3

—

fiz0 + yy0,

dcc/dt

=

fiQ3

—

$22,

BdQ2/dt = (C

—

A)Q1Q3

—

yx0 + ccz0,

dfi/dt

=

yQ1

—

ccQ3,

CdQ3/dt = (A

—

B)Q1Q2

—

ccy0 + fix0,

dy/dt

=

ccQ2

—

fiQ1.

=

(B

(3.7)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

9

Before Kovalevskaya’s investigations two specific cases, due to Euler (1750) and Lagrange (1788), respectively, x0=y0—z0=0;

x0=y0=0(z0>0,A=B)

(3.8a,b)

(besides the straightforward case A = B = C), were known to be integrable and the associated solutions were given in terms of meromorphic (Jacobian elliptic) functions. The connection between integrability and meromorphism in the above integrable cases had presumably prompted Kovalevskaya to find all the parametric choices of (3.7) for which the general solution is meromorphic. Kovalevskaya [13, 14] approached the problem by demanding that the solutions of (3.7) be single-valued, analytic functions in order that they might be integrable, and by expressing each of the six functions as Laurent series in the neighbourhood of a movable singular point. After a detailed investigation, Kovalevskaya concluded that apart from the two existing possibilities, found already by Euler and Lagrange, there is one more parametric choice, with y0=z0=0,

A=B=2C,

(3.9)

for which the general solution admits the required number of arbitrary constants and is meromorphic. Kovalevskaya went on to prove the integrability for the above parametric choice by constructing a sufficient number (four) of integrals of motion as Ii

=

C[2(Q~ + Q~)+ Qfl

—

(cc/C)x0,

2 + fi2 + y2 , 2(ccQ1 + fiQ2) + $23 , 13 = cc 14 = [Q~ Q~+ (cc/C)x 2 + [2Q 2 0] 1Q2 + (fi/C)x0] with respect to the Poisson brackets, ‘2

(3.10)

=

—

{ Q,, Q

3}

=

~tJk~k

,

{Q1, e~}=

—clJkek

,

{e1, e~}= 0

where ~•jk, i, j, k = 1, 2, 3 is the usual Levi-Civita tensor [6]. In fact, she went on to integrate eqs. (3.7) explicitly for the choice (3.9) and obtained the general solution in terms of hyperelliptic functions. Thus, Kovalevskaya’s discovery of a new integrable case for (3.7) provided strong evidence for the close connection between meromorphic solutions and the integrability of nonlinear dynamical systems. 3.2. Second-order nonlinear odes and Pain/eve transcendental equations [48,49] The next important development was at the turn of the present century due to the efforts of the French mathematician—politician Paul Painlevé and his coworkers. By considering the secondorder odes having the form 2w/dz2 = F(z, w, dw/dz) , (3.11) d where F is rational in w, algebraic in dw/dz, and locally analytic in z, they found that there were only fifty canonical equations possessing the propety of having no movable critical points. Among

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

10

them, forty four can be integrated in terms of elementary functions including the elliptic functions. The remaining six, which are now referred to as Painlevé transcendental equations, needed special treatment and introduction of new transcendental functions. Their explicit forms are as follows [48]: 2 + z,

P1:

w”

(3.12)

6w

=

P

3 + zw + cc 11:

w”

111:

w”

=

(1/w)(w’)

w”

=

(1/2w)(w’)

=

(3.13)

,

2w

P

2

P

—

(1/z)(w’

—

ccw2

—

fi) + yw3 + (5/w ,

2 + ~w3 + 4zw2 + 2(z2 1w:

Pv:

w”=(—+ \2w w

1 —

—

(3.14)

cc)w + fi/w,

(3.15)

\ 1 (w—1~2 w w+1 )w’2——w’+ 2 / +y—+öw lj z z (ccw + fi/w) z w 1

,

(3.16)

—

P~ 1: w”=~(!+ 1 + 1 ~~2(1 2\w w— 1 w—zJ +W

1 + 1 z— 1 w—zJ

2 ~cc+fi z (w—1)(w—z)f’ z—1 [z(z—1)] 2+Y(1)2+t5z()2)~

z—1

‘\

(3.17)

where prime denotes differentiation with respect to z, and cc, fi, y and (5 are constants. In some sense eq. (3.17) for P~may be considered as the most general Painlevé transcendental equation which by repeated limiting and coalescence procedures can be related to the other Painlevé equations. The treatise of Ince [48] contains greater details of the Painlevé transcendental equations. For newer results concerning transformations among Painlevé transcendents one may refer to ref. [52] for example. Finally, attempts have also been made by Bureau [53] and Chazy [54] to classify third- and higher-order odes in the same way as was done for the second-order odes at the turn of the century, though the analysis remains rather incomplete. 3.3. Painlevé property of odes: strong and weak P-properties The classical work of Kovalevskaya, Painlevé and his team did not draw much attention for several years probably due to the advent of quantum mechanics in the late 1920’s and the ensued interest in linear odes. However, the situation has changed dramatically after the discovery of solitons in mid 1960’s in the Korteweg—deVries equation by Zabusky and Kruskal [55]. It was shown that the solitons possessing evolution equations are completely integrable through the inverse scattering transform (1ST) method [56]. Furthermore, it was realized that a deep connection exists between the soliton possessing nonlinear evolution equations solvable by the 1ST method and the odes of P-type (possessing the P-property) and Ablowitz, Ramani and Segur (ARS) [17] conjectured that every ode obtained by an exact reduction of soliton possessing pde solvable by 1ST is of P-type. This has been verified for a large class of nonlinear pdes by Lakshmanan and Kaliappan [57] using the underlying Lie point symmetries. The term strong P is now used in the

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

11

ARS sense meaning that the solution of an ode (strong P-type) in the neighbourhood of a singularity z0 can be expressed as t=(z—zo)”,

z—+z0

(3.18)

where p is an integer. The fact that odes under suitable transformations can exhibit P-property, for example PVI, has also given rise to the definition of weak P-property. Ramani, Grammaticos and Dorizzi [58] have suggested that the existing P-property (ARS) can be generalised to include the so-called weak P-property by which it is meant that the general solution in the neighbourhood of movable singularity z0 can be expressed as an expansion in powers of =

(z

—

z0)’~ ,

(3.19)

where n must be “natural” and depends purely on the dominant behaviour of the singularity as well as the nature of the nonlinearity. Indeed a number of integrable Hamiltonian systems having weak P-property have been identified during the past few years [27—30].We will see more about the strong and weak P-properties in the following sections.

4. Painlevé method of singular-point analysis for odes For an ode to be of P-type, it is necessary that it has no movable branch points, either algebraic or logarithmic. We do not consider in our following analysis the presence of essential singularities, whose treatment appears to be much more complicated and the theory is probably not complete to locate them. In the following, we describe the ARS algorithm [17], which provides a systematic way to investigate the presence of movable critical points of branch-point type, and to determine whether the given ode is of P-type or not. To be specific, we consider an nth order ode, 5w/dz5

F(z; w, dw/dz,

=

...

,d

1w/dz”’)

,

i

,n

(4.1)

d or equivalently n first-order equations dw~/dz= F 1(z; w1, w2,

...

,w5) ,

=

1,2,

...

or equivalent higher-order equations. Here F and the F1 are analytic in z and rational in their other arguments. We expand the solution of(4.1) as a Laurent series in the neighbourhood of a movable singular point Z~.Then the ARS algorithm essentially consists of the following three steps. (a) Determination of leading-order behaviour of the Laurent series in the neighbourhood of the movable singular point z0 (b) determination of resonances, that is, the powers at which arbitrary constants of the solution of (4.1) can enter into the Laurent series expansion; and (c) verification that a sufficient number of arbitrary constants exist without the introduction of movable critical points.

M. Lakshmanan and R. Sahadevan. Coupled nonlinear oscillators

12

At the end of the above three steps one will be in a position to check the necessary conditions for the existence of a P-type solution and integrability of (4.1). Whether these conditions are sufficient is a further intricate problem; one has to show that the associated Laurent series does exist. In typical dynamical problems this is assumed to be so, as the actual analysis becomes too complicated, or the existence is proved by other means. We now give briefly the details of the above three steps. 4.1. Leading-order behaviour Considering the nth order ode (4.1), we assume that the dominant behaviour of w(z) in a sufficiently small neighbourhood of the arbitrary movable singularity z0 is algebraic, that is, w(z) ~ a~(z z0)~i as z —

—+

z0

(4.2)

,

where (ai, q~)are constants to be determined from (4.1) and Re q~< 0. Each allowed pair (as, q~)is said to belong to the jth branch or possibility of the solution. Note that, in principle, one can also allow for the possibility of logarithmic leading-order terms, but for simplicity we do not consider them here. Using (4.2) into (4.1) one can see that for certain values of q~,two or more terms may balance each other, while the remaining terms can be ignored as z z0. The balancing terms are referred to as leading-order terms. Knowing the qj, the a~can easily be evaluated. From the leading-order behaviour, we may observe the following. (i) If at least one of the q~is an irrational or complex number, then we can conclude, at this stage, that (4.1) is of non P-type leading to nonintegrability. (ii) If all the q~are negative integers, then (4.2) may represent the first term of a Laurent series, for each q~valid in a deleted neighbourhood of z0. This can then be an indication of the strong P-property and further analysis will be necessary to confirm this. (iii) If any of the q~is not an integer, but a rational fraction, then from the dominant behaviour (4.2) of w(z) near z0, we find that the solution will have a movable algebraic branch point. This may then possibly be associated with weak P-property. In either of the latter two cases, the solution w(z) may be shown to take the form —.

m w(z)=(z

—

,

0< Iz

—

zoI

<

R

(4.3)

.

z0)~J~aj,~(zz0) —

Let us illustrate the leading-order behaviour terminology with some typical examples. Example 1.

w”

+ ww”

—

2w3 + w2 + ~uw= 0,

(4.4)

where a prime means differentiation and ~t is a parameter. Assuming the dominant behaviour to be w(z) ~ ~ ~ = (z z 0), with z0 an arbitrary movable singularity, eq. (4.4) becomes —

3+ a~q(q 1)t2~2 —

a0q(q

—

1)(q

—

2)t~

Then there are two possible choices (branches).

—

2a~z~ = 0

.

(4.5)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

(i) q

13

1, a0 = 3, the leading-order terms are w” and ww”. 3. (ii) q = 2, a0 = 3, the leading-order terms are ww” and 2w Example 2. w” + w” + ww’ a(w + w’) = 0, =

—

—

—

(4.6)

—

where a is a constant. As before, the leading-order behaviour is obtained from balancing the terms w” and ~w’, a

+ 3 a~qt2~’= 0 0q(q 1)(q 2)~r~ leading to the values —

q

=

—2,

a 0

Example 3.

(4.7)

—

=

—12.

w” + 5mw

(4.8) 0.

=

(4.9)

Since the nonlinearity of (4.9) is nonalgebraic, it is convenient to transform eq. (4.9) into a different form: 2 + ~(v3 v) = 0, v = eR~~ (4.10) vv” (v’) From the leading-order analysis, that is v(z) = a 0(z ~ we find —

.

—

—

q=—2, Example 4.

a0=—4.

(4.11)

w~= w2w3,

w~= ii’3w1,

w’3

=

(4.12)

w1w2.

5, we obtain

Again with the leading-order behaviour w1(z) p=q=s=—1,

=

a0t~,w2(z)

=

b0~,w3(z)

=

c0t

a 0=bo=co=±1.

(4.13)

4.2. Resonances In the Laurent series solution (4.3), z0 is an arbitrary constant which is also the position of the singularity. In addition, if n 1 of the coefficients a~in (4.3) are also arbitrary, there are totally n constants of integration of the ode (4.1) and so (4.3) is the general solution in the deleted neighbourhood of z0. The powers at which these arbitrary constants enter inside the summation in (4.3) are called resonances. In order to find the resonance values, for each leading-order pair (q~,a~)or branch, one constructs a simplified equation that retains only the leading terms of (4.1). Substituting (hereafter we omit suffix j in the a and q for simplicity) —

w(z) = a~t~ + fi~~+r

r

=

(z

—

zo)

into the simplified equation, to leading-order in fi one can obtain 4=0, ~q+r—n, Q(r)fit

(4.14)

(4.15)

M. Lakshmanan

14

and R. Sahadevan, Coupled nonlinear oscillators

where n is the order of the ode (4.1). Here if the highest derivative of (4.1) is a leading-order term, then Q(r) is a polynomial of order n. If not, ~ > q + r n. Then the roots of Q(r), given by —

Q(r)

=

0,

(4.16)

determine the resonance values. We may note the following characteristics at this stage. (i) One root of eq. (4.16) is always —1, representing the arbitrariness of z0. This may be inferred in the following way: by perturbing the complex variable t = z z0 in (4.2) by t + ~, ~ ~ 1, and expanding it in terms of c, one can find that the first term appears at ~ (see also Bessis and Chafee [59]). (ii) Suppose that the constant a~in (4.2) is arbitrary, then one of the roots is at r = 0. (iii) Roots with Re r < 0 can be ignored because they violate the leading-order hypothesis. The occurrence of such resonances indicates that the associated Laurent series-expansion solution is a singular one*). (iv) Any root with Re r > 0, but not a real integer, indicates a movable branch point at z = z0, in general. The associated solutions, in general, are of non P-type. (v) Any root with Re r which is a positive rational fraction r = p/q, with q as in the denominator of dominant behaviour, indicates in general a movable algebraic branch point and this may then be associated with the weak P-property. The integrable systems admitting such weak P-property are also discussed in this review. (vi) If at least one of the roots of (4.16) is an irrational or complex number, then we can conclude that the given ode (4.1) is of non P-type. (vii) If for every possible set of values (q, a) in (4.2), excluding 1 and possibly 0, all the remaining roots are positive real integers, then there are no algebraic branch points. For the Laurent series expansion (4.3) to be the general solution of the ode (4.1), Q(r) must have n 1 non-negative distinct roots of real rational numbers including integers. If for every allowed (q, a), Q(r) has fewer than n 1 such roots, then none of the local solutions is general. Let us illustrate all this, again with reference to the four examples cited in the previous section. Example 1. Substituting (4.14) into eq. (4.4) we obtain for the branch q = 1, a0 = 3, —

—‘

—

—

—

—

[(q + r)(q + r

—

1)(q + r

—

2~r2

+ a0(q + r)(q + r

—

1)r

+a

2~+r2]fi =

0q(q

—

0,

(4.17)

1)~

which gives (r+1)(r2—4r+6)=0

or r=—1,

On the other hand for the second branch q

=

r=2±i~/~. —

2, a 0

r=—1,

(4.18)

=

3, we find

r=6.

(4.19)

Thus the first branch leads to complex resonances, thereby showing the presence of movable critical points. So, eq. (4.4) is not of P-type. Example 2. Considering eq. (4.6) the representation (4.14) leads to the resonances r=—1, *)

r=4,

r=6.

However, see ref. [93] for recent works on the implications of negative resonances.

(4.20)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

15

Example 3. For eq. (4.10) we obtain the resonances r=—1,

r=2.

(4.21)

Example 4. Extending the above analysis to eqs. (4.12) we obtain a system of three algebraic linear equations for the resonance parameters and for a nontrivial solution we require that r—1

c0

c0

r

b0

b0 1

—

a0

a0

=

0.

(4.22)

r—1

Expanding (4.22) and using (4.13), we obtain r=—1,

r=2,

r=2,

(4.23)

and so no movable algebraic branch points exist. However, we need to check against the presence of logarithmic branch points, which we will do so in the next section. 4.3. Evaluation of arbitrary constants The final step in the ARS algorithm consists of verifying whether or not in the Laurent series expansion (4.3) a sufficient number of arbitrary constants exist (without the introduction of movable critical points) at the resonance values. For a given leading-order set (q, ao) or branch, let r1 r2 ... r5 denote the positive rational roots of Q(r) = 0, s n 1. So we substitute —

w(z)

=

~

+

(4.24)

k1

into the full ode (4.1). Requiring now that the coefficients of Q(k)ak

—

Rk(zO; a0, a1,

...

,akl)

=

0

~q+kn

shall vanish, we have (4.25)

.

For k r1, eq. (4.25) determines the coefficients ak. At k = r1, Q(r1) = 0. Then there are two possibilities. (i) If Rr, = 0, a,., is an arbitrary constant and we can proceed to find the next coefficient. The procedure can be continued until the Laurent series solution possesses n arbitrary constants so that the ode (4.1) is of P-type. (ii) If Rn 0, then (4.25) cannot be satisfied identically and a,., is not an arbitrary constant and there is no solution of the form (4.3). In order to capture the arbitrariness of an,, we must introduce logarithmic terms as follows: r,

w(z)

=

001q +

—

k=1

~

+ (a,., + b,.log~)~~+I1 +

...

(4.26)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

16

Now the coefficient of Q(ri)bn,

=

~

+ “

“log t is

—

0,

(4.27)

but bn, is determined by demanding that the coefficient of ~ n, 1 vanishes; a,., is arbitrary. Suppose the resulting coefficient an, is not arbitrary, then again we have to introduce more singular terms like log log t in (4.3) and repeat the procedure until the coefficient a,., becomes arbitrary. In this case, (4.3) signals the presence of a movable logarithmic branch point which shows that the ode (4.1) is of non P-type. Having analysed the nature of the Laurent series (4.3), it is necessary now to verify whether the series converges and if so what its radius of convergence is. However, due to the highly nonlinear nature of recurrence relations between the various coefficients involved, the analysis is too difficult to carry out in general. Usually the convergence has to be checked by other indirect means; for example, for Hamiltonian systems by proving the existence of a sufficient number of integrals of motion. The above general procedure of the P analysis is succinctly described in fig. 4.1 for equations of motion ofnonlinear dynamical systems. We now illustrate the last step on our examples considered earlier. Example 1. As noted earlier, eq. (4.4) is not of P-type, as it admits complex resonances indicating the presence of movable critical points. +

-

Equation of Motion : Solution expressed Laurent series around a movable sinQular point

Rational fractien

Leading Order behaviour

Irrational fractioni

°‘~~“I I Nonlnt.ger (Nonratlonol) Resonances

Resonances / at which arbitrary

~

parameters con .me!

1

I

~.

I

Integer Resonances

>

I

.~

i

f~nsufficient/Inconsist.ncy1

Palnievê Properiy

Complicated singularity structure in the complex time plane

I __________________

Integrability

________________________

Fractai dimension, chaos

~amics Fig. 4.1

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

Example 2. Here we have q

=

2 + ~akz~2

—2, a0

—12 and r

=

17

—1,4, 6 and so (4.3) becomes

=

(4.28)

.

w(z) = —12t Following the procedure described in this section, we find a 1

=

12/5,

a2

a + 1/25,

=

a3

=

a + 1/125

(4.29)

.

But when j = r~= 4, Q(rj) = 0 and R4(zo, a1, a2, a3) ~ 0, which means we have to introduce logarithmic terms into (4.24) as 2 +~r’ +(a +~)+(a +th)r +(a w(z)= —12r and hence eq. (4.6) is not of P-type. Example 3. Again an arbitrary constant exists for r eq. (4.10) and the associated integral of motion is P(v, v’)

=

2+

...

,

(4.30)

4 —~logt)t

v’2 + v3

v

—

—

kv2

k

,

=

constant

=

2 in the Laurent series solution for

(4.31)

,

so that eq. (4.10) is completely integrable. Example 4. Proceeding in a similar manner for eqs. (4.12), we find that the Laurent series does admit arbitrary constants at r = (2, 2) and so the general solution is free from movable critical points. Thus the solution is of strong P-type. One can also check that the system is integrable and that the Laurent series is convergent by showing that the following integrals of motion exist for eqs. (4.12): cP 1(x1, x2, x3) so that dcP1/dt 4.4.

=

x~ x~

=

—

thP2/dt

=

,

~2(x1, x2, x3)

=

x~ x~ —

,

(4.32a,b)

0.

Further remarks

We end this section with some brief remarks on the other recent developments on the P-property and the integrability properties of nonlinear dynamical systems governed by odes. One of these developments is related to the concept of algebraic complete integrability of nonlinear odes introduced by Yoshida [34]. We may say that the given system of odes of order n is algebraically integrable if it admits n 1 independent algebraic integrals of motion. By an algebraic integral of motion ~ i = 1, 2, n) it is meant that for any given constant a, the relation —

...

=

,

(4.33)

a

can be rationalised to an algebraic equation of the form a”

+

‘~k1(w1)a”’

+

+

fl’k(Wi)

=

0,

(4.34)

M.

18

where

Lakshmanan and R. Sahadevan. Coupled nonlinear oscillators

rational in their arguments w1, w2, ,w,,. Let us consider an autonomous system of n first-order nonlinear odes defined as in (4.1) and assume that it is invariant under the similarity transformation z

~,

—*

~2,

...

,~Pk are

cc~z

w,

,

—s

...

cc~’w1

i

,

1,2,... ,n

=

(4.35)

,

where cc is a nonzero constant, and g1, g2, g~are fixed numbers. Also the functions F,(w~)are rational but homogeneous ones. By investigating the nature of the solutions in the neighbourhood of a movable singular point Yoshida [34] concludes that there exists an important connection between the weighted degree of the integrals of motion of(4.1) and the resonances discussed earlier [which are called Kovalevskaya exponents (KE) by Yoshida] which characterise the singularity nature of the solution. A function G(w1, w2, w5) is said to be weighted homogeneous of weighted degree m, if the identity ...

,

...

,

9~w G(cct’w1,

...

,cc

5)=ccmG(wi,

...

(4.36)

,w5)

holds. Then, Yoshida proves the following. Theorem 1. Let ck(w1, i = 1,2, ,n) be a weighted homogeneous integral of motion of weighted degree m for the similarity invariant autonomous system (4.1). Suppose that the elements of the vector ...

grad fP(c) = (é~(c~)/~w1, ,é~(c1)/éw~) ...

are finite and not identically zero for a fixed choice of the set c1, i = 1, 2, n. Then the quantity m becomes a KE. Here the c~are constants associated with the similarity solution of (4.1). ...

,

Theorem 2. Let P~and ~i2 be two independent weighted homogeneous integrals of motion of the same weighted degree m. Suppose that the vectors grad ~1(c) and grad ‘P2(c) are both finite, not identically zero, and these two vectors are linearly independent for a fixed choice of the c1. Then the quantity m becomes a KE with multiplicity at least 2. Theorem 3. Let us assume that eq. (4.1) is a similarity invariant Hamiltonian system with weighted degree h. Let fji be a weighted homogeneous integral of motion of weighted degree m. Suppose for a choice of the c1, the vector grad P(c) is finite and not identically zero. Then, in addition to m, the quantity h 1 m also becomes a KE. —

—

Furthermore, let ~2 be two independent weighted homogeneous integrals of motion of the same degree m. If the two vectors grad cP1 (c), grad ~2(c) are both finite, not identically zero, and are linearly independent then the quantity h 1 m becomes a KE with multiplicity at least 2. ~,

—

—

Theorem 4. In order for a given similarity system (4.1), which may be Hamiltonian or nonHamiltonian, with rational functions F. to be algebraically integrable, it is necessary that every possible KE becomes a rational number; or in other words, if there exists at least one irrational or imaginary KE, the similarity-invariant system is not algebraically integrable.

