“Rigid body in torque-free rotation” pattern in a model of quartic coupled oscillators and its bifurcations

Volume 146, number 7,8

PHYSICS LETTERS A

11 June 1990

“RIGID BODY IN TORQUE-FREE ROTATION” PATfERN IN A MODEL OF QUARTIC COUPLED OSCILLATORS AND ITS BIFURCATIONS Sebastian FERRER, Maria J. MARCO, Carlos OSACAR and Jesds F. PALACIAN Departamento de Fisica Tedrica, Universidad de Zaragoza, 50009 Zaragoza, Spain Received 23 February 1990; accepted for publication 2 April 1990 Communicated by V.M. Agranovich

Related to aquestion put by Vorob’ev and Zaslavsky on a quartic coupled oscillator, we show that the normalized flow is similar to the “rigid body in torque-free rotation”. Three bifurcations occur corresponding to parameter values of integrable cases. This analysis could help in understanding quantum chaos and the level distribution of this model.

1. Introduction The present paper is devoted to the study of the regular regime in the classical case for the Hamiltonian with 1: 1 resonance 2) + cW(x, 3)), (1) .~= ~(X~+ Y where ~W(x,y)= ~n2(x2+y2) + ‘/3x2y2

+

~a(x4+y4) (2)

with parameters a, fi with dimensions [L 2T ~]. This model of two coupled isotropic oscillators has been considered by several authors (see ref. [11) and recently it was analyzed by Vorob’ev and Zaslavsky [21. The particular case a = 0 is well known in the literature. It corresponds to the non-integrable Hamiltonian system which Pullen and Edmonds [3] introduced as a model of a system with chaotic motion. It is also derived by Matinyan et al. and Savvidi [4,5] in the Yang—Mills quantum field model. In galactic dynamics it is called the Ollongren case [6]. In our case, we have a scaling parameter a and the coupling parameter/i. When there is no coupling parameter, fl=0, we have an integrable system; the other two integrable cases correspond to fl=a and fl= 3a. The paper is organized as follows: First, we borrow the integrable cases ofthe system defined by (2) from the study of Bountis et al. [7] using the Pain—

levé property. Then, following Vorob’ev and Zaslavsky we focus our research on the domain a>0 and fJ~ a. In order to study the system defined by the Hamiltonian function (2) in the regular regime, instead of relying on the numerical Poincaré surface of section technique, here according to the approach proposedwe by proceed Deprit and Miller [8] forper—

turbed harmonic oscillators and Keplerian systems, based upon the qualitative understanding of the normalized global flow. Assuming the terms in a and/i as perturbations ofthe isotropic oscillator, and using the Lissajous variables [9], we make a normalization by a Lie transform [10]. After the reduced phase space for perturbed isotropic oscillators (“elliptic oscillators” in Deprit’s terminology) is set up as a two-dimensional sphere, a key issue not fully exploited yet for the qualitative understanding of the global flow, the first order reveals sufficient to show the features of the system: its “rigid body in free rotation” structurefor all the values of the parameters. We give the analysis of the equilibria as well as the bifurcations which are related to the integrable cases. The closed form solution in elliptic functions for the averaged system when 3a
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by the dynamical properties of the corresponding classical system”, and we think this communication could help in understanding quantum chaos and the level distribution of this model. In this sense it is worthwhile to point out that Vorob’ev and Zaslavsky only make the distinction of two regions related to /3~3a and /1> 3a. No attention is paid there to the global change connected to the bifurcation when /3= a. They are more interested in what they call “regular motion”; and this bifurcation lies in the region where, irrespective of a global shift on the phase space, the stability of the circular orbits does not change through this bifurcation, nonetheless separatrices are present in the averaged problem. From figs. 2 and 3 we conclude that the division of Vorob’ev and Zaslavsky has to be refined in the sense that for 0< /1< 3a, separatrices are also present there. It has been a bit striking, and gratifying at the same time, to discover that the model considered in this paper has the same phase space as the averaged ground state of the Zeeman effect (see ref. [11]), another simple model showing quantum studied in recent years by many authors (seechaos, ref. [121), irrespective of having in this case a six coupled oscillator given by u2v2(u2+v2).