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

19

Another interesting remark on the Painlevé expansion is due to Weiss [59, 60] and Newell Ct al. [61]. They have shown that the Laurent series expansions are not only useful to test the integrability nature but they can also be used to derive some of the intrinsic properties of the system as well, such as Backlund transformations, Lax pair, etc. There exist other developments also on the Painlevé expansion; for example, Chang et al. [44], Weiss and Tabor [44a] and others [44b] have shown how to derive the intrinsic properties of nonintegrability such as natural boundary, fractals, etc., from the P-analysis; details are given briefly later in this review. Similarly, Bountis et a!. [62] have reported that there exists a connection between the transversal intersections of homoclinic trajectories associated with the Melnikov function approach [63] and the multiple-valued solutions with infinitely many Riemann sheets in the complex time plane. The relation between the singular-point structural analysis and Ziglin’s approach [64] is pointed out in ref. [65]. Recently, Kruskal [65a] has also developed two direct algorithms namely poly-Painlevé and recursion methods to study the singularity structure of solutions of nonlinear dynamical systems. However we do not discuss these developments in this article.

5. Painlevé analysis and integrability of two coupled nonlinear oscillators 5.1. Quartic anharmonic oscillators [19,26—29,43] We consider a set of two quartically coupled nonlinear oscillators defined by the Hamiltonian (1.1) with ~thepotential (1.2). It is well known that the above Hamiltonian system is widely used as model in lattice dynamics [66], condensed matter theory [67], field theory [68], astrophysics [69], etc. The associated equations of motion of the two coupled quartic anharmonic oscillators are

~+2(B+2fiy2+ox2)y=0.

~+2(A+2ccx2+(5y2)x=0,

(5.la,b)

For clarity, we will describe the three steps of the ARS approach separately in the following subsections. 5.1.1. Leading-order behaviour

Considering (5.1), we assume that the leading-order behaviour of x(t) and y(t) in a sufficiently small neighbourhood of the movable singularity t0 is x(t) ~ a1,i.~”,

y(t)~ ~

,

~

=

t

—

t0—sO

.

(5.2)

To determine p, q, a0 and b0, we use (5.2) in (5.1) and obtain a pair of leading-order equations 2+ 4cca~r3”+ 2(5 aop(p

—

1)t~’

—

1)x~

2+ 4fib~~3~ +~

b 0q(q

2~ 0,

=

(5.3a)

0h~~~”’ =

0.

(5.3b)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

20

From eqs. (5.3), we identify the following two distinct sets of possibilities. Case 1. a~=

p=q=—1,

((5

2

2fi)A1

—

Case 2a.

p

=

—1,

q

=

Case 2b.

p

=

—

1

q

=

,

=

((5

—

2cc)A,

=

,

(4ccfi

—

(52)_i

(5.4)

.

b0

,

~ + ~[1 + 4(5/cc]112,

a~=

~

a~=

—

~(1 + 4(5/cc)112 ,

—

—

1/2cc,

b~= arbitrary.

(5.5a)

1/2cc,

b~= arbitrary.

(5.5b)

Due to the complete symmetry between x and y variables in H possibility q

=

1, p>

—

—

2 or in (5.1), we do not treat the other 1 as distinct from (5.5) in the following.

5.1.2. Resonances For finding the resonances, we substitute x(t)~aot”+Qit”~’,

y(t)~b0r~+Q2r~~+r,t’—sO,

(5.6)

into (5.3). Retaining only the leading-order terms, we obtain a system of linear algebraic equations, M2(r)Q

0,

=

Q = (Q,, Q2)

(5.7)

,

where M2(r) is a 2 x 2 matrix dependent on r. In order to have a nontrivial set of solutions (Q1, Q2) we require that detM2(r)

0

=

(5.8)

.

For case 1, the form of M2(r) is 3r + 8cca~ 4(5a 1 2 0b0 3r + 8fibo~j’ [4aobo r 2

—

[r

M2(r)

=

(5.9)

—

so that using (5.4), eq. (5.8) becomes (r2—3r—4)(r2—3r+Xo)=0,

Xo=4[1+2(ccao~+fibo~)].

(5.10)

Thus for case 1, the resonances occur at r= —1,

r=4,

r=~±+(9—4Xo)112.

(5.11)

For the leading orders (5.5) of case 2, the expression for M 2(r) degenerates to 2 M2(r)

=

diag[r

—

3r + 8cca~,r(r + 2q

—

1)]

.

(5.12)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

21

Furthermore, for P-property, all the resonances must be non-negative rationals, including integers and thus the following sets of resonances are isolated along with specific parametric restrictions. Case 1(i).

r=

Case 1(u).

r

Case 2b.

r

=

=

1, 1, 2,4

—

—

—

(5 = 2[cc + fi ± (cc~+ fi2

,

1, 0, 3, 4, cc

1, 0, 2, 4,

=

3cc

fi,

= 4(5

(5=

2cc

—

ccfi)”2]

(5.13)

.

(5.14)

.

(5.15)

.

Note that case 2a is omitted, since the resonance values contradict the leading-order singularity nature, q

~.

As usual the resonance at r =

1 is associated with the arbitrariness of t

—

0. 5.1.3. Evaluation of arbitrary constants

Introducing now the series expansions 4

x(t)

4

~ ~

=

y(t)

=

k=0

~

(5.16)

,

k=0

into the full equations of(5.1), one can obtain a pair of recursion relations. For example, for case 1, they are 2)a~+ 2Aa~_ (j 1)(j 2+ 4cc ~ ~ aj_j_ma,am + 2(5 ~ ~ bj_,_ma,bm = 0, (5.17a) —

(j

—

2)b~+ 2Bb~_ —

1)(j

2+ 4fi ~ ~ bj_i_mbibm + 2(5 ~ ~ aj_i_ma,bm

—

Im

= 0,

Im

0l,mj4.

(5.17b)

Solving (5.17) successively one can obtain the various a~and b~explicitly. For example, for case 1(i) we have the following system of algebraic equations: 2cca~+ (5b~= forj

=

—

=

1

,

(5a~+ 2fib~=

—

1

(5.18a,b)

,

0,

(4cca~ for]

—

1)a1 + 2(5a0b0b,

= 0,

2(5a0b0a1 + (4fib~ 1)b1 —

=

0,

(5.19a,b)

1,

Aa0 + 6cca0a~+ 2(5b0a1b1 + öa0b~+ (4cca~ 1)a2 + 2(5a0b0b2

= 0,

(5.20a)

Bb0 + (5b0a~+ 2(5a0a1b1 + 6fib0b~+ (4fib~ 1)b2 + 2(5a0b0a2

= 0,

(5.20b)

—

—

for] = 2, and similar equations for] = 3, 4. Now one can easily check that the coefficients a0 and b0 satisfy eq. (5.4). Similarly from (5.19), we find that a1 or b1 becomes arbitrary if cc=fi,

(5=2cc,6cc.

(5.21)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

22

However, from (5.4) we infer that for the values cc

=

fi, (5 = 2cc either a0 or

b0 becomes arbitrary, which is not indicated by the resonance values in (5.13), and hence this choice is omitted. Proceeding further, it is straightforward to check the arbitrariness of a2 or b2 which requires an additional constraint on the parameters, that is, A = B. In a similar manner, from the equations for j = 3, 4, we can uniquely determine a3 and b3, while either one of the coefficients a4, b4 is arbitrary without any further restrictions on the parameters and thus leading to a full four-parameter branch of the solution. We call such branches the main branches (MB) of the solutions x(t) and y(t). For the P-property to hold, it is necessary to verify that the remaining or subsidiary branches (SB) of the solution are also free from movable critical points (within the strong and weak P-concepts). We do verify explicitly that this is indeed the case for the above choice, namely

cc=fi,

(5=6cc,

A=B.

(5.22)

One can proceed in a similar manner for the cases 1 (ii) and 2 and obtain all the possible P-cases. The details are summarized in table 5.1. 5.1.4. Integrals of motion In order to substantiate the results of the previous section we derive explicitly the timeindependent second integrals of motion for each one of the P-cases by employing the procedure enumerated in Whittaker’s treatise [36]. In general, a given integral of motion is of the form I = I(x, y, p~,ps,). Then restricting ourselves to integrals which are of fourth degree polynomial in momenta p~,p,~,,,for the quartically coupled anharmonic oscillators (5.1) the second integral of motion may be written as 4

1

+f2p~p~

+f3p~p~

+f4p~p~

+f~p~

+f6p~

+f

“fipx

7p~p~

+fspy~

+f9,

(5.23)

where the .f~,i = 1, 2, 9, are functions of (x, y) alone. To obtain the j, we demand that the Poisson bracket [I, H2]PB vanishes. Equating now the coefficients of each monomial of the momenta p.’p~,m, n = 1, 2, 5 separately to zero, we obtain a system of overdetermined pdes. The first set of these pdes is ...

,

...

f1~=0,

,

f15+f2~=0,

f35+f4X=O,

f25+f3~=0,

f4V+fsX=O,

(5.24)

f5~=0,

where subscripts denote partial derivatives. By successively solving (5.24) we find 4 + ~,y3 + ~2Y + ~3Y + ~ = c0y 4 f2

=

3 + 3e 2 + 2c 3+ 1y 2y + ~3)x + ~loY

= (6e

2

1y

4Coy + —

e0x

+

e 1)x

(

4 =

+

772Y

+ 773,

2 + a,y + a 1y + 772)x + a0y 2, 2 (2a 0y + ~~1)x 0y + a1)x + eoy + ei

2 e2)x 3 + (3;7

+ 3e

0y

=

fliY2

(4c0y

—

—

7

3 0X

+ aox2

—

(377~y2+ 277

—

—

6 0X

+

62

(5.25)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

~~S

+

~

>~

N

0~

~.v

+

“t.~

N (~

+ +

N ,ç’N,~

~.

+

~

:~

o

N

-e a

‘—.5

N

~

~

++ NIl

0

~

~ ,—,—,r~l

N

N

N

N

N

.

~ +,~4-

++

.~

~

+~.sN

5_5j~

.~.

U

.~

~ ~

~

a

o

~

r~’

N

~t,~:;-’ S~ ~,rsi

++

‘—S N,,~,

~ ~

+~ ~.

~+ +~

~\C~

++~

~—

÷r

+~

~.

NIl

~

;;~S,~

~!

II II

II

Cs-’

-~--~‘

-~

I

NIl

N

~

IC

NIl+~

;;~ II

~+

I

II

C-7-~•

II

-~

I-

o ‘~

~.

r~,

~

C’-I

~

C’~ ~

,fl

U ~‘

Cr,

~

~

r’1-~

~

.0

~

i—E

~

~I

-~

~

—~,,ç ~

00

00

~ ~

I

I

~

~N~N

~N~N

=_;Ie?

=~re.1

,.~ 0000

,~ 0000

==

~

~.,

~.

~I

i E V

Cr,

‘~r~ ~r’~f~ r’~—’ ~4—’—’

00

==

ICC V .0 5,-

o V

•

I~.

a

a

,~

‘0

SIC

no no.

~

II ‘0 —

II ~

,.~‘

no. ‘0

II

no. 00 II

a

a II

no

~ ~

V SIC

a

II

ii

II

“~

a

“~

, ,~, —

~, —

23

M. Lakshrnanan and R. Sahadevan, Coupled nonlinear oscillators

24

where the c~,~ a, and 6, are integration constants. The next set of pdes is 3f2~+2f39+f6Y+f

4fi~+f29+f6X=0,

7X=0,

(5.26) 2f3~+3f49+f7~+fs~=0,

f4~i+4f59+f8~=0,

where the dot stands for differentiation with respect to t. From eqs. (5.26) we obtain the compatibility condition

(f~+ 4f~9)~ (2f3~+ 3f49)~~~ + (3f2~ + 2f3~3)~~~ (4fi~+f29)~~~ = 0. —

—

(5.27)

The final set of pdes is 2f6~+f7)7+fO~=0,

f75~+2f59+f9y=0,

(5.28)

and so the second compatibility condition becomes (2f6~+f79)5

—

(f~+ 2f~9)~ = 0.

(5.29)

Therefore it is clear that an integral of motion which is quartic in momenta exists whenever the equations of motion satisfy (5.27) and (5.29). Following the above procedure, we find that for the two quartically coupled oscillators (5.1) the compatibility condition holds only for the four parametric choices possessing the P-property and the explicit forms of the first (Hamiltonian) and second integrals of motion are also given in table 5.1. 5.1.5. Separability Now, it is of interest to investigate which of the above integrable systems are separable under coordinate transformations. In this connection we can summarize the results as follows. (i) If we make a linear transformation u = x + y, v = x y in (5.1) with the parametric restrictions A = B, cc = fi, (5 = 6cc, case 1(i), the equations of motion decouple into one-dimensional anharmonic oscillators, 2u/dt2 + 2Au + 4ccu3 = 0, d2v/dt2 + 2Av + 4ccv3 = 0, (5.30) d —

whose solutions may be readily given in terms of Jacobian elliptic functions u =s,cn(Q 1t+(51), v=s2cn(Q2t+(52) (5.31) 2 + ccs,2), and i = 1, 2 are integration constants. The corresponding Hamilwhere = 4(A tonian Q,~ becomes a sum of two independent Hamiltonians (see table 5.1). (ii) For case 1 (ii), if we transform the cartesian coordinates x and y to polar coordinates x = r cos 0, y = r sin 0 then for the parametric values cc = fi, (5= 2cc and A = B the Hamiltonian H is independent of 0, H = ~(p~ + 2Ar2 + 2ccr4 + I~/r2), where the angular momentum 12 = (xp 2is an integral of motion. 5 yp~) 5,

—

(5,,

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

25

For A ~ B, the Hamiltonian H is not separable in polar coordinates. In terms of the elliptic coordinates x

=

ei~/c,

y

=

(1/c)[(e2

—

c2)(c2

—

772)]112

c2cc

,

=

B

—

A

the Hamiltonian becomes H=

~2

772(C2

—

2c2A(e2

—

—

c2P~+

772)

1

~

+ 2cc(e6 _776) + 2(2A + B)(~4_,~4)

(5.32)

.

The corresponding Hamilton—Jacobi equation becomes

282

c2cc

772(22

=

—

c2

+ c2

772~2 +

2cc(c6— 776) +2(2A + B)(c4

—

774)

—

2c2A(e2

B—A,

—

~i2))

=

(5.33)

where E is the energy and the subscript denotes partial derivative. Obviously eq. (5.33) is separable. (iii) Similarly, for the parametric restrictions of case 2b(i), if we transform the cartesian coordinates to parabolic coordinates x = ~(s2 772) y = e~,then the resulting Hamiltonian becomes —

H

=

[1/(e2

+

172)]

+ p~)+ B(e6 + 776) + fi(~1~ + 171~)]

[1(~2

(5.34)

,

and hence the Hamilton—Jacobi equation is separable [69a, b]. (iv) Finally we note that for the parametric choices cc = 8fi, (5 = 6fi and A = 4B, case 2b(ii), the Hamiltonian seems to be nonseparable [69c], where the second integral of motion is quartic in the momenta. 5.2. Henon—Heiles system [20,44] The equations of motion of the Henon—Heiles system (1.3) are d2x/dt2 + Ax + 2ccxy

= 0,

d2y/dt2

By + ccx2

+

—

fly2

(5.35a,b)

= 0.

5.2.1. Leading-order behaviour, resonances, and evaluation of arbitrary constants The leading-order behaviour of (5.35) gives a set of two possibilities as in the case of quartic oscillators, Case 1.

p

= q =

—

2,

a 0

Case 2. p

= ~ ± ~(1

—

=

±

48~)h/2,

(3/cc)(2 + q

=

—

~

1)112

2,

a0

b0

,

=

=

—

arbitrary,

3/fl, e = cc/fl. b0

=

—6/cc

(5.36) .

(5.37)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

26

The resonances associated with the branches (5.36) and (5.37) are Case 1.

r

=

Case 2.

r

=

—

—

1,6,~±~[1 1, 0, (1

—

—

2p), 6

24(1 + 21)]u12

(5.38)

,

(5.39)

.

Furthermore, the requirement that the resonances be non-negative integers, apart from rise to the following subcases: Case 1(i).

e=

—

1 ,

cc

=

—

fi,

Case 1(u).

e

=

—

,

cc

=

—

~fl

Case 1(iii).

e=

—

~,

cc

=

Case 2(i).

2

=

—

~,

cc

Case 2(u).

2

=

—

~,

Case 2(iii).

e

=

—

Case 2(iv).

2

=

—

—

1, gives

r

=

—

1, 2, 3, 6

,

(5.40)

,

r

=

—

1, 1, 4, 6 ,

(5.41)

—

~fi,

r

=

—

1,0, 5,6

(5.42)

=

—

i~fi, r

=

—

1, 0, 4, 6,

cc

=

—

*fi,

=

—

1,0, 3, 6

,

(5.44)

~,

cc

=

—

-~fi r = ,

—

1, 0, 2, 6 ,

(5.45)

~,

cc

=

—

~fi , r =

—

1, 0, 1, 6

(5.46)

r

,

(5.43)

.

As before we use the series representation x(t)

=

~ akri’,

y(t)

=

~ ~

(5.47)

in (5.35) and proceed to evaluate the coefficients ak and bk, k = 0, 1, , 6 for the MB as well as SB associated with the resonances (5.41)—(5.46). One finds that the necessary conditions of the P-property are satisfied only for a set of three distinct parametric restrictions. They are ...

Case 1(i).

A

Case 2(u).

A, B arbitrary,

Case 2(iii).

=

B,

16A = B,

cc

=

—fi. 6cc 16cc

=

(5.48) =

—fi.

—fi.

(5.49) (5.50)

All the remaining cases are ruled out, as a sufficient number of arbitrary constants cannot be introduced into the Laurent series without the introduction of logarithmic movable branch points, hence violating the condition of P-property. The second integrals of motion associated with the above three P cases can be obtained as in the previous section to establish complete integrability. A summary of the details of the P cases along with the explicit forms of the integrals of motion are given in table 5.2.

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

~.‘ ~5Il

N

~

.~N

~

~

~N

).~

N~

*‘~

.~

+~.

+~

~ ~+

~

.~

0 U

0

E

N

~

a ~

.~‘—.

.0

1..

a o

N N

N++ >~ ~

~

~
‘~+_~

+

a

NS,5~ ~

N ~

NIl 0.0. L—C IC .IN 0.

N c~”i ~ ‘—‘~-~‘+ ~

+,~, NIC+a 0. L..I~IC I ..~N ~ II ii

~

-;•-:~‘

~‘~:~‘

++ ii

~

~

~ ~

—a ~ +~

a

al +~ ~

~

+N N ~

a

+~ N

II

~

+I~’

II

II

E

;~.;; ;~.

~ ~

V V

~I

—

—

II

~L

I

0

~

V

~

II

~o

—

t~-

N

III r.i ns,

0

N ‘a~ .0

~II

‘N

V .~ I0

a

,-.~I

I

V

00

U

~, ~

~ sO 00

o

—

~ sO 00

‘o.

U

I II

sO 0000

,-.,

a

‘0

—

II

.‘~

,~,

,~,

—

N

IN

II

II ~‘

a

a

..~NN

II

~

V SIC

N

I

N

27

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

28

5.2.2. Separability (i) If we introduce the linear transformation u = x + y, v = x y in the Hamiltonian together with the parametric restriction A = B, cc = —fi, then the equations of motion (5.35) are separated into that of one-dimensional anharmonic oscillators of the form —

d2u/dt2 + Au + ccu2

d2v/dt2 + Av + ccv2

= 0,

(5.51)

= 0,

and the solution can be readily obtained in terms of the Jacobian elliptic functions. (ii) For the parametric choice (5.49), one can easily check that the associated Hamilton—Jacobi equation is separable in terms of shifted parabolic coordinates [69d] defined by ~

y=~(2— 77)+(1/cc)(4A—B), 0
(5.52)

,j
(iii) However, it appears that the integrable Hamiltonian system is nonseparable for the parametric choice (5.50), where the second integral of motion is quartic in the momenta. 5.3. Sextic anharmonic oscillators [45] We consider a set of two coupled sextic anharmonic oscillators defined by the Hamiltonian (1.1) with the potential (1.4), the corresponding equations of motion become 2x/dt2 + 2(A + 3ccx4+2(5 2y2 +(5 4)x=0, (5.53a) d 1x 2y d2y/dt2 + 2(B + 3fiy4 + (5,x4 + 2(5 2y2)y = 0. (5.53b) 2x 5.3.1. Leading-order behaviour, resonances and arbitrary constants From the leading-order analysis, we obtain the following two sets of possibilities: Case 1.

p

= q =

—

,

3cca~+ 2(5

2b 2 + (5 1a0 0 2b~=

(51a~+ 2(52a~b~ + 3fib~=

—

—

(5.54)

.

112, Case2. p=—L q=~±~(1+(51/cc) a4= (1/8cc) , b~= arbitrary 0 The corresponding resonances are computed along with parametric restrictions. —

.

Case 1(i).

F =~,

r=

Case 1(u).

F =0,

r=

—

—

~

(5.56a)

.

1,0,2,3,

F

= ((52

—(5

1)a~b~ +(3fi —(52)b~ .

Case 2b. 4(5k

=

5cc, q

=

—

~,

r

=

(5.55)

—

1, 0,

~,

3

.