2. Symmetries and integrals As Vorob’ev and Zaslavsky point out, the chosen form of the potential energy is the most general under the following assumptions: (1) The potential (2) is a polynomial of order not higher than four; (2) ~W(x,y) possesses C 4~,symmetry, i.e.,

11 June 1990

To be self-contained we briefly outline the integrable cases. The particular cases /3=0, /3=a and fl=3a above are not only integrable but also separable. In the case /3= 3a, the second integral of motion is (3) ~ =XY+n2xy+axy(x2+y2). The form of fl~has been noted by Bountis et al. (see also ref. [13]). The equations of motion obtained from the Hamiltonian can be decoupled in terms of the linearly transformed variables u=x+y, v=x— y, to obtain ü+n2u+au3=0,

ii+n2v+av3=0.

These simply comprise the one-dimensional anharmonic oscillator equation of motion whose solution may be readily given in terms of Jacobian elliptic functions, u=s 1 cn(w1 t+ô1) cn(w2t+ô2) 4 + as?, s v=s2 and ô,, i = 1, 2 are integration where a~=The n associated energy integrals are constants.

i~=~~(U2+n2u2+~au4) so that ~W= ~(~+ ~), while the second integral of motion given above is J’~=~i~—.*~). In the case /3= a, if we transform the coordinates x and y to polar coordinates x—_p cos 0, y=p sin 0, then the Hamiltonian is independent of 0. Then it is straightforward to check that the angular momenturn is

~/I(x,y)= ~1f(y, x)= ~i/(—x,y)~

J~=xY—yX=const.

In other words, the full symmetry group ofthe Hamiltonian is the C4,,. F, point elements (2) of this group: C group. There are eight 2, C4, C~,a,, a2, a’,, a~. E is the unit element, and C2, C4 and C~are rotations by 1800, 90°and 270° respectively; a’1 and a’~are reflections about the x and y axes and a~and a2 are reflections about the same axes rotated by 45°.In the a = 0 case the parameter /1 can be eliminated from the Hamiltonian with the aid of a scaling. In our paper we study the nature of the classical motion of the model at different a values, which leads to the appearance of an additional parameter in the problem.

From the integration we have 2(T’t)] 1/2, p(t) =a1 [1 —/3~sn

412

(4)

where a,, /3~and rare constants (see ref. [14]). Of course, when /3= 0, it is just separated in Cartesian coordinates as a pair of Duffing oscillators.

3. Lissajous variables and normalization From now on, following Vorob’ev and Zaslavsky, we restrict ourselves to the domain a >0 and fl~ a. —

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In those conditions the potential (2) guarantees the finiteness of the motion. Moreover (see appendix), we only have the origin as an equilibrium. We restrict to the region where analytic investigation of the motion is possible, i.e. in the region of small energy values Emax(a, /3) << 1 in which the problem is considered can write as a perturbed oscillator system. Then we (5)

where

11 June 1990

where

c0 =~[(6+1)(3_,12)+ 3(2—2)e2 cos 4g], C

2 = e [ — (6 + 1) + (1—2) cos 4g]

=~e(2sin2g—lsin4g), 2 (1 +2) (1 + ~2) cos 4g] C4 = [3+2 3~ S4 = ~,i(2—2) sin 4g 52

—

~

—

ByaLietransform [10] 0: (1,g,L, G)—~(l’,g’,L’, G’ ), we normalize the Hamiltonian We are lucky because the new Hamiltonian up to first order is sufficient to show up a second formal integral with the main features of the dynamical system. In general .~°.