(5.56b) (5.56c)

Here again, with the series representation x(t)

=

~ akr’”,

y(t)

= ~

~~~k/2+~

,

(5.57)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

29

we identify the required arbitrary constants associated with the resonance sets (5.56) for a set of three distinct parametric constraints. Case 1(i).

A

Case 1(u).

A

Case 2b.

A

=

=

B,

cc

B,

cc cc

= 4B,

fi,

=

=

= (5~ =

(51

fi’

= 64fi,

(52

=

(51

~

15cc.

(5.58a)

3cc.

(5.58b)

=

(52

= 80fi,

=

(5.58c)

24fi.

For these choices both the MB and SB possess the P-property. Explicit time-independent second integrals of motion are also constructed for all the P cases to ensure their complete integrability. Brief details of the above P cases and the forms of the integrals of motion are displayed in table 5.3. 5.3.2. Separability (i) For the parametric choice (5.58a), case 1(i), the equations of motion (5.53) decouple into one-dimensional anharmonic oscillators, d2u/dt2 + 2Au + 6ccu5

d2v/dt2 + 2Av + 6ccv5

= 0,

= 0,

(5.59)

where u = x + y, v = x y. Equations (5.59) can be solved straightforwardly in terms of Jacobian elliptic functions [69e], —

u

=

s,sn(Q 1t + (5~)({1+ dn[2(Q1t + (5~)]}{1+ F1cn[2(Q,t +

(5i)]_1})h/2

112 v=s2sn(Q2t+(52)({1 +dn[2(Q2t+(52)]}{1 where (5~and

(52

=

2,

= (1 +

,

+F2cn[2(Q2t+(52)]~})

are integration constants,

fl

a(1 + e,)~ ,

5,

(5.60a) (5.60b)

,

=

(c,

—

l)(c, + 1)’

6cca4/A)(1 + 2cca4/A)”’

,

i~,

Q,

,

= [(1

+

=

6cca4/A)(1 + 2cca4/A)]”2

,

i

=

1,2

Here, the square of the modulus of the Jacobian elliptic function is given by = (l/2t~,)[,~,

—(1

+

3cca4/A)]

,

i

=

1,2

(ii) Case 1 (ii) is obviously separable in polar coordinates. (iii) For case 2b, the Hamilton—Jacobi equation becomes separable in the parabolic cylinder coordinates defined by x = ~(c2 772) y = i~ as in the case 2b(i) of quartic oscillators (section 5.1). —

5.4. Polynomial-potential nonlinear oscillators [45] The equations of motion for the polynomial potential nonlinear oscillator (1.5) are + ~(N

—

2k)cc~~xN_2k_ly2k =

0,

+ ~ 2kccNkxN_2ky2k_l

= 0.

(5.61a,b)

30

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

+

+~

N~ N

+

N

~N’NN N

+,.~la 5—

~

‘,~ ‘N

~

++

0

—‘ NO~

E

E

N

++

0

V

.

—--

a

I

+ ~

±N ++

+N ~ ~

IC

+1

II

+N

~

I

~5+

‘‘

N

+~ II

II

I~ II

-;‘-~

-..:.-~‘

-~:.

-.

~

~

~

‘N

‘N

~N

‘N

‘N —

—

‘IN

‘N

‘N

U

0

‘N

SIC

V.~

i-E a .0 a a

‘N

C.~ C~

‘N~ IA’

‘N~

“Vj’

‘N “s

~

.~I

~

~

‘-‘N II

II

—

I

‘IN

‘INNN

‘IN’IN

‘~

..4r5

I

U V SIC

o

~

-~

0.

o ‘~

I

I

‘IN.’IN

~

~ sO

sO 00

II

II

I

I

.‘IN

..SN

..~N

•~ sO

00

00

I

I

00 -

‘~ sO

V .0

a U

a

g ‘~

S.,;’

00

•~.m a

~

II

no

II a

‘N

-

U

I

5-

ii

II

a

a ~

II

II

II

—

—

.0 N

V

no-

V

a

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

31

5.4.1. Leading-order behaviour We again assume x(t) ~ a0’r”,

y(t) ~

,

x

0,

—.s

(5.62)

and obtain 21’2bO2’= —2s(2s + 1), Case 1.

p

= q =

—2s,

>~(N 2k)ccNka~ k —

~ kccNka~2”b~”2 = Case 2. p

=

—2s,

q

s(2s + 1)

—

= ~ ± ~(1

b

+

(5.63)

.

A,)”2

a~~2 =

,

—

2s(2s + l)/NccNO,

6ccN,/NccNO)s(2s + 1). 0

Here s

arbitrary,

=

(N

=

A,

2)’, N being the highest degree of the polynomial potential.

—

5.4.2. Resonances Case 1(i). B, =s(2s+ 1), Case

1(u).

Case 2. B,

B, =0,

N(s + l)ccNo

=

2k(k

~

(5.64)

(l

=

—

r=

r= =

1,2s, 1 +2s,2+4s

—

(5.65a)

.

1,0,1 +4s,2+4s.

—

4(2s + 1),

q

1)ccNka~’21’b~2 —

k

=

—

k(N

~

s, —

(5.65b) r

=

—

1, 0, 1 + 2s, 2 + 4s,

(5.65c)

2k)ccNka~~2”~2b~.

k

5.4.3. Arbitrary constants Substituting the series representations x(t)

~ a,t”,

=

y(t)

~ b,t”

=

I

(5.66)

I

into (5.61a, b), we verified that, for the following three parametric choices only, four arbitrary constants can be introduced without the necessity of movable logarithmic branch points. Case 1(i).

ccNk

Case 1(u).

Case 2.

=

(~) ccNo

=

2k!(N

ccNk = (~)ccNo~n

ccNk

=

~N.

2k)! ccNo.

(5.67a)

(5.67b)

=

2N_2k(k

—

k)

ccNN.

(5.67c)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

32 V .0

00

:3 N

N S

+

N

N

I

N

-‘——.55

I N

a

~ —

.9 0

U

N ~

~

:3

00

+ 0.

+ ~

II

— —

I 0

~

~

~

+

~

+

o~

)., ~L-..1i~

~

a +

ZN

~

— N

~‘

i~,

-.‘~

+

+

0.

0.

.~‘

, ,—.,~ —

~“

+

N ± N

N

-

‘‘~

‘r

0.

0.

+ ~ ~

,~)

~

.~

II

I

II

II

I

II

—

—

.-,

—

—

—

‘S1~

‘N

‘I~

‘N

~

‘N

VO

~

x

‘N

‘.5

N

;~: +

~

—

I

—

N

~ —

I

+

I

+

,~ ~

0,

~‘S N

‘.5 N

I

I

sO

00

—

-.

-~

I

~: I.~ ~0 0 a a

N

~ ~

I,, V~V

It

N I N

~ ~

‘~+

~‘

I

s-.. I

N

—

I

~.

I

7 N

~I N N

7N N

~N

I

I

I

I

00

sO

N

N

U

0 5a

01

a 0 a

00 .~

-~ sO

0 V 0.

N N

o 0

a ,‘~

0

—

0

0

‘.—...

a

V

V

~‘-......-‘

E

II

II

.~5N

II

a

a

~

a

—

—

~‘.

~

N

0 V V

L)

.0 N

00

00

M. Laksh~nananand R. Sahadevan, Coupled nonlinear oscillators

33

One can verify that both the MB and SB satisfy the required conditions of the P-property and hence they are expected to be integrable. This is indeed confirmed by finding the time-independent second integrals of motion through the Whittaker analysis elucidated in section 5.1. The essential features of the P-analysis of (5.61) and the explicit forms of the integrals of motion are given in table 5.4. 5.4.4. Separability (i) Under the linear transformation u = x + y, v = x y the associated equations of motion (5.61), for the parametric restrictions (5.67a), decouple into two one-dimensional anharmonic oscillators —

d2u/dt2 + NccNOu”~1=

0,

d2v/dt2 + NccNOv”~’=

0,

(5.68)

and so the system becomes separable [69a]. (ii) Similarly, case 1 (ii) becomes separable in polar coordinates x = r cos 0, y = r sin 0. (iii) The Hamilton—Jacobi equation becomes separable in parabolic cylinder coordinates x = ~(22 772), y = 277 for the parametric choice of case 2. —

6. Painlevé analysis and integrability of three coupled nonlinear oscillators [43,46] In this chapter, we wish to present our investigations on the identification of integrable cases of specific three coupled nonlinear oscillators defined by the Hamiltonian (1.6) by considering the potential V(x, y, z) to be either a quartic or a sextic polynomial. The analysis is a straightforward, though involved, extension of the previous section for the two coupled anharmonic oscillators. 6.1. Quartic anharmonic oscillators We consider first a set of three coupled quartic nonlinear oscillators whose potential is given by (1.7), V(x, y, z)

=

Ax2 + By2 + Cz2 + ccx4 + fly4 + yz4 + (5x2y2 + ~y2z2 + wx2z2.

The equations of motion become d2x/dt2 + 2(A + 2ccx2 + (5y2 + wz2)x

d2y/dt2 + 2(B + 2fly2 + (5x2 + 2z2)y = d2z/dt2 + 2(C + 2yz2 + wx2 + ey2)z

(6.la)

= 0,

0,

(6.lb)

(6.lc)

= 0.

Assuming that the leading-order terms of the solutions of (6.1) as 0

x(t) ~ a 0x”,

y(t)

~

b0~”,

z(t) ~ c0’r

‘r

—.s

0

,

(6.2)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

34

we find from (6.1) the following three distinct sets of possibilities [noting the complete symmetry between the coordinates x, y, z in (6.1) or in (1.7)]: Case 1.

2cca~+(5b~+wc~=—1,

p=q=s=—1,

(5a~+ 2flb~+ cc~=

—

a~=

=

(6.3)

.

2)~

—

b~= arbitrary,

,

1,

q

1/2cc,

—

1

—

2y)A 2

Case 3. p

wa~+ 8b~+ 2yc~=

,

q=~±~[1—8((5a~+2c~)]”2,

Case2. p=s=—1, a~= (w

1

—

= ~ ± ~(1

c~= (w

4(5/cc)”2

+

b~= arbitrary,

s

,

—

2cc)42

= ~ ± ~(1

+

A2

,

=

(4ccy

—

(6.4)

.

w

4w/cc)”2

c~= arbitrary.

(6.5)

As before, for finding the resonances, we substitute x(t) ~ a

+Q

5

0’r” +

y(t) ~ b0x’~+ Q~q+r

Q,tP~,

z(t) ~

tS’~

.~

—sO

,

(6.6)

C0T

into the leading-order terms of(6.1) and obtain a system of linear algebraic equations in Q 1, Q2 and Q3. Then for a nontrivial set of solutions we require that (r + p)(r + p det M3(r)

(r + q)(r + q

s

= q =

detM3(r)

=

=

—

—

1) + 8/3b~—

r’ + 8cca~ 4(5a0b0 4wa0c0 4(5a0b0 r’ + 8flb~ 4eb0c0 4wa 0c0 4sb0c0 r’ + 8yc~ 4

PQ

+

= 16~1/4(P

Q) + 4[(ccfi

2/4)b~c~ + (ccy + (fly

—

c

(6.7)

1) + 8yc~— 2

2 = 0

,

r’

=

—

4r

.

(6.8)

r

(6.9)

2(cca~+ fibs + yco~)], +

—

0.

is a root of (6.8), and so

(r’—4)(r’+P)(r’+Q)=0, = 4[1

(r + s)(r + s

=

1, and so (6.7) becomes

It is easy to check that r’ =

P+Q

4ob0c0

2

4ob0c0

4wa0c0

For case 1, p

4wa0c0

4c5a0b0

1) + 8aa~— 2

—

4öa0b0

=

—

—

(52/4)

(6.lOa) a~b~

w2)a~c~]} .

(6.lOb)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

35

From (6.9) we infer that the resonances occur at

~±f(9—4Q)”2. r= —1,4,~±~(9—4P)”2,

(6.11)

As before, the restriction that the resonance values (except

—

1) be non-negative rationals but must

depend on the nature of the leading-order singularity, leads to the following possibilities: Case 1(i).

P= 2,

Q =2,

Case 1(u).

P=2,

Q =0, r=

—

1,0,1,2,3,4.

(6.13)

Case 1(iii).

P =0,

Q =0, r =

—

1,0,0,3,3,4.

(6.14)

r= —1,1,1,2,2,4.

(6.12)

For case 2, we obtain the resonance condition (6.7) after omitting the coefficients of less dominant terms as r’ + 8cca~

0 2q

4(5a0b0 4wa 0c0

—

r(r +

4wa 0c0 1) 42b 0c0

2 = 0,

—

3r,

(6.15)

r’ = r

r’ + 8yc~

0

which on expanding gives 4,(1—2q), ~[3±(9—4R)112], R=4[1+2(cca~+yc~)]. (6.16) r= —1,0, Further, the consideration of non-negativ~rational resonances in conjunction with the leadingorder behaviour leads to the following possibilities: Case2(i).

q=0,

R=0,

(5a~+ec~=0, cca~+yc~=

r=

—~,

—

1,0,0,1,3,4. (6.17)

Case2(ii).

q=0,

R=2,

(5a~+2c~=0, cca~+yc~=

r= —1,0,1,1,2,4.

—~,

(6.18) Case2(iii).

q=

Case2(iv).

q=

—~,

—~,

R=0,

(5a~+cc~=

R=2,

(5a~j+ec~=

—~,

r= —1,0,0,2,3,4. (6.19)

cca~+yc~=

—~,

—~,

cca~+yc~=

—~,

r=—1,0,1,2,2,4. (6.20)

For case 3, in a similar manner we derive the following four sets of possibilities: Case3(i).

q=0,

s=0,

Case

q=0,

s=

3(u).

—f,

(5=0,

w=0,

(5=0,

w=~cc,

r=—1,0,0,1,1,4.

(6.21)

1,0,0,1,2,4.

(6.22)

r=

—

Case3(iii).

q=—~, s=0,

(5=~cc,

co=0,

Case3(iv).

q=

(5=~cc,

w=~cc, r= —1,0,0,2,2,4.

—~,

s=

—~,

r=—1,0,0,1,2,4.

(6.23) (6.24)

36

M. Lakshrnanan and R. Sahadevan, Coupled nonlinear oscillators

Thus we have isolated eleven distinct sets of integer resonances, namely (6.12)—(6.14), and (6.17)—(6.24). In order to compute the arbitrary constants for the above resonances, as usual we introduce the series representations x(t)

=

y(t)

~ a~~ k

= ~ k

b~tq+k

z(t) =

~ ck

(6.25)

k

into (6.1) and obtain recursion relations for ak, b~and ck, k = take the following forms:

0, 1,

...

For example, for case 1, they

(j— 1)(j— 2)a~+ 2Aa~_2+ 4cc~~aj_,_ma,am + 2(5~~bj_i_maibm + 2w~~cj_I_malcm

(j_ +

1)(]

= 0,

(6.26)

2)b~+ 2Bb3_2 + 4fi~~bj_i_mb1bm + 2(5~~aj_,_mambi

2e~~Cj_i_mbiCm

(]— 1)(]— 2)c~+

(6.27)

0,

=

2Cc~_ 2+ 2y ~ ~cj_,_mcj

Cm

+ 2w~~aj_,_majcm

+22~~bj_i_mbiCm=0, 0l,m]4.

(6.28)

Solving (6.26)—(6.28) simultaneously one can obtain the explicit values of the ak, bk and c~and we find that the necessary condition of the P-property, both in MB and SB, is satisfied only for a set of two nontrivial parametric restrictions, namely, Case 1(u).

cc=fi=y,

Case 1(iii). cc =fi=y,

(5=2cc,

e=w=6cc,

A=B=C.

(5=e =w =2cc, A,B,C arbitrary.

(6.29) (6.30)

Proceeding analogously as before for cases 2 and 3, we find that the P-property holds both in the MB and SB for six more nontrivial parametric restrictions. The details of the P-analysis are sketched in table 6.1. 6.1.1. Second and third integrals of motion In order to substantiate the investigations of the P-analysis of the previous section, in this sub-section, we investigate the explicit form of the second and third integrals of motion. For this purpose, we follow a method originally developed by Chandrasekhar [39]for systems having three degrees of freedom which admit integrals of motion with quadratic momenta. We extend his studies for a more general integral of motion with quartic momenta by assuming the following form: 65p~+ c9 I = 6,p~+ &2P~Py + ~ + t94pXp~+ 6p~p~ + 07p~p~ + &8PyP~ 613Px + 0l4PxPy + ~ + &9p~+ ~ + O,1p~p~ + 0i2P~Pz + + 0l6PyPz + &i7~~ + t9i 8pxpz + 19,9 (6.31) ,

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

37

where the 0,, i = 1, 2, ..., 19 are functions of x,y, z alone, and proceed in a fashion similar to the Whittaker analysis. However, for the sake of clarity, here we start with the case of quadratic momenta, t9,p~+ ~2P~ + 03p~+ 04PxPy + OSPyPz + 06PxPz + 07, where the 0, are functions of x, y, z alone. When the coefficients of p~p p~, r, s, Poisson bracket [I, H 3]pB = 0, are set equal to zero, the following pdes result: I

(6.32)

=

04x’.”O, ~

02x+04y0,

t = 1, 2, 3

in the

Olz+196x0,

Oiy+

+

03x

06z = 0,

035+052=0,

05y =

02y = 0,

02~+

03z”~°,

04z+05x+06y’°,

0,

(6.33)

where subscripts denote partial derivatives. Solving (6.33) successively, we obtain 0, =~cc

2+cc 2z

02

=

03

=

2)+fl 3y

1yz+y2z+(53y+010,

2+ cc 2) + fl 1z 2xz + y3z + (5,z + 020, 2+ cc 2) + fi f~cc1y 2x 3xy + y1x + (52y + 03~, -~cc3x

2—(5 04= —~3xy+fi2yz+fl,xz)+fl3z

3x—y3y+041z+ 040,

(6.34)

2— (5,y— y,z + 0 05

=

—(cc1yz + fl3xz + fi2xy) + fl,x

51x + 050,

2—(5 06= —(cc2xz+fl,xy+fl3yz)+fl2y

with the condition 041 + 05~+ ~61

= 0,

2z—y2x+061y+060,

where the cck, flk,

Yk, ~k

and 0~oare constants.

The other set of pdes read 20,~ + 049 + 06z +

~7x

06x + 059 + 203~+

07z = 0.

=

04~+ 2029 + 0s~’+ 07y

~,

= 0,

(6.35)

From (6.35), we can derive the following two compatibility conditions: (20~~ 04~)x+ (04~ 202~)9+ (06y —

—

0~~)~’ + 20,($Z)~

—

202(9)X + 04(9)~ 04(x)~ 0~(I)~ + 06(~)y= —

(2o,~

—

—

—

06~)~i + (194Z

—

—

Osx)9 +

(96~

—

(6.36a)

z03~)I+ 20~(.~)~

203(I)X + 04(9)~ 0~(9)~ + 196(±’)Z —

0,

—

06(x)~=

0.

(6.36b)

38

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

Thus an integral of motion quadratic in momenta exists if (6.1) satisfies the compatibility conditions (6.36).

On the other hand, if one assumes that the integral of motion is quartic in momenta as in (6.31); then the following set of pdes results: 0ix0,

Oiy+O2x~”.O,

192y+&3x’.”O,

O4x+O3y~’.O,

04y+O5xO,

05y”°~

05z+O6yO,

062+07y”O,

67Z+198y’.”.O,

08z+199y0,

09z°,

09x+OlOz”°,

Oi2x+Oiz°,

194z

+

196x = 0

OiOx+Oiiz°, 19iiy°~

°3z°,

,

°8x

+

(6.37)

Olix+012z°,

07x°,

02z+012y°,

0,0~ = 0

40~~ + 029 + 0, 2z + 3~12~ +

20~~2 +

0i3z

016y

+

+ 2039 +

302k

+

04~+ 4059 + 06Z + 015y = 0, 203~ + 3049 + 014y + ~15x = 0,

Oi8x = 0,

~i5z

= o,

°13y

+

013x = 0,

014x = 0,

(6.38)

3069 + 207±+

0io~+ 089 + 469k

2079 + 30~~ +

+ 0172 = 0,

20

0i6x + 618y

115i + 30,~1+ 0i7x + 19~=

0i6z

+

+

°i7y = 0,

~9i4z = 0.

0,

The final set of pdes reads

2013k + 0149 + 19,

8z + 019x

= 0,

0~4~ + 20~~9 +

016Z

+ 019y =

0,

(6.39) 018.~

+ 0~~9 + 2017E +

019z

= 0,

By successively solving the above three sets of equations (6.37)—(6.39) together with the equations of motion (6.1), we find that the consistency conditions hold only for the eight P cases displayed in table 6.1. We verified that all three integrals of motion (1,, 12 and 13) satisfy the Poisson bracket relations [Ii, Ik]PB = 0,], k = 1, 2, 3 in each of the P cases and hence they are in involution, thereby establishing the complete integrability in these cases.

6.1.2. Separability In this section we report the separability property of the integrable cases listed in table 6.1. (i) If we transform the cartesian coordinates x, y and z to spherical polar coordinates x = r sin 0 cos 4, y = r sin 0 sin 4 and z = r cos 0, then we find that for the parametric values cc = fi = y, (5 = = w = 2cc, and A = B = C, case 1 (iii), the equations of motion become separable. For A B ~ C, the system is not separable in spherical polar coordinates. However, Wojciechoski [69] has shown that the associated Hamilton—Jacobi equation becomes separable in terms of generalised elliptic coordinates (for details see ref. [69f]). ~‘

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

39

(ii) Similarly, for the case 3(iv)-1, if we transform the cartesian coordinates to parabolic coordinates, x

= ~(e2

y=

_772),

277 cosO,

Z = 277

sinO

(6.40)

then the resulting Hamiltonian becomes H

= (22

+ n2)’[~(p~+ p~)+ g,(e) + g2(77)] + I~/c2772

(6.41)

,

where g,(c) and g2(77) are functions of e and 77, respectively, and 12 is the second integral. So, the Hamilton—Jacobi equation becomes separable. (iii) For the remaining P cases, where the third integrals of motion in each of the cases are quartic in momenta, the system seems to be nonseparable. 6.2. Sextic anharmonic oscillators The equations of motion corresponding to the potential (1.7), 2 + By2 + Cz2 + ccx6 + fly6 + yz6 + (5,x4y2 + (5 2y4 + (5 4z2 V(x, y, z) = Ax 2x 3x + (5 2z4 + (5 4z2 + (5 2z4 + (5 2y2z2, 4x 5y 6y 7x are

(6.42)

d2x/dt2 + 2(A + 3ccx4 + 2(5 2y2 + ~2)’ + 2(5 2z2 + (5 4 + (5 2z2)x = 0, (6.43a) 1x 3x 4z 7y d2y/dt2 + 2(B + 3fiy4 + (5 4 + 2(5 2y2 + 2(5 2z2 + (5 4 + (5 2z2)y = 0, (6.43b) 1x 2x 5y 6z 7x d2z/dt2 + 2(C + 3yz4 + (5 4 + 2(5 2z2 + (5 4 + 2(5 2z2 + (5 2y2)z = 0, (6.43c) 3x 4x 5y 6y 7x From the leading-order behaviour of (6.43), we isolate the following distinct possibilities: Case 1.

p=q=s=

~

—~,

~

+(5 (6.44)

6c~+2(55b~c~ =

—~,

(53a~+

(57a~b~ + (55b~+ 2(54a~c~ + 3yc~+ 2(56b~c~ =

—f,

Case2. p=s=

—

q=~~f[1_8((51c4+(57a~c~+(564)]1l2,

3cca~+ 2(53a~c~ + (54c~=

—

~,

(53a~+ 2(54a~c~ + 3yc~=

—

~,

(6.45)

b~= arbitrary, Case3. p= a~= 1/8cc

—~,

—

,

2, s=~±~(1+(5 2 q=~±~(1 +(5,/cc)~’ 3/cc)” ~ = arbitrary, c~= arbitrary,

(6.46)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