~ =~(X2+Y2) +~n2(x2+y2), = x4 + y4 +2x2y2,

(6) (7)

and 1=2/1/a. Our aim is to study the influence of the parameter 1 on the equilibria of the system. According to the Hamiltonian ~, the system presents a 1: 1 resonance, therefore, in order to avoid zerodivisors and other complications, Lissajous variables (1, g, L, G) are introduced (for details, see refs. [8,9]):

higher order may be needed; for instance Deprit [9] has shown that the Hénon—Heiles case requires the normalized second-order Hamiltonian. In order to simplify the notation we drop the primes in the Lissajous variables. 4. Global representation

X= —J~iJ~-G)nsin(1+g) Since l is not contained in the reduced system, it is an ignorable coordinate to the first order; the cor-

..J~(L G ) ~ (1— g), Y= .J~T+ G) n cos(1+g) +

~‘i

—

+ .,/~TL — G) x=

~

responding integral is the conjugate momentum L. If terms independent of the variables g and G as well as higher order terms are dropped from the nor-

n cos (1— g),

(L+ G) /n cos (1+ g)

malized Hamiltonian, we are thus left with a Hamiltonian

—,J~(L—G)/ncos(l—g),

L2 y=~/~E~-G)/nsin(l+g)

~=a~-

+.J~L—G)/nsin(1—g).

The Hamiltonian in Lissajous variables is such that ~ reduces to nL, while Jt’ is a Fourier series in land g. Moreover, it proves convenient, in order to implement the Lie transform, to introduce the auxiliary functions ~ G) and e=—e(L, G) such that ~=G/L,

e2+,~2=1.

= nL

/

L2

2j

2

(,,c

0+

(C2J cos 2jl+S

~

2f1))

[—(6+l)~

(8)

The equations of motion are dg&ii’~ L dt ~ =—a~—j[(6+1)+6(2—1)cos4g]~, —

dG -~-

Implicit in these definitions is the assumption that e be real, which implies that G be ~ L. After some computations we get the following expression,

+

2+3(2—1)e2cos4g]. 4--j

OK =

—

3L2

i— =a.~-~--~ (2—1)e2 sin 4g.

The ~quation 1= 8,t°/OL can be integrated by a quadrature when the rest of the equations have been solved; thus the equations for land L disappear from the reduced system. It must be observed at this point that the map (g, G) does not cover the entire phase space, for it excludes the points e= 0 at which the argument g is 413

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not defined. This pole-like singularity, inherent to the Lissajous map, does not belong to the system itself: it disappears when the system is handled in variables like ~1=~cos2g,

1~2=~sin2g,

~

11 June 1990

have that the equations of motion on S2(L/~,/~) are 4 = (~ .r) = ~ dr n

~

where one recognizes the Cartesian components of a vector similar to the Runge—Lenz vector in perturbed Keplerian systems and the norm of the angular momentum. We will not go into details here; see for instance refs. [15,161. In the global map ~ ~2’ 1~3), since ~2 + ~2 + ~2 (9) —

3

—

2

‘

the reduced phase space reveals itselfoas bundle of 2 (L/,,/’~), ne aabove each two-dimensional spheres S point of the axis of the formal integral L. In geometric terms, the reduction performed in the preceding section has achieved a fibration of the phase space. The points such that ~3>0 which comprise what we shall call here the northern hemisphere of S2 (L/~,/~) stand for ellipses travelled in the direct sense while those for which ~ <0 in the southern hemisphere represent ellipses travelled in the retrograde sense. Any point on the equatorial circle ~ =0 corresponds to a segment of length L~h along a straight line passing through the origin having precisely the origin as its midpoint. The north pole (c~,=~2=0, ~ 3=L/~J~) is the circle of radius L/~,/~ travelled in the direct sense, and the south pole (~= = 0, = L/~J~), the same circle but trayelled in the retrograde sense. Let us notice that due to the symmetries of the problem, with these variables we only consider the range 0~g~~t of the motion. In the global coordinates, the Hamiltonian (8) is now the function 1= —i [(6—2)~ +2A~] (10) ~

—

.

on account ofthe Liouville theorem, and after the change of independent variable dr= ~a dt, we Then,

414

=(~ifl=

(~2~~3)=I~1,

(~,~ ) = ~ ~,

2

sphere

—~—

with the Poisson brackets (~l,~2)=~~3,

each

(2—2)~.

These are similar to the equations of motion of a rigid body in torque-free rotation about a fixed point: A/= (C—B)qr, Bq= (A—C)rp, C— B A r— ( )pq, A, B, and C designating the principal moments of inertia while p, q and r stand for the components of the angular velocity in the principal frame of inertia. Our task now is to show that there is here more than an analogy of form. In any ~= momentum, const, wherethe ~ stands for the norm of manifold the angular motions of the Euler—Poinsot problem are represented by the level contours of the Hamiltonian ~(Ap2 + Bq2 + Cr2) on the ellipsoid ~2 =A 2p2 + B2q2 + C2r2 likewise, the orbits of the reduced coupled oscillator in the integral manifold L = const are the level contours of the Hamiltonian i~on the sphere S2 (L/,J~). Furthermore, as we shall now see, the phase flow is topologically identical in both cases. Simple observation shows that when the value of the parameter A is 0, 2 or 6 we have three special cases. In fact, they are associated with the division of the domain of parameters a and /1 into four parts: (I) —a~fl<0 (—2~2<0); (II) 0~fl
1<2); (III) a~/3~<3a(2~2~6)and (IV) 0~ 3a < /1 or equivalently either 2>6 if a ~ 0 or 0< /3 when a = 0. 5. Equilibria and bifurcations The equilibria ofthe2 reduced coupled oscillator are (L/~,/~). Introducing the Lathe extrema of ~ on S obtain them as the extrema grange multiplier y, we of the function ~ ~ ~2 ~2 =

~‘,l~’~2

~3,

Y/—

~

~

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11 June 1990

Table I Equilibria on sphere S2(L/.,,,r2) when L#0 (2#0, 2,6). E

y

E

0

E E1 E2 E3 4

0

0 0 L/~J~

L/~.J~ 0

—L/~/~

L/~ñ 0 0

0 _L/~J~ 0

0

.f

g

G

0

0 2/8n2)(6—2)

0

2—6

(L2/4n2)A (L 2/8n2)(6—2) (L 2/4n2)2

0 L 0 0 0 —L

—22

1—6

0 0 _L/,.J~

0(L

—22 0

K

~

~L_—~

0—+--------——--------F-~’---._~~~26

maximum

minimum

Fig. 1. This picture isjust to show the nonmonotonic variation of the extrema of ~( as a function of2; the values of L and n are set to I. The dashed line belongs to the separatrix. Notice the very narrow range for ~ in the domain of2 where the bifurcations occur. Table 2 This table helps us to draw the variation of the energy in fig. 1, showing the nonmonotonic variation of it as a function of2.

minimum saddle maximum ~ .:*~,,J

2<0

1=0

0<2<2

E ,E4 E0, E5 E1, E3

C1

E0, E5 E2, E4 E1, E3 [0, 1(6—2)1

—

[12,1(6—2)1

E1, E3 [0, fl

=2(6—l+y)~,,

0=

=2(

o

~

2l+y)~2, O=~+~+~—~L2, in the four unknowns

~, ~,

and y. Visibly, there

l~3

are the equilibria shown in table 1 on any sphere S2(L/.J~) when L is not zero (1~0,2, 6). The equilibria and for E5 circular respectively at The the north and south polesE0stand orbits. characteristic equation at these points being

2<2<6

2=6

2>6

E0, E,

E0, E5 E1, E3 E2, E4 [0, 12]

C2

E1, E3 E0, E, E2, E4 [1(6—2), 12]