40

“C

+

S

C’C

NIl

+

N N

N

N

‘l-

N N

~

+

N

+

+

~,

,~‘ r1 N . ~

N

+

a

+

+

N N0. ~‘ —

NN

~11~

+

N N~ 5-1 ‘~ çj

N

++C-.5

~

~

+~

~

~N

LJN~

~

~

N’....

NS.1

+

+NN

~j

++

NIC’I

...

0~

~

+ V

~‘.

+~ ~

+

,...,I

I -~

.5~

so

II

5 0.N

0.N

NN

+~ I

N~

I

II

~

+

N~..’.

+~

+~

N~’~’~’

II

no.°°

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~ ~

•

Ni

—

‘~-..

~

N

I~

II

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~

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SIC

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+

~

+~N

~i..

~

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~

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~

+

+~ ~‘

~

~

~‘‘—“ ~

+‘~N ‘—+ ~N

++

+ ~++

N~N~

‘....N~N

NN

~s’~’.N

~

~

+~no.

+

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-~

.-~~-::‘

Cr,

~

50

‘0

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sr,

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00

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NftN~ NN+

+

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±

NIC

II II

±~

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+‘~‘

II

+ N+ +

N

4~ ~

~ N0~no.

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~

-::~-.T

0. a ~II

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aa

—

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‘N

0’N

—

‘N Sr,

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=

‘IN’IN

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—

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—

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III —

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~

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‘I~ 0

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—

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III

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—

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—

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sO

Cr,00C#~

III

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III

,-IN’N sO

000000

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sO

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g Ci~

nZ:~ II ‘~ ‘~

II II a no

~ II

II

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a no S

II ‘~

II II

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a no S

II ‘~

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II

a no

II

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,.‘. ‘~

II

— LI

~‘

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—

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a no

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0

II a

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I ,~.

a a

V V .0 —

~

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±N

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no.

0.

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a

‘I

+

~

-7-~-~

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‘r,

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no. +

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no. +

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++

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I

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

-

N

C.’;-’. N

+

0.,’.-’. ~N.C

I

N

r~±

~

~ + h-’. N +

+

++

N

~‘~N

~ ~‘1 +

N

N~,

~

+ ~

N’C+

N

N ~‘.~‘ N

+

~ C.N~5-4 0.N

SIC

,..~

~ N~ —

+~

++‘~N

+

EN ‘5

+

N

N

N

+

N~’N ~

+

N

+ +

NN+

N~ 00

+

••

N

,,~....~

~I

~

+ N 0. ~

1’~N

+

N1~N N,~N

I

~5,

IC

N ~

+00~+~~N~ ~

N

N IC ~.

CC

—

N.+~x

II

00 V.....

II —

II —

SC

‘I~

II —

II

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— —

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~

r.I

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a a

8 -

‘a (~‘.l

ICS

~

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‘,,

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—

——

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II

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no.

41

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

42

Then we can derive the resonances for each of the leading-order sets listed above, (6.44)—(6.46), in a straightforward way. For instance, for case 1 they become r= —1,3,1±(1—P1)”2, 1±(1—Q,)’’2 P, + Q, = 8(~ + X,), P,Q, = 64[3(~ + X,)

(6.47) + X 2]

X,

=

X2

=

22 3cca~+ 3fib~+ 3yc~+ ((5, + (52)a~b~ + ((53 + (54)a~c~ + ((55 + (56)b0c0,

(9ccfi— (5,(52)a~b~ + (9ccy— (53(54)a~c~ + (9fiy— (55(56)b~c~ 3~(53— (5~)c~ + (3cc(54— (5~)a~]a~c~ + [(3cc(52— (5f)a~+ (3fl(51— (5~)b~]a~b~ + [( + [(3fl(5 3y(5~—(5~)c~]b~c~ (6.48) 6—(5~)b~ +( + {[3((55 + (5 6)cc + ((5k + (52)(54 + ~2~3 ((52 + (54)(57— ~5~]a~ + [3((53 +

(54)fl + ((5~+ (56)(51

+

(52~6— ((52

+

(55)(57—

~5~]b~

+ [3((5, +

(52)Y

+ ((53 +

+

(53(56— ((54

+

(56)(57—

~5~]co2}a~bo2c2o

(54)(55

Here again, considerng non-negative rational resonances leads to the following subcases: Case 1(i).

P

= 0,

Case 1(u).

P

= 0

1

D

= ~4,

~‘

=

Case 1(iii). Case 1(iv).

~

,

~

Q

= 0,

r

=

—

1,0, 0, 2, 2, 3

Q

=

r

=

—

1 0

Q Q

= 0,

r r

=

—

=

—

=

,

~,

(6.49)

.

~‘~ “3

(6.50)

1 , 0,2,2,~, ~ ‘ 3 1 ~ ~ 3

(6.51) (6.52)

4’

,2,2,2,2,

In a similar fashion, we compute the associated resonances for the cases 2 and 3:

— f,

X0

=

—

X0 =

s

=

—

s

=

—

s

= 0

Case 2(i).

q

= 0,

s

=

Case 2(u).

q

= 0,

s

Case 2(iii).

q

=

—

Case 2(iv).

q

=

—

Case 3(i).

q

= 0

,

Case 3(u). Case 3(iii).

q q

= 0

,

=

—

Case 3(iv).

q

=

—

X0

= 8[~

,

~,

,

s= s=

,

s

=

~,

,

—

~,

= 0

X0 =

~,

,

~,

,

—

0

X0

= 0,

~

,

,

,

+ 3cca~~ + ((53 + (54)a~c~ + yco~]

r=

—

1, 0, 0, 1, 2, 3

r

=

—

1 , 0, ~2’ 1 ~2’ 3

r

=

—

1, 0, 0,

r

=

—

i , 0,252,2, i ~3

r

=

—

1, 0, 0, 1, 1, 3

.

—

1, 0, 0, 1, ~, 3

.

—

1, 0, 0, 1, ~, 3

.

(6.58) (6.59)

—

1, 0, 0,

.

(6.60)

r= r= r

=

.

(6.54)

,

~,

~,

2, 3

~,

3

(6.53)

.

(6.55) (6.56) (6.57)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

43

Having isolated the above resonance sets (6.49)—(6.60), as usual we substitute the series expansions x(t) =

akx’”

~

k

,

y(t)

=

2+~

~

b~~Idl

,

z(t)

=

k

~

CAT’

(6.61)

k

into (6.43) and obtain a set of three recursion relations in the ak, bk and ck. Solving them successively we obtain their explicit values. Continuing further, we find that the Laurent series expansion (6.61) admits six arbitrary constants without the introduction of movable logarithmic branch points for a set of four nontrivial parametric combinations. The results are displayed in table 6.2. Following the procedure for explicitly constructing the second and third integrals of motion outlined in the previous section, we have succeeded in finding their forms for the cases 1 (ii) and 3 (iv). However, for the remaining cases, 1 (ii) and 2 (iii), the third integral of motion could not be located with a quartic form in momenta, so the corresponding forms are being investigated now. The resulting integrals of motion are as follows (I, = H 3): Case 1(i). 2+ccr6, r2=x2+y2+z2, I,=4(p~+p+p~)+Ar (6.62) 7 r2 r r2 r2 r2 —

‘-‘xy

—

~—~xy 1

1

~—‘xz

Case 1 (ii). I,

=

f(p~+ p~+ p~)+ A(x2 + y2 + z2) + cc(x2 + y2)3 + ccz6 + 15cc(x2 + y2)z4 + 15cc(x2 + y2)2z2

r

r2

r

,

(6.63)

C

1

121-~xy~

3—noLloun

Case 2 (iii). I,

=

y

2 + z2) + By2 + 64fi(x2 + z2)3 + fly6 ~(p~ + p~+ p~)+ 4B(x + 80fl(x2 + z2)2y2 + 24fl(x2 + z2)y4,

(6.64)

r2

1

‘2~xz,

3—noLtoun

Case 3 (iv). I,

=

‘2 =

2 + B(y2 + z2) + 64fix6 + fl(y2 + z2)3 ~(p~ + p~+ p~)+ 4Bx + 80fi(y2 + z2)x4 + 24fl(y2 + z2)2x2 L~ 2,

(6.65) 2 + z2)x

13

=

L5~p5+ L2~p2+ 2B(y + 2fl[16(x2 + y2 + z2)x2 + 3(y2 + z2)] (y2 + z2)x.

M. Laksh,nanan and R. Sahadevan, Coupled nonlinear oscillators

44

50Cr,

‘I~

50 ‘r,

‘N

‘N~

‘N’I~

~0

~

U

~I —

0.5 5-

N

I rsi

r’I

NN ‘N NN’N©

~

,,,~ I~ -*1 — N ~I I

.4 s$ .,f

III

ICC 015-

SO Cr, ~

‘N’N~

r.~0’

~

I~

‘I~

‘N

‘N~

~ —

50 Cr1 Cr,

‘N

‘N~

SIC

‘I~

~

~

.4 —

~I —

III

s.41

I ~

II ‘IN

N

III

~ ~

—

‘IN ‘IN

~I —

II ‘IN’IC.C ‘IN

1111

.0 .~5N..4N.~IN’IN

‘5

1

N

E

~

V

0

II

-IN’IN’IN

.-SN’IN’IN

I .0 —

III

II

..~1’-SN’I’*C’C*

I

II

II

00

a

0.

-4N-~N-4N

-IN-SN-IN

-IN-IN-SN-SN

III

III

1111

-SN-SN-SN-SN

1111

I-

I —

~

N ‘C V

sO

N

sO 00 00

—

~I

N

sO 00 00

—

N

~

‘N

sO 00 00 00

—

N

U

‘3

jo

S

‘N

a

II

a

N

II

a II

I

0. o 0

— -

U V

II

no a ‘N

LI

I

‘N

II

no I

“~

..~“

Ii

II

00 N

..

a

‘5 N

.~i V U

a ~‘.

I

no

‘I ‘N

II

,~

II

II

iL) ~II II~ ..~II -

no

~ I’.

II

11~

no.

no.~

II a

I

~..-

II~. no.~

II no’a~ 1111 ~.II II~ no. 00

~ II

uno

V

II uno

.0

N

a LI

00

~0’ —

—

~0.

.0 N

‘N

sO 00 00 00

U ‘N

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

45

6.2.1. Separability

(i) The integrable case 1(i) is obviously separable in spherical polar coordinates. (ii) Introducing cylindrical coordinates p = (x2 + y2)”2, ~ = tan” (y/x), z case 1 (ii) can be reduced into an effective Hamiltonian system of two degrees of freedom, I,

=

2

—

P4’

~(p~ + p~+ p~/p2)+A(p2 + z2)+ ccp6 + ccz6 + 15ccp2z4 + 15ccp4z2, —

12

—

(6.66)

r2

—

When ‘2 = 0, the system is obviously formally equivalent to the integrable coupled sextic oscillator Hamiltonian with two degrees of freedom, case (5.58a). When p 4, ~ 0, we have not yet found the third integral of motion. (iii) In a similar way, for case 2(iii) in the axisymmetric coordinates 2 + z2)112 p =

I,

,

=

tan’(z/x), y

(x

=

f(p~+ p~+ p~/p2)+ 4Bp2 + By2 + 64flp6 + fly6 + 80fip4y2 + 24fip2y4, 7

—

—

12

—

2

(6.67)

r2

Again when P~= 0, (6.67) is formally the integrable system (5.58c). For P~ 0, we have not yet found the third integral of motion. (iv) Similarly, for the case 3 (iv) we find that the Hamilton—Jacobi equation becomes separable in terms of the parabolic coordinates x=

f(e2—

772)

y=

277 cos0,

z

= 277

sin 0,

(6.68)

as in the case of three coupled quartic anharmonic oscillators (section 6.1.2).

7. Application of Painlevé analysis to N coupled anharmonic oscillators 124, 43, 461 This chapter is concerned with the P-analysis and direct search for integrals of motion of a system of N coupled anharmonic oscillators defined by the Hamiltonian (1.9) and we consider the potential V(x~,i = 1,2, ... , N) as either a quartic or a sextic polynomial given by (1.10) and (1.11), respectively, for our present discussion. 7.1. N Coupled quartic anharmonic oscillators The equations of motion for the N coupled quartic anharmonic oscillators with V(x 1)

=

~ A.x? + ~ cc~xt+ ~ ~ (ij)

fl1~x~x~,

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

46

(1.10), are 2

N

d ~—~+2A~x 1+4cc1x~+2

~ fl~x1x~=0, i= 1,2,

...

,N.

(7.1)

ji

Assuming now (the index i hereafter varies from 1 to N unless otherwise specified) that x,(t)

~

a0’r’~’,

t

0

-.+

(7.2)

in (7.1), we isolate N distinct sets of leading-order behaviour.

p~= — 1,

Case 1.

2cc1a~+ ~ fl~a~ = — 1.

(7.3)

ji /

p,

Case 2.

a20

=

=

1,

—

N

= ~ ±

P2

\1/2

fi2ia~o)

8 ~

j2

2cc1a~+ ~ fi~a~ =

arbitrary,

i

—1,

= 1,3,

...

,

N.

(7.4)

j2 /

Case 3. p,

=

1

—

P2 = ~ ±

N

~(, 1

\1/2

8 ~

—

fi2ia~o)

j2, 3 /

j

2cc~a~0 +

\1/2

N

p3=~±~(,1—8~

~

fi~a~o)

,

a20,a30=arbitrary,

2,3

fi~aJ~ =

—

1,

1

1,4,

...

,

N,

(7.5)

j2. 3

and so on. Thus for case N we have Case N.

a30

=

p,

=

—

1, p~=

arbitrary

,

j

1/2

f ±~(i_

= 2, 3,

...

,

N

8fij,a~o)

.

,

a~0=

—

1/2cc,,

(7.6)

In order to determine the resonance values for the above leading orders, here again we substitute

x,(t) ~ a0t~’+

Q.tPI’{’r

~

0

(7.7)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

47

into the leading-order terms of(7.1) and obtain a system oflinear algebraic equations for the Q1. To have a nontrivial set of solutions, we require detMN(r)

MN(r)

= 0,

o

2

=

AN + DN(r),

,iø

rp12a10a20

occ,a,0

(7.8a)

AQ -‘p,NaioaNo

...

4fl2Na20aN0

AN

4fi21a20a,0

=

4flNlaNOalO

4fiN2aNoa 2o

DN(r)

=

(7.8b)

,

8cc2a~0 8cc~~a~

...

Diag[(r + pj)(r + Pi— 1)— 2,(r + p2)(r + P2—i)— 2, (7.8c)

...,(r+pN)(r+pN—1)—2].

Then we derive the resonances for each leading-order behaviour isolated algebraic 2 3r)above, ‘N~ 50 using that det MN(r) 3r. Thus the coefficients of this polynomial can be expressed on the one hand in terms of its roots flo, ?7~, , 17,P.J and on the other hand in terms of the quantities tr(A~,i = 1, 2, ... , N). Using eq. (7.3), we can prove that there is a root, 77o, equal to 4. As a result, for case 1, eq. (7.8a) reduces to —

manipulations. For instance, for case eq. (7.3)[AN] implies thatvariable DN(r) = r’(r = r2 is the characteristic polynomial of the1, matrix in the

—

...

detMN

=

(r + 1)(r— 4)(r’ + t~,)(r’+

(r’ +

172)

77N—,) = 0,

..~

(7.9)

so that the resonances occur at r

=

1,4, ~ ±~(9.....417,)112 ,

—

Here the quantities 77~’,j’= N-i ~

= 4(1 +

~..

1, 2,

...

,

...

,

N

~ ±~(9 —

417N—1)”2

(7.10)

.

1 satisfy the conditions

~cc~a~ 0),

j’=l

N-i

~

f(77rn17n)42(~

~

j’=~

m,n1 N~1

77f+

1(177777) =

43[.L

m,n,p=1

711

i,j=1

N~1 ~(77m77n) m,n

—

cctcci_~fl?ia?oa~o)~

+ 16(

ftcclccfcck

i,j,k=1

~

~(ccIfi~k+ cc~fl~ + cckfitJ

~

—

flljfljkflkl)] a?oa~oato)]~

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

48

and so on. Continuing further we can finally write down the value of the product (17,772 ... , directly from the expansion of the determinant (7.8a). Then the allowed values of resonances for case 1 are]’ of the ~ = 2, N 1 —]‘ of the 17~= i,j’ = 1,2, , N— 1, and so —

0,

...

r

=

—

1,0,0,

...

,

N—]’

—

1, 1,

1 times ,

...

times

,]‘

(7.12) 2, 2,

...

,]‘

times ,

3, 3,

,

...

N—]’

1 times

—

4

,

Continuing the analysis for the leading orders of case 2, we obtain the following possibilities: (i) P2=°,]’ of the ~ N—2—j’ of the ~ i,]’ = 1,3, ,N— 1 in (7.11), and so ...

r

=

—

2, 2, (ii)

...

P2 =

r=

—

1,0,0, ,]‘

...

,

N—]’— 1 times,

times ,

3, 3,

,

...

1,1,

N—]’

+ 1 times

,]‘

...

2 times ,

—

4

(7.13a)

.

~,] of the n1=2, N—2—]’ of the n,=0, i,]’= 1,3, 1,0,0,

...

,N—]’— 1 times,

2,3,3,

...

...

,N— 1 in (7.11), and so

,N—]’—2 times,

In a similar fashion we derive the resonances for the remaining cases 3, 4, resonance values of case N become 2P2,

r

=

—

1,0,0,

...

,

N— 1 times ,

1—

1—

2p

2PN,

3,

...

,

1—

4. ...

4

N

,

(7.13b) —

1. Finally, the (7.14)

.

Next, in order to identify the arbitrary constants a,p, we substitute the series representation

x~(t)=

a~ 0

(7.15)

,

into eq. (7.1) and obtain a system of N recursion relations in a,,,, i = 1, 2, , N and = 0, 1, 2, 3, 4. Solving them we obtain their explicit values. For instance, for case 1, the recursion relations become ...

(j’— 1)(]’— ~

+ 2A~,a1,~._2 + 4ccj~,~aj.j_j’_maj,taj,m

2~~~fljjaj.j’_i.maj,,aj,m = 0,

0

l,m ]‘

4.

(7.16)

+ At

each ~u,we obtain an equation of the form [MN,,]

[as,,] = [S,,(a,(,,_,))]

,

(7.17)

M. Lakshmanan and R. Sahadevan. Coupled nonlinear oscillators

49

where [MN.,,] is a N x N matrix, [a,,,] and [S,,] are N x 1 column matrices. The matrix [S,,] depends only on the a,,,,_, coefficients. The application of Cramer’s rule for determinants yields a unique solution to (7.17) a,,,

=

detMN,,,/detMN,,,, detMN,,, ~ 0,

(7.18)

where the N x N matrix [MN,,,] is obtained by replacing the ith column in [MN.,,] by the column matrix [S,,]. At the resonance j~, det MN,,, may vanish and eq. (7.17) may not have a unique solution. Suppose further that det MN.,, also vanishes at the same resonance; then one of the coefficients of a,,, may be arbitrary. Furthermore, if m rows of [MN, ji] are identical and so also the corresponding m elements of [S,,], then m 1 of the a,,, will be arbitrary. Making use of the above technique, we identified 4N 4 nontrivial distinct parametric choices, two from each of cases 1 and 2, and the remaining from the other cases 3, 4, ... , N, for which the general solution x,(t) in (7.1) possesses the required number 2N of arbitrary constants, and is free from movable critical points both in the MB and the SB, so that the P-property holds. The salient features of the P-analysis are summarized in table 7.1. In order to demonstrate the previous discussion, we again extend Chandrasekhar’s procedure [39] for finding the integrals of motion to N-degrees of freedom, described in section 6.1. We explicitly derived all the required N 1 time-independent integrals of motion, which are in —

—

involution for each of the 4N Casel(i).

cc,=ccN,

—

4 P cases. The integrals are the following:

fi,~=2cc,,

i,]=1,2,...,N—1,

A,=AN,

6cc,,

fl,N=

i~].

The integrals of motion are I~= H

=

k=1

=

P~2

L~,

~

~N-~

[

2=

Uk

(xjpxk— xkpX.),

=

N-i

~ ~

xt + cc,(~ x~)

—

k=

1(

2, 3,

...

,

:~:xt)x2~, N— 1

(7.19)

+2r2)(~x~)xN](>x~)

~ xj ~ j [~(,‘. ~ j=~ j=’

IN-i

~ x~ j=1

j~i

r

k1

PXNxJPXJ)+2(Al

IN=[(~

+

2+ 4cc 0+ A,

\21r/N-i

1

\-1

xj2)

PX N +

8cc~x~~‘ J

~N x~. k= 1

Case 1 (ii). cc,

=

cc,

,

fi~=

2cc,

,

i,]

= 1, 2,

...

,

N,

cc,(Ak—AJ)’L~k+p~+ 2A~x~ + 2cc,x~r2,

i j

]=

1,2,

...

,N,

k ~J,

50

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

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—

—

+

N

11

—

—

+

—

11

—

—

+

N

11

V

~ V

...

J

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~

E

‘I~

~

I N

‘N

— ‘N

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+

Cr, ~‘ ‘I~

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1

—

+

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~

‘~ -

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I

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.:..N

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0

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~‘~I

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‘N

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N

III

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—

‘N

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d~ .4,4 d’II

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0

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a no.,4

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M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

NN

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+

II

+

00 Cr~

00 Cr,,,f

‘I.

~ I

‘I~”N

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~ I,~ç

.,r.’._~’ if~I ,~IZ

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a

a

‘IN

III

V

:

.0

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a 3-

:

:

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..4N5

:

—

:

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III

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III

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—

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N

CI N

III ‘I N

—

“C N —

II

II

1151

III

I

III

I‘

I’’ ‘all

(ID

sO0000

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a

Ii

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(ID

~

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C.~

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~ ~‘~~II U

~

a

N~.

II

II

II

.-.,

II 1 11 ~‘no’

51

M. Laksh,nanan and R. Sahadevan, Coupled nonlinear oscillators

52

and the Hamiltonian H

=

~~IJ

=

cc,

Case 2(i).

p~+ ~ A~x~ + cc,r4,

~

16cc

=

2

,

i,j=1,3,...,N, I,=H=.,

=

fl23

12cc2

r2

fl,3 =

,

(7.20)

x~.

=

32cc2,

A,

= 4A2

i~], N

~p~,+4A2

\2

/N

~ xt+A2x~+16cc2(\ ~ xt)

k=1

k=1

+cc2x~

+12cc2(~x~2)xL k~2,

=

~ L~+,, r

k= \

/N

IN=[x2Px2(\

2,3,4,

~

...

7N

N

\/

k=i

[(~ x~)( cc,

Case 2(u).

=

k~

N

\12/

N

\—~

(‘. ~

x~)

,

A.

= 4A2

/N

\2

x~)j

k=i

P~k)

(k~,

—

xkPxk)]

~~)-‘+ 8cc2$]~ k ~ 2,

ki

8cc2,

i,]=i,3,...,N,

(7.21)

k=1

/

x [(x2P~2)2(

2,

x~)~~2

+2x~(A2+4cc2 ~ xt+2cc2x~)(\ ~

+

]

\

XkPx~~)_”(5,~

k—i

,N —1,

fi2J

=

f3,~= 16cc2

6cc2,

i], N

~ p,~,+4A~ ~ x~+A2x~+8cc2(,, ~ x~) k=i

/N

\

+cc2$+6cc2(5~xt)xL

k2,

k= ~

k

~ L~,÷, , j= , r

‘N =

P~2+

4x~I

L

k=2,3,4,

...

,N— 1, ]2, /N

\1

Ip~2 /J

A2 + cc2x~+ 6cc2( ~ x~) \k=1

—

(7.22) /N

16cc2x~(\~ xkPxk)Px2 k=1

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators \

/N

/

53

N

+ 4cc2x~~ P~) + 4A2(’.A2 + 4c~2~ x~+ 2cc2x~)x~ k—i

k=i 2

N

+4cc~(x~+2~xt)xL

k~2.

2cc

32cc

Case3(i).

cc,=l6cc3,

i,k=1,4,...,N, I, =H=

fl,~=6cc2, i~k,

fi,k=

fl~~~=

2,

2, A,=4A2,

]~j’,

j,]’=2,3,

p~+4A2~x~+A2(x~+x~) \2

/N

\

/N

+16cc2(,’. ~ xt) +cc2(x~+x~)2+12~2(5 ~ xt)(x~+x~)~ k~2,3, k=i

k=i N

I2=U~3,

~ U~k÷i, k=3,4,...,N—1,

‘k=

j2,3,

i=i

‘N =

{(x2P~2+ X3P~3)(k=i XkPxk) r

/

+2[A2+4cc2(\ r/N

—

(p~2+ P~3)(k=1

\

N

1

/

\

\/N

N

\~2/

I(x~+x~)~ ~ x~jf

~ x~)+2~2x~+x~ k=1 J

k=i

N

(\ ~

\—~

xt)

k=i

(7.23)

\2

/N

x~)(5~ P~k)(5 ~ xkPxk) k=i

k=1

k=i

(

2 x

[(x2P~2+ x3p~3)

k=i

xt)’

+ 8cc 2(x~+ x~)2],

k ~ 2,3.

6cc Case3(ii).

cc,=8cc~,

i,k=i,4,...,N,

fi~~=6cc2, fl3k=l i~k, ],j’=2,3,

2,

fl3~.=2cc2,

A,=4A2,

]~]‘,

N

=

H

=

~

>

p~,+ 4A2 k=i xt + A2(x~+ x~) \2

/N

IN

\

~ x~) +cc2(x~+x~)2+6cc2(5~ xt)(x~+x~)~ k~2,3, k=i

I2=U~3,

‘k=

~L~k+l,

k=3,...,N—1,

]~2,3,

(7.24)

M. Lakshrnanan and R. Sahadevan, Coupled nonlinear oscillators

54

‘N

=(p~2+ p~3)2+ 4(A2 + cc2(x~ + x~)+ 2cc2 k=1 x~)(x~+ x~)(p~2 + p~3)

(

2 + 16cc2(x2p~2+ x3p~3)

N k1

x~)

—

16cc

N

2(x~+ x~)(x2p~2 + x3Px3)( k~

IN

\

~/N

~

xkPxk)

1

~ k=1

2, k

2,3.

+ 4cc~[2( ~ x~)+ x~+ x~](x~ + x~)

Case 3(iii).

cc,

l6cck,

=

fi,k =

12cc

i,]=1,4,...,N,

4Ak,

fi~= 32cc2, k,k’=2,3, k~k’, 2

i~],

fikk’

,

N

~ p~+4A

=

/N

6cc2,

A,

=

\2

2 ~ xt+A2(x~+x~)+16cc2(~,~ x~) k=1

k—i

+cc2(x~+x~)+4cc2x2x3+i2cc2(~xk)(x2+x3)~ k~2,3,

‘k

‘N-i

~

=

k=2,3, ...,N—2,

[(x2Px3

—

x3Px2)(

k=i

k)

/

—

N

j~2,3,

(~2

+ P~3)(k=1 x~)

\

/N

\—1

\1/N

+2(A2+2cc2r2+2cc2 ~ x~)(x~_x~)(’.~

x~)j(~ ~ x~) k=1 k=i

k=i

N ~

Xj,

/N

~ Pxk(\ k=i

k=1

~

XkPxk)

k=1

( ~ ~~)‘ +

2

x [(x2Px2 x3p~3) —

2

‘N = (p~2 p~3)

+ 8cc

8cc 2(x~ x~)2], —

(

2(x~p~3 + x~p~2)k=i

+ 8cc2x2x3(r2 + ~ k=i x~)px2px3

P~k) +

8cc2(

k

2,3,

k=1 Px2k)(x22x32)

(7.25)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators / —

/

N

16cc2x2x3(x2p~3+ x3p~2)(,,~

+ 16cc2(~r2+ ~ xt) x~x~

xkPxk)

k—i

+ A2[4~~2P~3x2x3 + 4A2x~x~ + 16cc2(r2 + 2,

fi,k=

i~k,

i,k=1,4,...,N,

x~)x~x~]~ k ~ 2,3.

,~

fi,~=6cc

2cc,, cc,=8cc3,

\2

N

k=i

Case3(iv).

55

],j’=2,3,

fi~=O,

A,=4A3,

]]‘,

N

IN

\2

~ p~+4A2 ~ x~+A2(x~+x~)+8cc2(,~ ~ x~ j=1

k=i

k=i N

~

k~2,3,

N

~ ~

k=2,3,...,N—2,

]~2,3,

k= i /N

1N_i~(X2Px3+X3Px2)(,,

r 2[(Px2Pxs

—

~ PXk k—i I

2A2x2x3

—

N

k=i

2+ 4cc ‘N =

(p~2 p~3)

2+

k=i

2

—

(7.26)

/

2)

k=i xt)(Px2

2(x~ x~)(A2 + r

—

\

\1IN

2cc2(5r2 + ~ xt)](,, ~ xk

—

P

—

\

/N

IN

+ 16cc 2(x2p~2 x3p~3)2(,, ~ xt) —

—

16cc2(x~ x~)(x2p~2 X3Px3)(,, —

—

k=i

+4{cc2( k=i P~k)+ / N +cc~(~r2+ ~

2 + 2(

k~i xe)]

A2[A2 + 2r

\2~

x~)

x~_x~)2,

k~2,3,

k=i

and so on.

Case N(i). cc,

=

l6cck,

12cc

4A~,

2

k,k’=2,3,...,N, 1N

I,=H=~

flik =

~

i=i

~ k=i

,

fikk’ =

2cc2,

A,

=

k~k’..., N k=2

/N

\2

XkPxk

56

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

+ 12cc2(~xk2)x~.

= i2

‘N =

Pxi

U~k+,,

k=

k=2 xkpXk

—

cc,

Case N(ii).

=

~i

8cc,,,

...,N —1,

2,3,

(~

PX~~) +

fiik =

6cc2

2

(7.27)

flkk’ =

,

)

+ 2cc2r

2cc2,

A,

(k~2

2 + 2cc 2xf)x,

=

k,k’=2,3,...,N,kk’, 1,=Hr’r~ ~

+ 6cc2

= i2

)

(k~2

U~k+,, k = 2

N

1N(~P2xk)

x~,

...,N —1,

2,3,

N

N

+4(~xt)(~p~k)(A2+cc2r2+cc2x~) \2

/N

+ 16cc2x~ ~ xkPxk)

~

flik =

j~]’,

\2

/N

Ik~Uj.k+2,

~ xt) k=2

XkPxk

k=2

2 + cc 12cc ,

fl~= 6cc2

,

/

+4(,’.

N

~

fiji’ =

k=2(~3)

k=2,3,...,N—2,

16cc2x~ /N

x~)x~+12cc2(\~

2+ xi). 2x~)(r

k=2,3,...,N,

1~=H=~~p~,+4A2x~+A2( ~xt)+

+cc2(\

~

+ 4cc2 k2 x~(2A2+ cc2r 2

],]‘=2,4,...,N,

N

k=2

+ 4(A~+ P~1)(~ cc~ = 16cc,,,

\/

/N

16cc2x,p~1 ~ x~)(\

—

k=2

Case N(iii).

(7.28)

\

2cc2,

A1

=

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillatOrs

—

~

~i

—

(X

3PXS

57

Pxi

—

(7.29)

+2(A2+2T2+2~~xjx3)1 /

\j=2

\

fN ‘N =

/N

\

~ P~)P~+ 8cc2(x~p~+ x~p~,) ~ x~)+ 8cc2

~

k2

kS2

2) + 8cc2x3 ~(3x~ + r

~

+ 4A~(~2

+

(

~—2 ~2

Pxk) Pxs

Xk

~ + 4A2 (~2

cc,

X~) Pxs

+

X

Pxk)~}

3 (~2

k~3.

+X~)X~,

+cc2x21)(r

fl,,,6cc

=Scck,

~£

X~P5k) X3

2

N l6cc2(Zxt)~2~2r2

Case N(iV).

2x, Pxi

—

x~x~

0~

2~

fljj’

~2cc

2,

~3J

A1

4A,,,

JfC=2,4,...~N~3~J’ k=2,3,...,N, N

I1=H=~~ p~,+4Az

+A2 ~

j’i

k2 /

\2

N

\

/N

~ 3) j~2(*

k2

I,,~LJC~Z~

IN-i = (x 2 Pxi

Xi

—

p~2)P~3 +

~m

Pxm~t)Px

[A2 + cc2(r

(,,~2xkXk’+1)

/

/

\2

N

‘N =

N

~ p~) + 4cc

+

N

Pxi) ~i Px~

j

3,

(7.30)

\

/

N

2 + x~)~P~s + ~

2

JNZ

/

(~

2 + xii)],

N-i

+ 4x,

—

X~+ \2

i2 ~ xj )(A2 + r /

N

~Xj

j=2 \2

2 + xi)]

+ 16 ~—X

3

PS3 + j—2 ~ x~~~3) x~+ 4ccX~ + j=2 ~ x~) [P~~ + A~+ 2A2(r

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

58

+ 16cc2(x~ /

~J)[(~jj)P3]1 2+xf)2,

\2

N

]~3.

—4cc~(~x~_ j—2 ~ x~) (r After a detailed but straightforward calculation, one can convince oneself that the Poisson brackets Urn, ‘n]PB = 0, m, n = 1, 2, , N vanish identically, so that the N-integrals are indeed in involution for all the 4N 4 P cases. For example, in order to show the involutive property of the N-integrals derived for the P case corresponding to case(ii), we exploit the following relations: ...

—

~

PX,]pB

=

pX, ~ + 2A, x, x~+ 2cc 1 x~

=

Y,

~

PXJ]PB =

2 x~, ~

Pxk]PB

= 0

,

i,j

k,

r2

—

=

P~,= ~p~,+ A, x? + cc, r

Thus, the complete integrability of N coupled-oscillator system (7.1) for all the above 4N 4 parametric choices stands established. Regarding separability, case 1 (ii) has been shown to be separable in generalised elliptic coordinates by Wojciechowski [69f]. Whether a similar possibility exists for the cases 3(i) and N(i), which admit quadratic integrals of motion only, has not yet been explored. All the other cases admit at least one integral of motion involving quartic power in momenta and these cases are unlikely to be separable. —

7.2. N coupled sextic

oscillators

anharmonic

Now the equations of motion, for the N coupled sextic anharmonic oscillators whose potential is given by (1.11), are N

t

N

+ 2A,x, + 6cc,x~+ 2 ~ fi,~x,x~+ 4 ~ fi~,x?x~ +2 ii

ji

N ~ ~ i..j,k= 1

= 0,

(ijk)

i

= 1,2,

...,N

(7.31)

.

From the leading-order behaviour, we isolate the following distinct choices: Case 1. p,

=

—

3cc,a~+ ~

fi,~4+ 2

~ fi~,atoa~o + ~ ji

y,jka~oa~.o =

_~,

i

= 1,2,

...

,N.

j,k=1

(7.32)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

59

Case2. p,=—~,

1r

1 /N N N p2=~±~I1—8( ~ fi2~a~o+2~fl2,a?oa~o+ ~ ~

\j*2

arbitrary

=

\11/2

Y2~kaJoako)]

j,k=i

i*i

(7.33)

,

N

N

3

N

3cc3a~+ ~ fl~a1~ + 2 ~ a~a~0+ ~ ji

j~

~

a~0=

—

~,

i

= 1,3,

...,N,

j,k=i

and so on; finally 2, CaseN.

p,=—~,

j

arbitrary,

=

a~

p~±~(1+fl,~/cc,)” = 2, 3,

...

,

N

0=—1/8cc,, (7.34)

.

Having obtained the above leading-order sets, and proceeding as in the case of quartic anharmonic oscillators, one can derive N sets of distinct resonances for each of the leading-order cases. One then proceeds to compute the arbitrary constants of x,(t) corresponding to the resonances along the lines of the problem of the quartic anharmonic oscillators. The calculations show that the introduction of N arbitrary constants in the solution x,(t) is possible for a set of N + 1 nontrivial parametric combinations. One can further verify that the associated SB does not admit movable logarithmic branch points and hence the P-property holds for N + 1 cases. The results are displayed in table 7.2. The associated other N 1 time-independent involutive integrals of motion are as follows (I, = H): —

Case 1(i). ~

1

N

N

~x~+cci(,

i=i

/N—i

~

\3 xt) +cc1x~

i=i N-i

N—i

2

+15cci(~xt)x4~+15cci(~xt)

x~, (7.35)

~ U3~,,, k=2,3,...,N—1,

Lik=xjpxk—xkpxf,

i= 1 ‘N =

not yet found.

Case 1 (ii). 2 + cc,r6 I~=

i=i ~

p~,+ A,r

,

r2

=

~N i=i

x,~

‘k =

~k U~,,, k =

2,3,

...

,N.

(7.36)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

60

— N II ~

.

N

~

II

—

+

+

N N N

N N N

‘N

‘N

N N

‘I~

~ — ~e

-~

.

~~,-,..,I — E ~r

T

‘

II

—

+

N

!~ V..~

‘N

~

a

N

~—

iCC

~

~“—~~‘i

~

~II —~ ~ $ III

‘5

ic

‘:‘~-~‘

—

~I

—

“~

I

a

©~

~ssc

z

~

:

:N

.r.~’if

~

—ic

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~

T

5011

501-

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: c$ I

‘IN

-SN-SN-SN

‘IN

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:

r.i I

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~

r.i I c$ I

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—

I

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‘I’I.~

‘~

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~

~

N~

I

-El

T

~ -

~ ‘IN~

‘N

.,.,~I

:r~.1

0

+

‘N

Iz:

~.

—

N N N

I ~

+

‘N~

~E

II

—

N N N

‘I,’’N

~fl ~

0

N

—

II

—

.‘.

‘~‘

~ ~

I~cI~

I

~.

I —

.-~

:

~II ~A .4 CIA ~‘sA ‘IA III I

:~I ~.4ssA ~II .4 “sA ~A III

‘IA I ~A I

~ ©.-~‘I ‘4 ~A ~A III

I ~A

-IN “IN-IN

-SN-SN-IN

‘IN

.-SN.N*’.IN

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V .0 N

N -SN-SN-SN

III

III

‘IN

III

III

III

.0 3—0

.

a

:

:

:

:

:

:

:

:

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.-SN.-IN’IN

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II

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,~

‘IN

I

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sO

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III

l~II

I(ID

sO (ID(#D

~

‘IN

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sO 0000

00

‘L

~

II,~ii

“H-N

.-SN.-SN’IN

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‘IN

-SN-SN-SN

-SN

II ,,~ ~

_~~--,13. a no, ~ ...~ ...

II

~

‘-SN-SN-IN

-SN’.SN’.SN

III

l~II

~ I

I~ ~l

I~

l~II

sO (ID (ID

~ ,~,

Ii

II N

II II E •N..,~~13. a no. ~. ..r .~ .s~

N

00

sO 0000

A~-~

N

•~‘-S13. a ~. ... ....

-SN

I

~ ~

II

III

(ID

‘IN

-IN-IN’IN

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‘IN

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II

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5~~3. a no. a’.. ...- -~ .‘.

‘N

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-SN-IN-SN

II

II

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‘IN

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a 0.,

.-SN’IN’IN

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Ii no. ,~

‘IN

III

III

,i~ :~ ~

-SN-SN-SN

I

-SN-IN-SN

~

‘IN

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o 800a

-SN-SNCIN

I

a no.

II ~ “~

‘-SN

~ I

(ID

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

61

Case 2. N ~ p52, + 4A2 ~ ~ + A2x~+ 64cc2

=

i=i

\3

~ xt)

k=1 \2

/N

+ 80cc2

/N

(,

/N

~ xt) x~+ 24cc2 k=i

\

~ xt) x~, k s’~’2,

(7.37)

k—i

k=2,3,...,N—1,

1k~Li,k+i,

]2,

IN=notyetfound.

and so on. Case N. N

/N

\3

~ i=1

k=2

k=2

N

N

2

+80cc2(~xt)xt+24cc2(~xt)

x~,

N

V’

7 —

~

r2

‘-~j,k+i

~

,

?(

TtT —

L,

.)~ ~

—

1 I ~

j—2

‘N =

~,

(k~2 Xk

Pxk)

/N

—

~

(~

P~)+ 2A2 (k~2 \/N

)~

\2

+2cc2(53 ~ x~+16x~+16x~)(,,~ xt) xi. k=2

k=2

8. Invariance analysis, Lie symmetries and integrability of Hamiltonian systems In this section, we wish to discuss how the integrability properties of the coupled nonlinear systems (which are of finite-dimensional Lagrangian type) discussed in the earlier chapters follow in a natural way through a study of the symmetry properties. In fact, Lie advocated almost a century ago the study of differential equations through invariance analysis under one-parameter continuous transformation groups associated with symmetry properties as a means of analysing them. This method has in recent times been further developed by Ovsjannikov [41], Bluman and Cole [40] and others [70, 71]. Conventionally the theory ofone-parameter Lie groups of continuous transformations is applied to dynamical systems, particularly to Lagrangian systems, by assuming the infinitesimal generators of the group to be functions of the independent and dependent variables alone and not their derivatives. However, this approach has a rather limited range of applications as it deals with point symmetries alone. The limitations in the above approach can be circumvented by including the

M. Lakshmanan and R, Sahadevan, Coupled nonlinear oscillators

62

velocity terms in the argument of the infinitesimals. Such transformations operate on trajectories in space—time and are called dynamical symmetries [18], if they preserve the equations of motion of the given system. The study of the invariance of differential equations under derivative-dependent transformations has drawn wide attention in recent times [72—80].Recently, Lutzky [18] considered the one-dimensional harmonic-oscillator problem and derived the associated dynamical symmetries using Lie’s theory of extended vector fields. Subsequently, several Lagrangian systems have been identified to possess dynamical symmetries by many authors [70, 77—82]. In the following, we present a brief outline of Lie’s extended theory of one-parameter continuous transformations applicable to two-dimensional Lagrangian systems and then apply the method to the nonlinear oscillators of our interest. At the end, the application of Lie’s invariance analysis to higher-dimensional coupled Lagrangian systems is also briefly discussed, along with the method of direct integration of symmetries. 8.1. Invariance conditions, determination of infinitesimals and first integrals

of motion

Let us consider a two-dimensional Lagrangian system related to the Hamiltonian H=~(p,~+p~)+ V(x,y),

L=~(~~2+5’2)—V(x,y),

(8.1)

where the dot means derivative with respect to time. The Euler—Lagrange equations of motion are =

~L/~x

=

9=

cc,(x, y),

aU/3y = cc 2(x, y).

(8.2)

For eqs. (8.2) to be invariant under the action of the infinitesimal transformations of a oneparameter (6) continuous Lie group, X

t

-.4

X

—+

T= t+

=

x

+ 677,(t, x, y, 61~(t,x,

y, i,

~,

9)

i))

+

+

0(62),

0(62)

,

6 ~

y

—~

1

,

Y

=

y + 6772(t, x, y,

.~,

9)

+

0(62),

(8.3)

we require the following invariance conditions to be satisfied [18]: (8.4)

22’~(t~2),

~,—~—2~cc,=E(cc,),

12Y

where the infinitesimal operator E is given by E = ~ ~/~t + 1J~~/8x +

772

~/8Y + (ti,

—

~) ~

+ (‘~ ~9) ~/a9. —

(8.5)

From eqs. (8.4), it is clear that the invariance conditions form an incomplete system in ~ 772 and This suggests that one will have to assume specific forms for 77i~ 172 and ~ in order to solve (8.4) consistently. A trivial choice is ~.

~1i=

772 = 0,

~ =

C

=

constant,

(8.6)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