—

C3 [0, fl

satisfying the constraint in (9). In other words, we solve the system made of the four equations 0=

2=2

—

E2, E4 [0, ~]

2 —0 —

1(1—6 2n4 L

—

the equilibria are stable if 0~1~6.All other equilibria, E,, E 3, E2 and E4 lie on the equatorial circle, hence they correspond to linear solutions. We show in the appendix that such linear solutions in the reduced system are also solutions of the original systern. At E, and E 3, the characteristic equation is 4

~2 —

—

4n

at E 2 and E4 it is 415

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—

PHYSICS LETTERS A

32(2—2) 2 = 0; 2 2 L

hence all four linear equilibria are stable if 2>6 or 1<0.

Fig. 2 gives immediately full account of the evolution of the flow as a function of the parameter A. The integral .5~is resolved in the constant ~ 2=const on the averaged problem, sliding the whole sphere. On each sphere the level curves are drawn for different values of .I~creating a “nicely distributed” family of curves. Of course, curves are drawn only with a geometrical point ofview, with no concern for the real levels of the energy. From the point of view of the evolution of the se-

11 June 1990

paratnx, we can thespheres case offig. 2g as the starting one, dividing thetake set of into two blocks. From

an orbital point of view, when 0<2<6 there are domains of initial conditions for elliptic type trajectories. When 2<0 or 1>6, all the trajectories go through the origin.

6. First-order solutions when 1>6 Having located the equilibria and assessed their stability, we now consider the phase flow globally. It is simple enough that it can be characterized in its entirety by producing the complete catalog of solutions outside the equilibria. We give here details of

*s~s.

a.

*0

Fig. 2. Orthographic projections ofthe orbital sphere for the coupled oscillators after reduction for six different values of the parameters A: (a)1=—2, (b)1=—1, (c) 2=0.05, (d) 2=0, (e) 2=0.5, (f)2=l.95. (d) Corresponds to the integrable case fl=O, where the bifurcation occurs. The others show the separatrices related to the onset ofchaos. (g)—(l) Orthographic projections ofthe orbital sphere for the coupled oscillators afterreduction for six different valuesof the parameter2: (g) 2=2, (h) 2=2.05, (i) 2=4, (j) 1=6, (k) 2=6.5 and (1) 2oc or a=0. The last one corresponds to the Yang—Mills model. (g) And (j) are the other two integrable cases where the bifurcations occur.

416

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the closed form solution of the normalized system in the range 1>6 (3a
~,we deduce that

2ir+1(L2—2 3(1—2) [—n

1~)]

,

2i(+(1—6)(L2—2~)].

n

K2 ~ 0

K2 2=

such 2i(that 2n 2~ 1 + (1— 6)L

K~~

3(1—2) 21

0

3(1—2)

2

~ K 2~

1—6

In those notations, the solutions in (9) and (10) take on the form = ±L

±L

=

~ \~I~~) ~ ,,J~

~2)K2Ldn(v~k). _______

~Lsn(v, K1

k).

In this class of solutions, the phase flow induces a clockwise circulation about either E 2 or E4. Geometrically speaking, the perigee librates about either take ~ with the upper or lower sign. Astronomers the E4 depending whether dewe viewequilibrium these orbitsE2 asor resulting from a on continuous formation of what they call an osculating ellipse. At v=0, this curve is a linear orbit making an angle 2g=arctan[.J~J6)/21/k] with the x-axis line. As

According to the inequalities (12), 0~

2

K~LCfl(v, k)

= ±

~=

~i 0 and

2

(12)

In consequence, we introduce the dimensionless K1

v=~K

so that

~2±

therefore 2—n2i(~(1—2)L2, 0~1L 0~2n2.f+(l—6)L2~3(1—2)L2.