63

and so E = cô/3t from which we may infer (see below) that the Hamiltonian H is an integral of motion. To determine the existence of the other nontrivial infinitesimal symmetries, we may assume for example fli, 772 and ~ to be polynomials in velocities i and 9 and then find the t, x and y dependence consistently. As an example, let us first consider the linear form a, +a2~+ a39,

=

,~, =

b, + b2~+ b39,

72 =

c, + c2~+ c39,

(8.7)

where the a~,b, and c,, i = 1, 2, 3 are functions of t, x, y only. Making use of (8.7) into (8.4) and equating the various coefficients of im9t5, m, n = 1, 2, 3, 4 to zero, we obtain a system of overdetermined pdes, a255

= 0

2a2~~ + a3~~ = 0,

,

b2~~ (a,~5+ 2a2~,)= —

0

a2yy

+ 2a3~~ = 0,

a3yy

= 0,

(2b2~~ + b3~~) 2(a,~~ + a25~+ a35~)=

,

—

(8.8)

0

(8.9) b2~~ + 2b35~

—

a,yy

—

2a3~~ = 0

b3~~ = 0

,

2a,~, a —

~

+ 2b25~

2cc,~—

—

5cc,a2~ a2~~ 2cc2a3~ cc2a2y —

—

—

= 0,

2(b,~~ + b2~,+ b3~~) (2a1~,+ a2cc,y + 4cc,a2~+ a3cc2~+ a3~f+ 3cc2a3~+ 3cc,a35) —

(8.lOa) = 0,

(8. lOb) + 2b3~~ + 2cc,a3~= 0,

~

(8.lOc)

2b,57 + b2cc,5 + 3cc,b2~+ b3cc2~+ b2~~ + 2cc2b3~+ cc2b2~ 2cc2a —

~

(a,7, + 4cc,a2~+ cc2a,y +

3f

+ 3cc,a,~) (b2cc,~+ c2cc,~)= 0, —

(8.lla)

+ b2cc,~+ 2cc,b2~+ b3cc2~+ b31, + 3cc2b3~+ cc,b3~

2cc,(a,~+ a3~) (b3cc,5 + c3cc,~)= 0 , 2cc,b b,,, + 2~+ 2cc2b3~+ cc,b,~+ cc2b,~ 2cc,(a,, + cc,a2 + cc2a3) —

—

(8.llb)

—

—

c2XX

(b,cc,5 + c,cc1~)=

= 0

0,

(8.12)

c2~~ + 2c35~ 2(a,~~ + a2y, + a3~,)=

,

—

0

(8.13a, b)

,

2c 2~~+

c355

—

(a,~,,+ 2a2~,)=

0

,

c3yy

—

~

+ 2a3~,)=

0

,

(8.13c, d)

~ + 2c2~~2cc2 a2~= 0, (8.14a) 2(c,~ 5+ c2~~ + c35,) (2a,~7+ a2cc,~+ 3cc, a25 + a3cc2~+ a2~~ + 4cc2a35 + 3cc2a2~)= 0, —

—

(8. 14b)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

64

c,yy

+ 2c3~, (2a,~~ + a2cc,y + 2cc1a2~+ —

~

a3cc2y

+ a3~l+ 5cc2a3~+ cc,a3~)= 0,

(8.14c)

+ c2cc,~+ 3cc,c2~+ c3cc2~+ c2,~+ 2cc2c3~+ cc2 c2~ 2cc2(a,~+ a2~) (b2cc2~+ c2cc2~)=

—

—

2c,5, +

—

c2cc,y

+ 2cc,c2~+ c3cc2y +

C3ff

0

(8.15a)

,

+ 3cc2c3~+ cc,c35

(a,,, + 2cc,a2, + 4cc2a3, + cc,a35 + 3cc2a,~) (b3cc25 + c3cc25) —

= 0,

(8.15b)

c,,, + 2cc,c2, + 2cc2c3~+ cc,c,5 + cc2c,5 (8.16)

—2cc2(a,~+cc1a2+cc2a3)—(b,cc25+c,cc2~)=0,

where subscripts denote partial derivatives. By successively solving the determining eqs. (8.8)—(8.16) together with (8.2), we can find the forms of ~ 772 and ~ explicitly. In a similar way one can start from a cubic form for ~, ~, and 772 in the velocities ~ and ,9 as 3

=

~

3

~

~,

i,j=0

=

~ i,j—0

3

~

172

=

~

~

(8.17)

i,j=0

where the ajj, b,3 and c,,, are functions of t, x, y only and obtain determining equations which turn out to be too many in number. Solving them consistently, one can again obtain the generalised symmetries. As the determining equations have a multiple number of terms, we refrain from giving their forms explicitly here, since they were presented elsewhere [45, 84]. In order to find the explicit form of integrals of motion using the dynamical symmetries derived earlier, we make use of Noether’s theorem [42], stating that, given the infinitesimal symmetries 77’, 172 and ~ and the Lagrangian U, the integral of motion, if it exists, may be written as l(t, x,v,~,9) = where

(~

—

77,)3L/~2+

f is a function of x, y E(L) + ~U =f.

(~9

—

772)~L/~9 ~L —

+f,

(8.18a)

only and is to be determined from the equation (8.18b)

As mentioned earlier for (8.6), one can easily check that I = H, the Hamiltonian of the system. We may add that the connection between the integrals of motion and the dynamical symmetries is not too direct; for instance, there always exists an infinite number of linearly independent dynamical symmetries which have no independent integrals of motion; see, e.g., the work of Sarlet and Cantrijn [73].

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

65

8.2. Applications 8.2.1. Henon—Heiles system [83] The Lagrangian of the Henon—Heiles system (1.3) is U=~(~2+92)—~(Ax2+By2)—(ccx2y—~fiy3).

(8.19)

From the equations of motion, d2x/dt2 + Ax + 2ccxy

= 0,

d2y/dt2

+ By + ccx2

—

fly2

= 0,

(8.20)

we have cc,

=

—(Ax + 2ccxy),

cc

2 2

=

fly2).

—

(8.21)

—(By + ccx

Now analysing eqs. (8.8), we obtain a

2+a (8.22a)

2 =a20y

21xy + a22x + a23y + a24, 2 a a2,x 20xy + a31x + a32y + a33, —

a3

=

—

(8.22b)

where the coefficients ajj are functions of t only. Also, from (8.9) and (8.13), we find 2y + â 2+â 2+b 2 +b b2 a, = a2, x 20xy 22x 20xy + ~b2,y 22x + b23y + b24, —

3 b3

=

—à2,x

—

ã

2+ b

2y + ~b 20x

(8.23a)

3,xy + b32x + b33y + b34,

30x

(8.23b)

2y à 2 + ã 2 + ‘~c 2+ c —á21x 20xy 32y 30x 3,xy + c32x + c33y + c34, 3 +á 2+’~c 2+c c2 =a20y 2,xy 20y 2,xy+c22x+c23y+c24, —

c3

—

a,

=

2b20 + b32

=

2(á23 + a3,)

=

c20 + 2c31

b2, + 2b30

,

= 2à32

,

c30 + 2c2,

(8.24a) (8.24b) = 2á22

where the coefficients a,j, b,3 and c~3are functions of t alone. Similarly, from (8.lOc) and (8.14a), we get 3 + (ä 2 b = (A + ~ ccy) (a20x a32) xy 20x b30)xy 3,y + B,0y + B,, , (8.25) —

—

—

4 [B(a 2y cc(a2,y + a22)x 2, + a22) + c21] x [fl(a 2y2 ê 2 C 22 + a21y) + a22]x 22x 10x C,, —

c1

=

—-~

—

—

—

—

,

(8.26)

where B,0, B,1 and C,0, C1, are functions of t, x and t, y, respectively, and the remaining coefficients are functions of t alone. Making use of the above coefficient values (8.23)—(8.26) in the remaining equations of(8.1O)—(8.12) and (8.14)—(8.16), one can explicitly find the values of the a3, b, and c,. However, for practical purposes, particularly for constructing the time-independent integral

66

M. Lakshrnanan and R. Sahadevan, Coupled nonlinear oscillators

of motion, it is sufficient to consider that the functions b, and c~are independent of time while the a vanish. As a result, one can straightforwardly check that the consistency condition holds only for the following three parametric choices (excluding the trivial choice cc = fi = 0). For example, for the parametric choice A

B,

=

cc

=

—fi,

b,

=

b2

(8.27a)

we have a

= 0,

=

c,

=

c3

b3,c2

= 0,

=

constant,

(8.27b)

and so the infinitesimal symmetries become i~, =k9,

~=0,

772=ki,

k=constant.

(8.27c)

The other two parametric restrictions are A, B arbitrary,

6cc

—fi,

=

16A

=

B,

16cc

=

—fi.

(8.28a, b)

The associated forms of the infinitesimals ~, i~, and 772 are displayed in table 8.1. Earlier, we pointed out that in order to construct the integrals of motion associated with the infinitesimal symmetries derived above, it is necessary to solve eq. (8.18b), which can be rewritten as 77,3U/0x +

772

aL/ny +

(t),

—

~) ~ + (~ 9~)9 —

+ ~L

= (3f/3x)~

+ (ôf/~y)9.

(8.29)

Solving (8.29), one obtains the explicit form off It turns out that effectively in all the integrable cases the explicit form of f can be chosen without much difficulty. For example, for the set of infinitesimal symmetries ~ = 0, ~, = k9, 772 = k~ together with the parametric values A = B, cc = —fi, eq. (8.29) gives 2 + ccy2), = 2~ =

—

2(A + 2ccy) x,

0f/~x =

29 =

(8.30a)

—(Ay + ccx

and so

f= —2(Axy + ~ ccx3 + ccxy2).

(8.30b)

As a result we obtain the second integral of motion as I

=

~9 +(Ay +kccx2

+ ccy2)x,

(8.30c)

which is identical to the one displayed in table 5.2 obtained from the direct method. The forms of the integrals of motion I for the remaining two sets of infinitesimal symmetries are given in table 8.1.

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators N

C::;~-~

N

N0~

N

+

a

~

N

N

~

—

,

N

N

a

N

+

N

N

I

N

~

a

.d4~

+ I +N

~‘‘I’

N

‘I~

I

~

‘~

+ N

N N

~

~,

N

N

N~‘

N0.

N

no,

00

N

+~•~no.

~‘

.1~ —

+

..SN+

~

N

+ N no,

~

++

I

‘5 ~

~

I

~ ~

+>~++I .;‘.,.4 N

—

~-“~

‘l.(

~

~

~,

~N

~‘

N

‘N

a

0.

____

+

++~.4~+I++ I N .~,

.~.

~

+

+

‘4

N

~

,.4

0

~

~,

~-‘~7~5

N~.3., C.4 ~

‘l.(

~‘

±r.1±~ ~

“~

N

+

“~ ~

NI I 1+

N

I

67

I

.).~

V1 + NIC

..4

~“I +

N

-~~

<

‘~‘~ ~

N

NI no.~ —

+.~

0

~N

)~

U

E 0 0’

a a a

-~

a

I

CI’

‘~ N

)‘~ N

N~Sr,

I

NIN

E

~,,

00no.

~ ~ -~

~~

+ CI’

N

~

I

‘.4 N

—1

.~ .~

~~

N

~‘

~

NI

‘l-( .3~

.0 ~‘

~10 V

VO N

3-~ ‘C0

E E

N

3~C “S ‘~

~

E

E

~‘

.~

a —

~ N

+

~N

N

I

.~, CC

0’

N

a

+

“

~

~a. ~

F~ ~1N

~0~5 ‘~ ,~ CI’

~).(

0

0

N

•—

N >~

+

I

~-4

~,

~

‘~ ~ -~

~ ~ 0., 00

0

0

0 0

0

‘4 NI

‘4

.a

~

0 0

0

.~

0

0

0

a

a

a’~

N

N I-

V

a

‘5 a ~

a

no.

noI

I I

~ no. II ‘0 II ‘0 no 11 II no a no~ono. no,

a

no.

‘0

I II

~‘

a

~a II

0.’,

a ‘0 —

N .0

E

no. I II

5-

~

-

~

no,

~‘

~no.1111

II .o a a- a~CI~CI’ -

“i~ ‘0

~

‘N

—

II

~

II

N N

111100

-—

II

V

‘N~~C~’

II

- CC no. nonoCI

0-

II

no.’no. II II aII a a ~~CI’ -

I

a

II II no

II

0

no. CI’ N

~ Ii II II no -

aN

-‘~“‘~ 0 ~

~ II

a

.a

.1~

0

aN

~..—.,,

N

I

a

—

N

N

aN

N

V

‘a

LI

— N

,~

0

N ‘N CI’

—

N

‘N

—

C 0

V — (/D

0

a

V

U 50

a 0

0’

.~ —

0.,

N

‘N

N<

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

68

8.2.2. Quartic, sextic and polynomial-potential nonlinear oscillators [45, 85] Proceeding in a similar manner, one can carry out the Lie symmetry analysis for the quartic, sextic and polynomial potential nonlinear oscillators whose Lagrangians are, respectively, L

= ~2

+

92)

—

(Ax2 + By2 + ccx4 + fly4 + öx2y2),

(8.31)

L

=

+

92)

—

(Ax2 + By2 + ccx6 +

(8.32)

L

= I(,~2

~

+

fly6

0

$~2)_ ~2ccNkxN2ky2k

+ ö,x4y2 + s5 2y4), 2x

k

[N/2]

(8.33)

.

The obtained results are displayed in table 8.1. The constructed forms of the second integrals of motion are in full agreement with the Whittaker analysis given in section 5. The above discussion and illustrations explicitly demonstrate that the approach by means of Lie symmetries extended vector fields provides a more direct, systematic and useful analytic technique in analysing the integrability properties of two-dimensional Lagrangian systems. In particular, this analysis seems to be quite useful to study the underlying nontrivial dynamical symmetries and to construct the associated integrals of motion exploiting Noether’s theorem. Now, we wish to point out further that this analysis is also quite powerful in predicting the integrable cases even for higher-dimensional systems. As an illustration, we consider the three coupled quartic and cubic nonlinear oscillators in the following section. 8.3. Three coupled quartic and cubic nonlinear oscillators [84, 86] The associated Lagrangians for the quartic and cubic nonlinear oscillators, respectively,’ are L

= I(~2+ 92

—

U=

+

~2)

(Ax2 + By2 + Cz2 + ccx4 + fly4 + yz4 +

~(~2

+

5,2

+

~2)

—

‘~(Ax2+ By2 + Cz2)

—

+ ~y2z2 + wx2z2),

5x2y2

(ccx2y + flyz2

—

~‘y3).

(8.34) (8.35)

The Euler—Lagrangian equations of motion, respectively, are = —2(A

+ 2ccx2 + oy2 + wz2)x

9 = —2(B + 2fly2 + ox2 + 6z2)y

=

=

cc, cc 2

2 + wx2 + cy2)z =

—2(C + 2yz

=

—(A + 2ccy)x

=

(8.36a)

,

,

(8.36b)

,

(8.36c)

cc 3

2 +

2’ =

—(C + 2fly)z

=

=

cc1 cc3

,

.

9

=

—(By + ccx

flz2

—

‘yy2)

=

cc 2

,

(8.37a, b) (8.37c)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

69

For eqs. (8.36) and (8.37) to be invariant under the one-parameter continuous transformation groups, x

X

-~

=

x + 8?7,(t, x, y, z, t, 5’,

~)+ 0(62),

(8.38a)

5’, ±) + 0(82),

(8.38b)

5’, ±) + 0(62),

(8.38c)

y —~ Y = y + 8772(t, x, y, z, ~, Z t—+

=

z + 6?73(t, x, y, z,

~,

T= t+ 83~(t,x,y,z,)~,9,~) + 0(62),

c

1,

‘~

(8.38d)

we require the following invariance conditions to be satisfied: ij,

—~—2~cc,=E(cc1), —

—

ij2—9~—2~cc2=E(cc2),

2~cc3= E(cc3),

(8.39a,b) (8.39c)

where the infinitesimal operator E is given by E = ~ 8/ôt + + (ii,

—

ii,

~/ôx +

~)

772

ä/~ +

~/ôy + ~

(~— ~5,)ô/ô5’ + (t~ — ~)

~/aL

(8.39d)

We repeat Lie’s invariance analysis described in the previous section, first by considering ~, ~,, 772 and ~ as linear polynomials in the velocities ~, 5, and ±, and then extend them to cubic polynomials having the forms 3

~

~ 93 ±“,

=

ba,,

77~ = i,j,k0 ~

3 72

3

3

= i,j.k=0

X

±k

y

3

~

i,j,k=0

c,~,,~‘

Sii ±‘~

7~=

,

~

i,j,k—0

d 3~,,~

3±“

S~

(8.40)

,

where the coefficients a,jk, b,~,,,c3Jk and d 3~,,are functions oft, x, y, z only. Then we find that, for the quartic oscillators, for a set of eight distinct parametric combinations, nontrivial extended symmetries exist [84]. The parametric restrictions (cases) in (8.34) are given below, and the associated symmetries are given in table 8.2.

fl =

Case (i).

cc

Case(ii).

cc=fl=y,

Case(iii).

cc= 16fl=y,

0=8= 12fl,

Case(iv).

cc=8fl=y,

O=e=6fl,

=

y,

0

=

c = cv

=

2cc,

O=c=6cc,

A, B, C arbitrary.

co=2cc, w=2cc, w=2cc,

A=B=C, A=4B=C. A=4B=C.

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

70

Case (v).

cc

Case(vi).

cc=8fl=8y,

Case(vii).

cc=16fl=16y,

Case(viii).

cc=8fl=8y,

= 16fl = l6y,

0 =

cv

c=

= 12fl,

O=w=6fl,

c=2fl,

0=w=12fl, O=w=6fi,

A

2fl,

.

(8.41)

A=4B=4C.

v=6fl, 8=0,

= 4B = 4C

A=4B=4C. A=4B=4C.

Similarly for the cubic oscillators (8.35), we identify that the nontrivial dynamical symmetries exist for a set of three distinct parametric choices. Furthermore, we can derive the corresponding integrals of motion for each of the isolated infinitesimal symmetries for both the quartic and cubic oscillator systems by exploiting Noether’s theorem. Details of the results of the three coupled anharmonic oscillators are briefly given in tables 8.2 and 8.3. 8.4. Integration of symmetries*) In this section we directly integrate [87] the infinitesimal Lie symmetries to show that the integrability and symmetry aspects discussed earlier through the Painlevé analysis, direct method and Noether’s theorem can be obtained in a straightforward manner. We also discuss how the separability aspects discussed earlier follow by integrating a part of the characteristic equation associated with the symmetries. 8.4.1. Two degrees offreedom Let us consider the one-parameter (8) Lie group of continuous transformations associated with the infinitesimal generator E of(8.5). Then the local invariant U(t, x, y, ~, 5’) of E is a solution of the first-order partial differential equation E(U)

= ~

+

hi

3(1 -~—

+

72

0U -~-

+

(ti,

—

.3U xc~) + ~--

(~2 —

.3fJ y~) -~-

= 0.

(8.42)

From the classical theory of partial differential equations, we can easily see that eq. (8.42) admits four functionally independent solutions U,(t, x, y, ~, 5’), i = 1, 2, 3, 4 [48, 70, 88]. These solutions are the four essential constants which appear in the general solution of the system of four first-order ordinary differential equations resulting from the characteristic equations, dt/~= dx/77,

= dy/772 = d~/(i),

—

~)

= d9/(~2

—

9~).

(8.43)

8.4.1.1. Method for constructing invariants. To find the function U~,i = 1, 2, 3, 4 we observe that any tangential direction through a point (t, x, y, i, 9) to the surface U,(t, x, y, ~, 5’) = C,, I = 1, ... ,4 satisfies the relation [89] 3 Ui 3 U~ 0U, 0U~ . 0 U, . —dt+——dx+-—dy+----dx+—---dy=O, Ot Ox Oy Ox Oy N)

This section has been written in collaboration with M. Senthil Velan.

z=1,...,4.

(8.44)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

71

If U3(t, x, y, ~, 5’) = C, is a suitable one-parameter system of surfaces, the tangential directions to the integral curves of the characteristic equation (8.43) through the point (t, x, y, ~*, 5’) are also tangential directions to these surfaces. Hence, OU,

011,

0U~

Then to find U, (i =

.

0U~

0(1,

.

we try to find functions P,,

1, 2, 3, 4),

(8.45)

Q., R,, S, and

~

T~such that

i=1,...,4,

(8.46)

0U,/0~, T, = OU,/09,

(8.47)

with the property P.

=

OU~/0t,

Q, =

0(1,/Ox,

R, = 0U~/Oy, Si

=

so that

Q, dx + R. dy + S, d~+ T, d5’ is an exact differential dU,. The interesting point here is that for all the systems with two degrees of freedom considered in this article, with ~ = 0, one can find at least two nontrivial invariants U, and U2 straightforwardly which turn out to be the required two involutive integrals of motion. This can be done by choosing the functions P,. Q,, R,, S, and 7’,, I = 1, 2 in (8.46) in one of the following ways: P, dt +

(i).

S, so

Q,

P~=0, =

=

—cc,

=

=

0U/0~, T,

—OU/Ox,

=

5’

=

R,

=

—cc2

=

—OU/Oy,

OU/05’,

(8.48)

that U, is the Hamiltonian for all our integrable cases. (ii).

P2

= 0,

=

Q2

—~,

R2

,

=

S2

~12

= h71

,

T2

= ~72

(8.49)

,

so that U2 is the second integral of motion in all the following examples. 8.4.1.2. Example: coupled quartic oscillator (see table 8.1). 2 + By2 + ccx4 + ccy4 + 2ccx2y2 Case 1.

A, B arbitrary.

,

(8.50)

H = ~ (~2 + 5,2) + Ax

Generalized Lie symmetries: =

0,

= 772 =

~,

=

2y(y~— x5’) + (2/cc)(B

25’(~y x9)

4(B

—

A)xy2

2~(x5,— y~)+ 4(A

—

B)x2y.

—

—

—

—

8(B

—

A) i, A)x3

772 =

—

2x(x5’

(4A/cc)(B

—

—

y~),

A)x,

(8.51)

72

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

+

.-~‘

N 0.,

N I

‘,

I C).)

‘~

,—,N~

— -~

±

~

a

.~ “

+ —

~ ,~

I

‘5

~

~ ~

~,0.

N

+ ~

~+ +~o,,

+N +N 0. i.~ .>.,‘,‘

~ ‘0.,~ ~+N C1) ,

+“

‘~

+,~‘

~‘0’~

$‘~0.,~

~

.~

.,‘>.)+ NM

“-

N

I

+00 ~+ ~N

+

Nil

~00+

‘0’, ~

0)

I

~F

,_,

— ~ N)..)0,

no,

~t

~+~-‘

MON

N

N

~““s-~

~

N

±~I

E .r~

N

+ +

>~

—

+ N+ N NN

~_,+LI

0

I

~

‘0.,

+

+

+

+~‘N~

~N+~__

~.

N

N

,~.,

I +

NM

+N

+

~

~

~

~

~

,,

~, ‘N

N

N

,~

NC,

I

.,.~

N0.

).)

N

I

~

.,++++

~,X~+++

N~.

IC

~ N

>1. ‘.4SC

‘N

‘0’, — N

~.

N

no.

NN 0.,

N

N

+

N

—+ IIN

>~L)

N

N

~‘

—