2,/~X~—6) L(r—r0) ~=am(v,k),

But, from the analysis of the equilibria, there appears that, on any sphere S2(L/%J~) 1>6,

quantities =1— ~

Three cases are considered. (a) In the interval 0.c1~<(1/n2 )L2, where 0< ,c~<1<#c~.<3(1—2)/(1—6),we introduce the modulus k= K 1 /K2 which, by assumption, is <1, and we set ~ = (K1 /...,/~)L sin w. In the Jacobian notations for elliptic functions, there follows from (13) that

6(1—2) [2n

~L2~4L2, 2n

11 June 1990

~

Hence the resolution of the equations of motions in the global spherical coordinates hinges on the elliptic quadrature

v increases, the linear orbit opens into an ellipse whose perigee moves away; when v passes through the quarter-period K( k), the osculaellipse reaches its minimum eccentricity e= 1 K~.Past

~/

—

the tually, quarter-period, at the half-period the ellipse v= 2K(k), flattenstheagain. ellipse Evendegenerates again into a linear orbit (e= 1) with a perigee at maximum inclination. Thereafter the evolution will repeat itself in the southern hemisphere on 52(L/~J~) solution — [(1—6) 2(b) ]L2 The < 1~ <0, where in 0< the K~
/

d~ 3

2L2J~L6)

inverse modulus k, = l/k=,c2/,c, is <1, and the solution is given by the equations

=

~

2~\

(13)

_______

~l=±\1I~)KlLdn(v~kl).

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11 June 1990

In those notations, the four asymptotic solutions are =

±~J6(2_2)

K2L cn(v’, k,)

,

given in table 3. The solutions I~and 13 having detached themselves slowly from the equilibrium E5 tend toward

K2

L sn (v k1),

=

the northern equilibrium E0 as t tends to + x. The solutions ‘2 and 14 do the reverse going asymptotically to E0 and E5 as t tends to ~ and + ~

the amplitude being now the function 2~J22(1—6)

—

+Ki

V

—

respectively.

L(t—r0)

2

In this case, the phase flow circulates counterclockwise about the equilibria E1 and E3 while the perigee librates about either E1 or E3 depending on whether the upper or lower sign is adopted for ~. (c) When ~f=0, then ic, =,c2= 1. Besides the equilibria at the north and south poles, the equations of motion admit four solutions asymptotic to these equilibria. To define them in a concise manner, we set 2

7. Conclusions Instead of relying on the numerical Poincaré surface of section technique, we proceed here according with the approach proposed by Deprit [91. First the dynamical system is normalized using the Lissajous variables in the domain of the bounded motions. Then, the phase space for perturbed isotropic oscillators is set up as a two-dimensional sphere, a key issue not fully exploited yet for the qualitative understandingofthe global flow. The flow of the normalized coupled oscillator on

~-_~_——~_——

v

—

~L(r—ro)

and we introduce the argument

~(w+ ir/2)

lambda ( v)

=

w such that

the orbital sphere S2(L/~/’~) after reduction is susceptible of several forms; it may present degeneracies in the form of manifolds of non-isolated equi-

arctan e°.

And the solution is given by the equations

libria (2 = 0, 2, 6), transiting through parametric bifurcations from one regime to another: in particular the existence of bounded elliptic type of motions when 2<2<6. In the face of such a diversity of possible behaviours, we are glad to report that the

I

—

2 Lcos~t’ ~J3(2—2) ‘

~J

=

± / 6(2—2) 2—6 L ~

=

L ± sm 2W V

quartic in its manifold orbits behavesoscillator simply like a rigid body of inaveraged torque-free rotation about its center of mass. On the ellipsoid of

(14)

given components angularofmomentum the angularinvelocity, the space theoflevel the three con-

Table 3

‘2 ‘2 ‘3

\/

_____

_____

\/6(22)LcosW

3(22)LCOSW

V

_____

_____

3(2—2) Lcos w

\16(1_2) LcosW [~J~

___

~J3(A—2) Lcosy, 14

_

V 3(2—2) 418

~sin~v

L cos

~

\J6(22)

—~sin w L.

LcosW

~s1nw

L.

_ —

V 6(2—2) L cos w

—

sin w

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tours of the kinetic energy are analogous to those of the reduced quartic oscillator on its orbital sphere. The linear orbits on the OX and 0 Y axes shift from stable to unstable in the domain 0<13< 3a. Very recently, in the analysis of the truncated Toda lattice of three particles [8] and its relation with Yoshida’s study of the problem [17], Deprit got a new insight into his approach. That came about after along effort taking advantage ofthe MACSYMA and

11 June 1990

Appendix. Equilibria and particular solutions

~

From the equations of motion (n2+fiy2+ax2)x,

x=

—

Y= Y, 1’ (n2 + fix2 + ay2 )y, results a series of elementary properties. (i) The center ofattraction is an equilibrium — denoted E 0 at the energy level = 0. (ii) By reason of the unicity theorem for the mitial value problem in the theory of differential equations, the system admits two singular classes 110 and fl~of linear orbits. We have 11~when Y and y= 0 permanently, whereas, in the phase plane (X, x), the system is reduced to 2 +ax2 )x. X X=of symmetry, (n By X~ reason the class 11, is the plane (Y, y), with the equivalent system. (iii) The system admits two families 112 and 113 of linear solutions along the axis y=x and y= —x respectively. For the singular class fl2 of orbits, it reduces to a Duffing oscillator whose Hamiltonian is the function —

—

the MAO algebraic processor working on a LISP machine in order to get the normalizations of the several truncations to the twentieth order; the result is the conjecture that the global phase space of the normalized system gives also a clue in the search for integrals ofthe original system. In this paper we gather further evidence for this case. In this quartic coupled oscillatorpoints the three integrable case are associated with infinite of stable equilibria in the normalized system, which correspond to the bifurcation values ofthe parameter. Off these values, the analysis ofthe first-order solution is sufficient to report the presence of separatrices, which means that there occurs in the vicinity ofthe separatrix a stochastic layer [18] that goes over into chaos for high energy.

.~

—

~=X2+n2x2+~(a+fi)x4, Acknowledgement

and the equivalent function for 113.

We are deeply acknowledged to Dr. Deprit of the U.S. National Institute ofStandards and Technology who, during his sabbatical at the Universidad de Zaragoza in 1988, introduced us to the Lissajous

(iv) When a <0 the ener~i manifold ~ = n4/4a> 0 contains four critical points: an equilibrium E, at the point (n/~JJ~, 0) and, by virtue of the symmetry, the equilibria E 2, E3 and E4 see table 4~case II. (v) When a ~ a + /3<0, the energy manifold .1t’= n”/2 (a + contains four critical points E5, E6, E7 and E8 given in table 4, case III. The particular —

variables, and pushed us to carry out this research. Partial support for this research came from the Comisión Interministerial Cientifica y Técnica of Spain, Project PB87-0637.

/3, /3)

—

Table 4

(x,y) casel casell case III

Va,fl a<0

ci+fi<0

E0

E E~,3 E2,4 3,7

(0,0)

±(n/~f~,0) ±(0,n/f.~) “

n

a=fl<0

“

—

..,./—(a+fl)

/

E,,,

(a,b) a

n ~/J~a+fl) 2+b2=—n2/a

E6,3

______

caselV

+

—

) )

0 4/4a —n4/4a n 2(a+fl) 2(a+fl) —n4/4cs —

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Volume 146, number 7,8

PHYSICS LETTERS A

cases a = 0 or 13=0 are included in the previous ones. For a = 0, we have case III if /3<0 or have case I when /3>0. For /3=0, we have case I if a>0; when ~ —n4/4a we have case land for .~=—n4/2ahave case III. (vi) The case a=fi<0 gives rise to a circle of equilibria E 2 +b2 = n 2/a. 0,,(a, b) such that a —

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