.~

+

N

0.,

N

+

+-“ I.)

N

N 154’

+

.

,,~

~-‘~

_II

0

I

0.

N

0’

N

~ ~ “ ~j. “C!.

C~

N N

I

~

+

,N~no.no.

>~

0’

‘N

‘4~ ).) 0’.,

).~

N N

‘4+

+~~‘,).)

•0.s

N

NMO

,,+~ ‘4:~+ )‘) 1......)

N).)

no.+.).)

N~,

+>~

~‘4

I I~

~-SN

N.,~

N~,,

~I

no.

0.., ‘C N ‘.4

N 00

,,~

~

.-,N

‘N

,,~++ 0.’, ,..,.

I

~

‘-IIC

N CI’

.0.,

N

N, 8 00

50., Cc—

‘4 OIl 0.

+

~,

UN

,

—~

E

I

a

+

>,

+

N

+

+

~

N

‘~,.)

no.

‘N +

N

~

N

+N

~

~

++

~

~

-

..L+++

N

‘I.)

N

0’

EN

0) ‘_N N

0 N

0’,

I

‘0.,

0 N

0 CI’

~I

N

0) no.

C,

N

~

‘4’S~’,+I

N ‘.4

N

~.,

N NCI’

IC “

‘.4

‘0’,

N

0)

IC

‘4

.i,2 0011

N N

++0.,

-~

~

+

+0” ~ N>~

,.~ N0.’,

~

‘5>)

0.

N

no.

~

C~

+IC

+

.

C,’>)

no

0.)

I

~‘

‘4’

N

I

.

NN

~‘~I L I I~

E

I~ ~+ 0’

“

~“

0.’,

0)

0.,LIM

‘C-I

N0.

LI

0.

a I

,

N

).)NN

—

+“~ 3$

I

Na

no.

+~

‘0.”0

N

~~“‘

‘N

-~

+

~

~

N L.~ N N

N N

N 00

0

0 0

0 0

0 0

0 0

N

0”,

I

+ IC

~+

N

no.

‘~, 0.,

.~

‘~

I

0 no. 00

‘0’, 0.,

‘0 —

0 0

0

0

‘0 S’,~

a

0 0

N

+ +

~‘N ~

N

,,+

C+~

C” N ‘0.’,

N

N

~

NM

a

)~ no,

M. Lakshinanan and R. Sahadevan, Coupled nonlinear oscillators

N 0)

N

a

— 0)

0.

~

~

0)

II

N

Cc~’~

-~

0.’,

0.’, ~ ~

N ~, N N

‘a

I N

+

N

C...,

-

I

~

~

NN

+N

S0.’,’0.S ‘:‘~, ‘IN ..

N

a

~

~,

+

‘0)~~ ~

‘N

‘N

+

‘.-

+

—

N 0’,

.i~

‘0

IM

I

‘N —

a+ + ‘O., N’i~

‘~‘, ,,+“IN

$:~,,+

L..~

0

N

N

N

N

I

N

~.

N

~no,I~’

~‘

‘0.,

0.’,

.,I~

$,~+

$~ +

I

+

~

~++ ‘a

N

~

0’

.0)

‘~‘

ON

I

‘~

OCI’

0

.~0.’,

+N

SC

CI’

~

‘0

N 0’, N

+1+

0.’,

‘C IC

‘~

N

E

N

HI

~~I+

a

0.,

2 a

NN’0 ~

—

+

0)

~~NI N

+

~+

,,‘~++

N~

+

“M~

I

N

‘N~N

I

~

~

I

Cc’

‘~

~o’,~

.,‘~ ~

N

~‘~‘ ~ IC0~IN

‘0) —

~N ~‘~N

‘0.,,,

LI +

N

0.,

~

0

0.’,

++I’~’I++

~H+

N

‘

+

N

I~

C0.’,

“~ ,,“~‘

+“~H :~—+ 0’~N 0’,

.i~

.—..N

N

-~‘

~

‘0.,

N’

‘~ 0.~ N 0.’,

~

~0.’,~

N

Cc~’

l1M~i,

N

+—

~ —C

+

no. N

N

~

73

—

N

I

— N

0’

IC

‘.4

N+I CI’

a

a

0

~

0

o~ ~

N

N

+

,~

‘~

~‘

0

0

I

~,

x

.~

~

~

~

a V V

~

“~‘~

~

+ -SNNN

I ~j.

Ii

H

I no. ~ II I a

N

(ii’;’.,

.~

~

+

II

~.

(Jc,~. II

II

LI II I a

CI’

NN

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

74

The characteristic equation is dt 0

—

dx 2y(y~ x9) + (2/cc)(B

29(y~* x5’) —

4(B

—

—

A)i

—

2x(x9

—

y~)

d~ A) [xy2 + 2x3 + (A/cc)x]

—

—

dy

—

—

—

— —

_______________________ 2~(x9 y~)+ 4(A B)x2y~ —

—

(8.52) (a) First

invariant. We choose

P,

= 0,

R,

=

Q, OU/Oy

—

=

=

—

OU/Ox

=

(2Ax + 4ccx3 + 4ccxy2),

(2By + 4ccy3 + 4ccx2y),

S,

=

OU/Oi

=

T,

~,

=

OL/05’ =

5’, (8.53)

so

that ~P, +

Q,

~,

OU,/Ot = OU,/Oy

+

~)

—

OU,/Ox

0,

=

R, + (~,

772

=

(~ 9~)T~=

S, +

(8.54)

0,

—

(2Ax + 4ccx3 + 4ccxy2),

(2By + 4ccy3 + 4ccx2y),

0U,/0~=

(8 55)

5’.

OU~/O5’=

i,

Integrating (8.55) we obtain the invariant U,

+92)+(Ax2 +By2 +ccx4+ccy4+2ccx2y2),

~

(8.56)

which is nothing but the Hamiltonian (8.50). (b) Second invariant. Now we choose =

0,

Q2 =

—~,=

25,(xj)

—

y~)+ (4A/cc)(B

R

2y, 2

T2 so

x5’) + 4(B

= ~

= 2~(y~

—

= 172 =

2x(x9

y~),

—

—

—

A)x + 8(B

A)x3 + 4(B

—

—

A)xy2,

S

A)x

2

=

=

2y(y~ x5’) + (2/cc)(B —

—

A)~, (8.57)

that ~P2 +

17,Q2

0U2/Ot

= 0,

+ 772R2 +

(th

—

-~~)S2+ (~2 —

5’~)T2=

0,

(8.58)

3 + 4(B

0U2/Ox = 25’(x5’ ~U2/OY

=

—

y~)+ (4A/cc)(B

2~(y~ x5’) + 4(B —

—

—

A)x + 8(B

A)x2y,

—

—

A)xy2,

0U 2/0~= 2y(y~ x5’) + (2/cc)(B —

0112/09

=

2x(x9

—

y~).

(8 59)

A)x —

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

75

Integrating (8.59) we get, U2

2+ (2/cc)(B (x9

=

A)[(~2/2) + Ax2 + ccx4 + ccx2y2]

—

(8.60)

,

y~)

—

which is nothing but the required second integral of motion, see table 8.1. It appears that finding the remaining two invariants (113 and U 4) is not a straightforward problem and we are working further on it. The above procedure can be repeated for the remaining three cases of the coupled quartic oscillator as well as the other examples of two degrees of freedom in table 8.1. However in order not to repeat the details, we have succinctly presented below the results for the coupled quartic oscillator case only. The details are exactly the same for the Henon—Heiles system and the system of two coupled sextic oscillators. 2 + Ay2 + ccx4 + ccy4 + 6ccx2y2), Case 2.

H=

~=0,

~(~2

+

~

3+l2ccx2y), 71=k5,,

772 =

(8.61)

+ (Ax

5,2)

772=k~,

tj1=—k(2Ay+4ccy

—k(2Ax + 4ccx3 + l2ccxy2).

Case 2(i).

P,

Q,

= 0,

S,=x,

(8.62)

(2Ax + 4ccx3 + l2ccxy2),

=

R,

=

(2Ay + 4ccy3 + l2ccx2y),

T,=9,

U

(8.63a)

2 + Ay2 + ccx4 + ccy4 + 6ccx2y2).

1

=

H=

+

~(~2

+ (Ax

92)

Case 2(u). P

3 + l2ccx2y), 2

= 0,

Q2

=

R

(2Ay + 4ccy

3 + l2ccxy2), 2

=

(2Ax + 4ccx

S 2=9,

T2=~*, 2 + y2).

U2

=

= ‘2

Case 3. H

+

= ~(~2

~=0,

92)

+ (4Bx2 + By2 + 16flx4 + fly4 + 12flx2y2),

77,=y5’,

—~9

4Bxy

—

Case 3(i). P,

—

l6flx3y

Q,

= 0,

=

(8.64)

2—2By2—4fly4—24flx2y2,

77 2=y~—2x5’,

=

(8.63b)

5c5’ + 2Axy + 4xy(x

—

i~1=5’

(8.65)

l6flxy3.

(8Bx + 64flx3 + 24flxy2),

R,

=

(2By + 4fly3 + 24flx2y),

S 1=.x,

T,=’, 2

U,

=

H

= ~(i

+

92)

+ (4Bx2 + By2 + 16flx4 + fly4 + 12flx2y2).

(8.66a)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

76

Case 3(u). P2 R 2

Q2

= 0,

~5,+ 4Bxy +

=

2 + 24flx2y2 + 4fly4, + 2By

= _92

3y + l6flxy3,

S

l6flx

2

=

y5’,

T2

=

—2x5’ + y~’,

(8.66b)

2 + 8flx3y2 + 4flxy4.

U2

(y± x,fl5’ + 2Bxy

= ‘2 =

Case 4.

H

—

+

= ~2

P7~ =

= 0,

= 16fly3~9

—

92)

8f3(yi

+ (4Bx2 + By2 + 8fix4 + fly4 + 6flx2y2), 2x9)y3,

—

48flxy25’2

—

172 = 45,3 + 8(B +

32Bflxy4

—92(8By + 16fly3 + 48fix2y) + 48fl2xy2i9

=

—

Case 4(i).

48Bfly5

—

64Bflx2y3

P 1

Q,

= 0,

5, =x,

T,

32fl2y7

—

= 0,

U,

Q2 =

R

l6flxy3i, (8.68)

16fly3~2 16B2y3

—

—

96fl2x2y5 + 320fl2x4y3.

—

3 + l2flxy2),

=5’,

—

32fl2xy6,

R,

(2By + 4fly3 + l2flx2y), (8.69a) +92)+4Bx2 +By2 + 8flx4+fly4+6flx2y2. =

~(~2

292 + 32Bflxy4 + 64fl2x3y4 + 32fl2xy6 2

—

16fly3~5’,

48flxy

3 + 48flx2y) = 5’2

—

fly2 + 6flx2)y29

(8Bx + 32flx

=

Case 4(u). P

2

64fl2x3y4

—

(8.67)

—

48fl2xy2~9+ 16fly3~2+ 16B2y3

(8By + 16fly

+ 48Bfly5 + 64Bflx2y3 + 32fl2y7 + 96fl2x2y5

—

320fl2x4y3, (8.69b)

52 =

8fl$~y4 16flxy35’,

T

—

25’ + 8fly45, + 48fl2x2y25’ 2

U

2 + fly2)y25,2 2

= ‘2

= 94 +

—

l6flxy3i,

8By

= 45,3 +

—

16flxy3~j+ 4fly4i2

4(B + 6flx

+ 4B(B + 4flx2 + 2fly2)y4 + 4fl2(2x2 + y2)2y4.

8.4.2. Three degrees offreedom The above procedure can also be directly extended to systems with three degrees of freedom. With the infinitesimal generator (8.39d), the partial differential equation for the local invariant U,(t, x, y, z, i, 5’, ~)becomes E(U,)

=

OU, ~-~—

OU, + 7i-~-—+

+(~3

OU,

—~)--~---=0, ,

+

~

OU,

= 1,...

+(77,

,6

.

—

. OU, x~)-~-+(772

—

. OU, y~)-~-

(8.70)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

77

The corresponding characteristic equation is dtdxdydz —

JI,

d~

7~ ;i~

—

(th

—

—

~)

d5’

—

(~ ~)

—

—

—

di

—

—

(871)

Solving the above characteristic equation we can find six independent invariants U,(t, x, y, z, ~, 9, ~),i = 1, 6, of the one-parameter group E. For this purpose we have to find six functions P,, Q,, R,, S,, T,, V1 and W~such that ...

,

,7,Q, + 772R, + 773Sf + (?~‘

~P, +

—

~JT, + (~2

—

5’~)V, + (,~ ~) —

K”,

= 0,

i=1,...,6,

(8.72)

with the property P,

=

Q

~,

=

R,

~

Ot

Ox

=

S,

~

=

Oy

T, =

~

Oz

V,

~

Ox

=

Oy (8.73)

i=1,,..,6, Oz so that P,dt + Q,dx + R,dy + S,dz + T,d~+ V,dj) + W,d±is an exact differential dl],. We have already noted in section 8.3 (tables 8.2 and 8.3) that for each integrable case there exist two sets of symmetries. For each set of symmetries, as in the case of two degrees of freedom we can find two sets of nontrivial invariants (U,, U2) following the same simple procedure [see eqs. (8.48), (8.49)]. The third invariant can be constructed by noting the following relations among the two sets of invariants given in tables 8.2 and 8.3 for each of the cases. Denoting the two sets of symmetries as (~= 0, ~, ‘72, ‘7~)and (~= 0, th~~ ‘73), we can verify that —

~

—

~

—

~

+~

+~

+~

= 0,

(8.74)

and vice versa. The details are given in the following example corresponding to case 5 in table 8.2 of the system of three coupled quartic oscillators. The details for all other cases as well as the system (8.35) are exactly the same and so we do not repeat the details. 8.4.2.1. An example: three coupled quartic oscillators

Case v (table 8.2). 2 + y2 + z2) + fl[16x4 + (y2 + H = ~(±2 + 5,2 + ±2) + B(4x (i) First set of symmetries. Generalized Lie symmetries: ~=0,

77,=O,

=9(y2—z9).

+ 12x2(y2 +

z2)].

(8.75)

‘7 2=z(z5’—y±),

173

z2)2

‘73=y(y±—z5’);

17,=O,

‘72=±(z5’—y±), (8.76)

M, Lakshmanan and R. Sahadevan. Coupled nonlinear oscillators

78

The characteristic equation is (dt)/0

(dx)/O

=

=

(dy)/z

(dz)/( y)

=

= (d±)/0 = (d5’)/± = (d±)/(

—

—5’).

(8.77)

(a) First invariant. We choose P,

= 0,

R,

=

Q,

—cc,

=

—cc

=

+ 24flx2y + 4flyz2,

3

2

=

2By +

—cc3

=

(2Bz + 4flz

8Bx + 64flx3 + 24flxy2 + 24flxz2, (8.78)

4fly

S

3 + 24flx2z + 4fly2z), 1

=

T 1=~, V,=y, so

W,=2,

that ~P, +

~

+ 772R, + 773Sf + (,j~ i~)T, + —

5,~)V,+ (~j, ±c~)W, = 0, —

0,

OU,/Oy

(2By + 4fly3 + 24flx2y + 4flyz2),

=

(8Bx + 64flx

0U

3 + 24flx2z + 4fly2z), 1/Oz

OU,/0.~=

OU,/09

,~,

(8.79)

3 + 24flxy2 + 24flxz2),

OU,/Ot = =

OU,/Ox

(~2 —

=

5,,

=

(2Bz + 4flz

OU,/0±= ±.

(8.80)

Integrating eq. (8.80), we obtain the invariant U,

= ~(i2

+ 5,2

+

±2) +

B(4x2 + y2 + z2) + fl[16x4 + (y2 + z2)2 + 12x2(y2 + z2)], (8.81)

which is nothing but the Hamiltonian (8.75). (b) Second invariant. Now we choose “2°~

R

Q2=—~,=0,

2=—~2=—i, T2=77,=0, so

V2=772=z,

S2=—i~3=5,,

(8.82)

W2=’73=—y,

that ‘~1~2+

0U2

‘11Q2 +

772R2 + 173S2 + (ii,

0U2

0U2

—

.

~T2 0U2

+

(~2

.

—

5’~)V2+ (;~~ ±~) W2

0U2

—

0U2

= 0,

(8.83)

0U2

—~—=O, —~-—=O, —~—=—z, -~-—=y, —~—=0~-~--=z, —~-=—y. (8.84)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

79

Integrating eq. (8.84) we get (after a sign change) U2

=

(y±

—

(8.85)

z9),

which is nothing but the second integral of motion. (c) Third invariant. Now we choose P3

2 + z2) + 24flx2(y2 + z2) + 4fl(y2 + z2)2,

Q~=

= 0,

R

~,

—

=

~9 + 4Bxy +

3

= ~72

=

3

=

= ~± +

±~ + 2B(y

—

3y + l6flxy3 + l6flxyz2, l6flx

(8.86)

3z + l6flxy2z + l6flxz3,

S —~

4Bxz + l6flx

T 3=i~,=y5’+z±, V3=~2=yi—2x5’,

W3=i~3=z~—2x±

(where the bar stands for the next set of symmetries, see table 8.2), so that i~P3+ P~~Q3 + 172R3 + ,7~S3+

(~,

—

‘~T3+

(i~~ —

9~)V3

+

(~

—

±c~)W3 = 0,

(8.87)

2 + z2) + 24flx2(y2 + z2) + 4fl(y2 + z2)2 0U3/Ot

=

0U

0,

0U3/Ox =

_5’2

±2 +

—

2B(y

2 + 4fly2 + 4flz2),

3/Oy =

~*5’ + 4xy(B + 4flx

0U

(8.88)

2 + 4fly2 + 4flz2),

3/Oz

= ~±

+

4xz(B + 4flx

OU 3/O~= y5’ +

0U3/09

z±,

=

y~ 2x5’,

0U3/0±=

—

z~ —

2x±.

Integrating eq. (8.88) we get 2 + z2) + 8flx3(y2,+ 113

=

5’(yi.~ x5’)+

±(z~—

—

z2)+

4flx(y2 + z2)2,

(8.89)

x±)+ 2Bx(y

which is the required third integral of motion. (ii) Second set of symmetries. Generalized Lie symmetries: ~,=y5’+z±,

~

2=yi—2x5’, ,73=z~—2x±, 2 + z2) + 4fl(y2 + z2)2 + 24flx2(y2 + z2)] , +

5,2

=

—~5’—4xy(B+4flx2+4f1y2+4flz2),

±2 —

(8.90)

[2B(y

=

i~= —~±—4xz(B+4flx2 +4/1y2 +4f3z2).

As mentioned previously, if we choose P,=0,

Q,=—cc 1,

R1=—cc2,

S,=—cc3,

T,=i,

V,=5’,

W,=z,

(8.91)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

80

then we get the Hamiltonian. If we choose ~2’°~

Q2=_4~, R2=—,12, s2=—,13,

T2=,~,, V2=i~2,

w2=~3, (8.92)

then we get the second integral of motion. For the third choice we can choose P3=0,

Q3=—~,=0,

S3=—~3=5,,

R3=—ij2=—±,

T3=77,=0,

V3=172=z,

W3=’73=—y

(8.93)

[where the unbarred symmetries are the previous set (8.76)], then we get the third integral of motion. Thus we can obtain three independent invariants for each set of symmetries. 8.4.3. Separability Next we will show that from the generalized symmetries, we can also find suitable coordinate systems in which the equations of motion or the Hamilton—Jacobi equation become separable. Since the separability is associated with the coordinate transformations, we will consider the corresponding part of the characteristic equations discussed earlier. 8.4.3.1. Two coupled quartic anharmonic oscillators

1. From eq. (8.52), we have 2~]= (dy)/2x(x9 (dx)/[2y(~y ,))x) + 2c This can be rewritten as Case

—

yk),

c2

=

(B

—

A)/cc.

(8.94)

—

—

i] + (x2

—

y2

—

c2)(dy/dx)

(8.95)

= 0,

which can be readily integrated to give (m + 1)(mx2

—

y2)

—

mc2

= 0,

(8.96)

where m is an arbitrary constant. Rewriting (8.96) in the elliptical form x2/~2+ y2/(~2 c2) —

= 1

,

,~2

=

c2/(m + 1)

,

(8.97)

p~2)]l/2 .

(8.98)

we have the obvious parametrization x = ~‘7/c,

y = (1/c)[(~2

—

c2)(c2

—

As we have noted in section 5.1.5, the Hamilton—Jacobi equation (5.33) is separable in terms of the coordinates (8.98), even though the equation of motion is not. However for the particular case A = B, the ellipse degenerates into a circle, in which case the equation of motion itself is separable.

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

81

Case 2. From eq. (8.62), we have (dx)/kj)

=

(dy)/ki

d(x2

or

—

y2)

=

0,

(8.99a,b)

so that x2

—

y2

constant,

(8.100)

leading to the linear transformation U=x+y,

V=x—y,

(8.101)

noted in section 5.1.5. In this case the equations of motion itself are separable. Case 3. From the characteristic equation (8.65), we have (dx)/y5,

=

(dy)/(y~ 2x5,),

(8.102)

—

which can be written as y(dy2

dx2) + 2xdxdy

—

or

= 0

y(dy/dx)2 + 2x(dy/dx)

—

y = 0.

(8.103a,b)

This can be integrated to give y2

—

2cx

—

c2

c = constant.

= 0,

(8.104)

Rewriting (8.104), we have y2/774

—

2x/’72

= 1

772 =

c.

(8.105)

Naturally (8.105) can be parametrised in terms of the parabolic coordinates x=~(i~2—772),

y=~,

(8.106)

with which the Hamilton—Jacobi equation (5.34) becomes separable. Case 4. In this case, from the characteristic equation (8.68) we have (dx)/8fl(pi

—

2x5,)y3

= (dy)/[493

+ 8(B

+ fly2 + 6f12x2)y25,

—

16flxy3~],

(8.107)

or dy dx

= 45,3

+ 8(B +

+ 6fl2x2)y25’ 8fl(y5c 2x9)y3 fly2

—

16flxy3~

(8 108)

—

Due to the 5,3 term in the right-hand side of the above, we are unable to find any separable coordinates, as noted in section 5.1.5.

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

82

8.4.3.2. The Henon—Heiles system Case 1. From the expression for the symmetries in table 8.1, we have (dx)/k9

=

(dy)/k~,

(8.109)

which is the same as (8.99a). So we obtain the known linear transformation (section 5.2.2). Case 2. We now have (dx)/(4ccx9

—

8ccyi + 8Ai

2B~)= (dy)/4ccx~,

—

(8.110)

so that x(dy/dx)2 + (k

—

2y)dy/dx

—

x

=

0,

k=

(4A

—

B)/2cc,

(8.111)

On integrating (8.111) we have —

k = cx2

—

1/c,

(8.112)

where c is an arbitrary constant, which can be parametrised in terms of the shifted parabolic coordinates, x

=

(~‘7)hI2,

y = ‘~(~ —

,~)+ (4A

—

B)/cc

,

(8.113)

with which the Hamilton—Jacobi equation becomes separable [69d]. Case 3. In this case we have from the characteristic equation, (dx)/[4~3 + 4(A + 2ccy)x2~ ~ccx35,] = (dy)/(~ccx3~), —

(8.114)

which we are unable to integrate as such, and no separable coordinates seem to exist in this case. 8.4.3.3. Three degrees offreedom. The method of finding the separable coordinates from the symmetries for the system with three degrees of freedom is a direct extension of the above procedure. First we illustrate the method for the case (v) (see table 8.2) which is separable in parabolic coordinates. Consider the system of three coupled quartic oscillators. Case (v). For the first set of symmetries we have, from the characteristic equation (8.77), (dx)/0

=

(dy)/z

=

(dz)/(—y)

(8.115)

.

Integrating (8.115), we get the circular symmetry in the y—z plane, y2 + z2

=

constant,

(8.116)

which is evident from the form of the Hamiltonian (8.75). For the second set of symmetries, we have from eq. (8.90), (dx)/(y5’ + z±)= (dy)/(y~ 2x5,) = (dz)/(z.~ 2x±), —

—

(8.117)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

83

which on integration yields (y2 + z2)/774

2x/q2

—

= 1

(8.118)

.

This can be parametrized as x

y = ~‘7cosO,

= ~(~2,_ ‘72),

z =

~77sinO,

(8.119)

which are the required separable coordinates (section 6.1.2). Case (i). From the expression for symmetries in table 8.2, we have (dt)/0

=

(dx)/(2cf~ 2yL~~ 2zL~~) = (dy)/(2xL~~ 2zL~~)

=

(dz)/(2c~,± + 2xL~~ + 2yL~~),

—

where c~cc= B

—

—

—

(8.120)

A and c~cc= B— C. Equation (8.120) can be rewritten as

xy(p~ 1) + (x2 —

—

y2

—

c~)p, yzp —

2 + xzp,p2 2

—

(8.121a)

= 0,

z2 + c~ c~)p —

xz(p~ 1) + (x

2 yzp1 + xyp,p2

—

—

= 0,

(8.121b)

where p, = (dy/dx) and P2 = (dz/dx). Solving eqs. (8.121) separately and then comparing the solutions we get 2/cc2 + y2/fl2 + z2/y2 = 1 , (8.122) x cc2 = A2/(m, m 2 = —A2, ~2 = A2, 2), fl A2 = [(m, + m 2)c~ m2cfl/(1 + m, + m2) (8.123) —

—

.

where m, and m2 are arbitrary constants. We can parametrize (8.123) in terms of generalized elliptic coordinates and the Hamilton—Jacobi equation becomes separable as noted in ref. [69f]. Turning now to the remaining cases we have for case (ii), from the first set of symmetries, (dx)/(zL~~) =

(dy)/0

=

(8.124)

(dz)/(xL~2),

which on integration yields 2 + z2 = constant, (8.125) x which is clear from the form of the Hamiltonian (8.34) with the parametric restriction cc = fl = o = = 6cc, cv = 2cc, A = B = C for this case. The second set of symmetries contains momenta with cubic powers, which do not seem to be integrable. Similarly for the cases (iii), (iv) and (vi) of table 8.2, we find that the first set of symmetries leads to the expected circular symmetry either in the x y [cases (iii) and (iv)] or y z [case (vi)] planes, while symmetries of the second set are nonintegrable. For cases (vii) and (viii) no separable coordinates seem to exist. A similar analysis can be performed for the system of three coupled cubic oscillators given in table 8.3. —

—

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

84

Thus we see clearly that the generalized Lie symmetry approach provides a very systematic method of finding the integrals of motion and separable coordinates of the Painlevé cases discussed in the earlier sections.

9. Nonintegrability aspects of the Henon—Heiles and the system of two coupled quartic anharmonic oscillators In the earlier chapters, we discussed the salient features of the P-analysis in detail and illustrated how this analysis plays a vital role in the determination of the integrability of nonlinear dynamical systems in general and in particular to two, three and arbitrary N coupled nonlinear oscillators with polynomial potentials. In this section, we wish to analyse briefly the nature of the singularity structure of the solution in the complex time plane for typical non-integrable cases. As prototypes, we consider two examples, namely the well known Henon—Heiles and system of two coupled quartic anharmonic oscillators. 9.1. Henon—Heiles system [44] To start with, we consider the Henon—Heiles system given by

H = ~(p~ + p~)+ ~(Ax2 + By2) + ccx2y

~fly3

—

(9.1)

,

whose equations of motions are d2x/dt2 + (A + 2ccy)x

=

0,

d2y/dt2 + By + ccx2

—

fly2

=

0.

(9.2)

From the leading-order behaviour and resonances one can isolate the following cases (see Section 5.2): Case 1. x(t) ~ ±(3/cc)(2+ fl/cc)”2t”2 , y(t) ~ —(3/fl)r”2, r=—1,6,

~{5±[1—24(1+fl/cc)]”2}.

Case 2. x(t) ~ a p

=

‘~

[1 + (1

—

,

y(t) 48cc/fl)”2]

0r”

~

,

2 —(6/cc)-r ‘r = t t

r

,

= —1,0,

(1

—

2p), 6,

(9.3)

—

0

.

(9.4)

It is evident from the resonances of case 1 and the leading-orders of case 2 that the singularities of the solution are multiple-valued in general, and hence the general solution x(t) and y(t) of(9.2) can be reexpressed as a double series, usually called the psi series, about the movable singularity t0 involving powers of (t t0 )~,r = rR + r1, rR = Re r, and r1 = Im r. For notational convenience, we will replace t t0 by t itself in the following. Then for case 1, the expansion becomes —

—

2~

x(t)

=

t~

y(t)

=

t~2~

~

k=0 j=0

a,

2~ 3t’t~+ t~

~ b,jr’~t~ +t

k=Oj=O

~ ã,f”t~

(9 5a)

1=~j=0

~ b~Itt3, k=1j=0

(9.5b)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

85

where the a,,3, a~3,b,,~and b,,~are constants and the bar denotes complex conjucation. Here 2};

=

t~, y

=

t~,

= ‘~{1 + [1

~ = ~{1

—

—[1—

24(1 + fl/cc)]”

(9.6)

fl/cc)]”2}.

24(1 +

Making use of (9.5) in (9.2) one can verify in the usual way that the coefficients a,

2, a,2 and a06 are arbitrary, and so they can be chosen to satisfy given initial conditions; further, (9.5) represents the general solution of(9.2) in the vicinity of a case 1 singularity. The various coefficients a,,J, bkJ and ~kj, b,,3 are found to satisfy the following set of recursion relations:

(yk

+1

—

2)(yk

3)akJ + Aa,,

+1

j

k

3_2 + 2e ~

—

~ ak_,,j_mb,m

i’O m0

+

28~ ~0(a,,+n,mbn,j_n_m+

(53k +j

—

+1

2)(53k

—

3)a,,~+

b,,+n,man,j_n_m) =

Aa,,j_2

k

j

+ 2e ~

~

(9.7a)

0,

ak_i,j_mbjm

z0 m0

+

28 ~

(~~+n.mbn,j_n_m+ b~÷n.man,j_n_m)= 0,

~

(9.7b)

n0 m0

(yk +j

—

k

3)b,,~+ Bb,,~_

2)(yk +j

2+ ~

—

j

~ (8a,,_,,j_ma,m

—

b~_i,j_mbim)

i’O m0

+

~(6a,,+n,mãn,j_n_m

~

—

b~+n,m~n,j_n_m)

k

(53k +j

—

2)(53k +j

—

3)b,,~+ BbkJ_2 + ~

=

0,

(9.7c)

j ~

(8~~_i.j_m~im — b~—i,j_mbim)

i0 m0

+

~

~(eä,,+n,man.j_n_m

—

bk+n,mbn,j_n_m)

= 0.

(9.7d)

Here 6 = cc/fl. The above recursion relations show that the various expansion coefficients in (9.5) do not possess closed recursion formulae, in general. However, one finds that a subset of recursion relations corresponding to the coefficients ~ and b3,2~or a323 and b3,23,j = 0, 1, 2, ... in (9.7a—d) is closed [44]. For example, setting =

~

W3

=

~

(9.8a)

M. Lakshrnanan and R. Sahadevan, Coupled nonlinear oscillators

86

one obtains

(yj + 2]— 2)(yj + 2j

—

3)0~+ 26~ ~9j—m~t1m = 0,

(yj + 2j

—

3)W~+ m~o(60j_m6m

2)(yj + 2j

—

—

(9.8b)

Wj_m~m)=

0.

(9.8c)

From these closed recursion relations (representing a well defined approximation to the full solution, as noted below) one can study the asymptotic properties (It I ~ 1) of the series solution in the vicinity of the singularity locally. For this purpose, one may associate a set of appropriate nonlinear differential equations having the same asymptotic property similar to eqs. (9.8b, c). We define 0(X)

=

~

W(X)

=

(9.9) X

=

t~2,

y

= ~{1

+ [1—

24(1 +

fl/cc)]”2}

Now multiplying the left-hand sides of (9.8b,c) by X3 and summing, and using (9.9), we obtain (y + 2)2X(X6’)’

—

S(y + 2)X0’ + 60 + 280~t’=

(y + 2)2X(XW’)’

—

S(y + 2)X~P’+ 6~P+ C02

—

(9.lOa)

0,

=

0.

(9.lOb)

where the prime denotes differentiation with respect to X. Equations (9.10) can be easily seen to mimic the local behaviour of the equations of motion (9.2) in the neighbourhood of t 0. To see this, we make the substitution 2

x(t)

) 0(X)

,

y(t)

= (1/t2) !P(X)

,

(9.11)

= (1/t

where the variable X is defined by eq. (9.9). Then it is easy to check that asymptotically as t 0, eqs. (9.10) reduce to eqs. (9.8b, c). We may now ask what is the singularity structure of the set of eqs. (9.10). Proceeding as above for eqs. (9.2), one can check that eqs. (9.10) have exactly the same singularity structure as (9.2) so that the functions 0(X) and ~t’(X)in (9.10) display exactly the same sort of singularities in the X-plane as do x(t) and y(t) in (9.2) in the t-plane. Therefore, close to a given singularity, the singularity structure of x(t) and y(t) may be determined by studying the singularities of 0(x) and W(X) through appropriate mapping from the X-plane to the t-plane. Let us assume that the singularity in the X-plane is situated at —~

X =X 0 = X0exp(2nin),

n

= 0,1,2,

...

(9.12)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

87

where exp(2irin) is the unit phase factor. Then the corresponding position of the singularity in the t-plane is given by =

[X0exp(2irin)]’~2’~

= X~

2’~exp{5irn+ iirn[24(1 + fl/cc)

—

1]h/2}

(9.13)

y=4{1 +i[24(1 +fl/cc)—1]”2}

,

n=0,1,2,...

Thus each pole in the X-plane yields an equiangular spiral of poles in the t-plane, one pole for each of n = 0, 1,2, ... Hence the poles have an angular displacement about the central pole which is given by

A4

=

*ir(2 + fl/cc),

(9.14a)

and the radial decrement is equal to

AItI

exp[—itn(23 + 24fl/cc)”2/6(2 + fl/cc)]

=

(9.14b)

.

In an exactly similar manner, one can show that case 2 singularities have an angular displacement =

~ir(1 + 2fl/cc),

(9.15a)

and the radial decrement becomes

~ItI =

exp[—irn(48fl/cc

—

1)1/2/12(1 + 2fl/cc)]

,

n

=

0,1,

...

(9.15b)

In order to substantiate the above findings, one can numerically integrate the equations of motion in the complex time plane. For this purpose, it is well accepted that the ATSMCC and also ATOMCC numerical programs developed by Chang et al. [44,90], which is a refined version of the Taylor series method of finding the singularity position nearest the point of integration, is a standard tool. Using the former program Chang et al. [44,90] have observed the following. When the solution x(t) and y(t) of (9.2) is expanded at various points along the real time axis there emerges a nonuniform row of seemingly isolated singularities. Note that one has to specify the initial data in such a way that the resulting motion is bounded for real time and the singularities are at a finite distance from the real time axis. On the other hand, when the path of integration is deformed into the complex time plane and passes between two of the singularities observed from the real time axis, there emerges a third singularity located at the apex of an (approximately) isosceles triangle whose base is the line joining the two singularities that are on either side of the path of analytic continuation. Furthermore, when one integrates between any pair of singularities that are observed to be neighbouring during the process of analytic continuation, the above construction is repeated. Several levels of structure are implied by this “self-similar” process. One is that the set of singularities consists of a perfect closed set with no isolated points on the multisheeted Riemann surface. Another is that about any singularity there emanates a double spiral, one clockwise, and the other anticlockwise. Since the base between the neighbouring singularities is contracting

M, Lakshrnanan and R. Sahadevan, Coupled nonlinear oscillators

88

geometrically at the successive stages of the analytic continuation process, it is impossible to continue the solution beyond more than a given finite distance in any direction beyond a pair of singularities. In other words assuming one does not retrace the original path, any path of analytic continuation between a pair of singularities would appear to be trapped in a geometrically converging web of singularities that creates a natural boundary of the solution and thereby produces a fractal. Using the self-similar nature of the above construction Chang et al. [90] have calculated the fractal dimension of the singularity set to be 1.1419, see also ref. [91]. However, one must be aware that there are cases where nice enough natural boundaries can lead to “integrability” as in the case of the third-order Chazy equation [92,93]. 9.2. Two coupled quartic anharmonic oscillators Here for the Hamiltonian H = ‘~(p~ + p~)+ Ax2 + By2 + ccx4 + fly4 + 0x2y2,

(9.16)

we have the equations of motion d2x/dt2 + 2(A + 2ccx2 + 0y2)x

d2y/dt2 + 2(B + 2fly2 + 0x2)y

0,

=

= 0.

(9.17)

From the leading-order and resonance analysis (see section 5.2) we get Case 1.

x(t)

=

a

2]

0t”’, a~=

(0—

y(t)

=

b,~’r”’,

r

= —1,4,ff3

± (9— 4Y0)~/

2fl)A 2,

b~=

2cc)A2,

(0—

A2

(4ccfl

=

—

02)_i,

(9.18) Y0=4[1+2(cca~+flb~)], ~=t—t0. Case 2.

x(t)

=

—

1t2] q

= ~[1

± (1 +

y(t)

(1/2cc)-r’, ,

40/cc)

=

b0~,

r

=

—

1,0,4, (1

—

2q)

b 0

=

arbitrary

(9.19)

.

Again we expand x(t) and y(t) of (9.17) in a double series in the neighbourhood of a movable singularity taking into consideration the multiple-valued nature of the singularities in both cases 1 and 2. For case 1, the psi series becomes ,

(9.20a)

1f”t~

(9.20b)

3

x(t)

= t~’ k=0 ~ j’=0 ~ a,,~r”t~ +

y(t)

=

t~1 ~

t~’k=1 ~

~ b,,jr’t~+ t’’ ~

j=0 ~ ãkjtt

~ b,, ,

k=0 j=0

= t~,

y

= ff1

+ i(4Y0

k=~j=0

—

9)1/2];

f

=

t~,

~=

ff1

—

i(4Y0

—

9)~/2] .

(9.20c,d)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

Here the coefficients a,1, (yk

+1

1)(yk +j

—

j

k

+ 2 l”O ~ k

m0

j

a,, and a04 are arbitrary, besides t0. —

I

Then the recursion relations become

2)a,,3 + AakJ_2 m

p’O q’O ~ I

89

2ccak_I,j_mal_p,m_qapq + Oa,,_,,j_mbi_p,m_qbpq) (

m

+2~

~ (2cca,,_j,j_ma,+p,m_qãpq + Oak_,,j_mbl÷p,m_q!Jpq)

1=0 m0 p.~ q0 k

j

I

+2~

~ (2cca,,+,,j_mã,_p,m_qãpq + 0a,,_i,j_mb~_p,m_qb’pq)= 0,

11 m0 p’1 q0

(~Yk+j

—

k

+ 2 11 ~ k

1)(~ik+j j

—

2

1

2ccã,,_i,j_mãi_p,m_qãpq + O~k_I,j_m_p,m_qbpq)

1

(

m

+2~

~ (2ccã,,_,,j_mã,+p,m_qapq + 0a,,_i,j_mb~÷p,m_qbpq)

1’O m0 p’l k

2)ã,,~+ AãkJ_

m0 pO q0 ~

j

(9.21k)

j

q0

I

m

+2

~ (2ccã,,+i,j_mai_p,m_qapq + Oak+I,j_mbl_p,m_q!~pq)=

0,

(9.21b)

11 m0 p=l q’O

(yk +j

—

k

+ 2 10 ~ k

+

1)(yk +j j

—

3 + BbkJ_2

I

2flb~_i,j_mbi_p.m_qbpq+ Ob,,_,.j_mai_p,m_qapq)

m0 pO q0 ~

j

(

1

2 ~

~ (2flb~_i,j_mb~+p,m_qEpq + Obk+I,j_mal÷p,m_qãpq)

10 m”O p1

k

2)b,,

j

q0

1

+2~

~

(2flb~+i.j_mä 1_p,m_qäpq

+ Ob,,_,,j_ma:_p,m_qapq)

= 0,

(9.21c)

11 m0 p~ q0

(‘~k+1

—

k

1)(~k+1 j

I

+ 2~

—

2)b~1,~ + Bb,,~_2 m

~ (2flbc_i,j_mb’i_p,m_qB’pq + 0Eic_l,j_mãI_p,m_q~pq)

10 m0 pO q0 k

j

I

+2~

~ (2fl&ic_i,j_m&+p,m_qbpq + 0~+,,j_mã,+p,m_qapq)

10 m0 p1 k

+2~

j

1

q0 m

~ (2fl~+I,j_mbl+p.m_qbpq+ Ob~+i,j_mäi+p,m_qãpq)=

11 m”O p.1 q’O

0.

(9.21d)

M. Lakshrnanan and R. Sahadevan, Coupled nonlinear oscillators

90

One can check that the expansions defined by (9.21a—d) are consistent and well defined. As in the case of the Henon—Heiles system, the analysis of the singularity structure of the solution for the non-Painlevé case can be carried out in two ways. The first one essentially makes use of the closed subset of recursion relations (here the coefficients aJ2J, b32~or ~ b32~)which actually represents a well defined approximation to the full solution corresponding to the limit It I ~ 1 and t finite. Or else, we introduce the transformation x(t)

(1/t)0(X)

=

y(t)

,

(1/t)~P(X)

=

X

,

=

t”’~’

(9.22)

in the vicinity of a given singularity into (9.17), and obtain 3 + 200~P2= 0, X(X0’)’

—

(9.23a)

3X0’ + 20 + 4cc0

X(X~/i’)’ 3Xifr’ + 2~’+ —

4fl~I’3+

200211,

=

0.

(9.23b)

In (9.23) contributions from linear terms (which vanish in the limit ItI —÷0)have been neglected. Now considering (9.23), the following points emerge. (i) The generating functions 0(X) and ~P(X)display exactly the same sort of singularities in the X-plane as do x(t) and y(t) in the t-plane. (ii) Suppose we consider some singularity at X = X 0 near the origin, then there will also be a singularity at X = —X0. Thus, overall, we can say that there is a singularity at X = X0 exp(iiw),

n

= 0, 1,2,

(9.24)

...

(iii) Then the location of the singularity in the t-plane is given by =

[Xoexp(i1tn)]”~’’~

y

,

= ff1

+ i(4Y0

—

9)1/2]

,

(9.25)

where Y0 is given by (9.18). After a simple manipulation, using (9.25), we find that the poles have an angular displacement about the central pole which is given by

A4

=

~ir[1 + 2(cca~+ flb~)],

(9.26a)

and the associated radial decrement becomes

L~ItI= exp[—nit(4Y0

2/Y —

9)”

0]

(9.26b)

.

(iv) In a similar way, we find that for the case 2 singularities, eq. (9.19) have angular displacement equal to =

4ir(2

—

0/cc),

(9.27a)

and the radial decrement is

AItI

2/2(2 =

exp{—m[—(1 + 40/cc)]”

—

0/cc)}

.

(9.27b)

M. Lakshmanan and R. Sahadevan, Coupled nonlinear oscillators

91

To augment these asymptotic results, one can carry out actual numerical integration with the help of the ATOMCC numerical program of Chang et al. [44] and confirm that the picture is correct indeed as in the Henon—Heiles system. The above study has indicated that there exists a direct connection between the occurrence of a certain type of multivaluedness of the solutions and the existence of self-similar natural boundaries. One hopes this kind of considerations will eventually lead to an understanding of the onset of chaos in a rigorous way.

10. Results and discussion In this review, we have tried to demonstrate in a systematic manner how the singularity structure analysis (of ARS for strong P- and Ramani et al. for weak P-) provides an invaluable analytical technique in identifying and classifying the integrable cases of nonlinear dynamical systems with particular reference to two, three and arbitrary N coupled nonlinear oscillators with polynomial interactions governed by generic Hamiltonians. The major handicap of this approach is that it determines only a necessary condition for integrability of a system of nonlinear odes, since it fails to locate the presence of movable essential singularities. Keeping this in mind, we established the complete integrability of each of the P cases of the above coupled oscillators by explicitly constructing the time-independent, sufficient number of involutive integrals of motion (barring an exception of 3 and N coupled sextic oscillators for which one integral of motion remains to be explicitly found). From our investigations, we find that the combination of the twin approaches, namely the P-analysis and the direct search for integrals of motion, plays a vital role in predicting as well as in establishing the complete integrability of nonlinear dynamical systems. The question of complete integrability of coupled nonlinear oscillators has also been examined through the Lie invariance analysis involving velocity-dependent vector fields and the obtained results agree with the Panalysis investigations. It appears that the Lie symmetry analysis provides another but more direct and rewarding analytical method to study the integrability properties of the nonlinear dynamical systems, particularly Lagrangian systems.

An attempt has also been made to illustrate the usefulness of the singular-point structural analysis in obtaining deeper insight in nonintegrability, such as the concepts of natural boundary, fractals, etc. through the examples of Henon—Heiles and two coupled quartic nonlinear oscillators.

Acknowledgements The work reported here forms part of a research project funded by the Department of Science and Technology, Government of India. It has also been supported through a grant from the Third World Academy of Sciences, Trieste.

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