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Normal form solutions of dynamical systems in the basin of attraction of their fixed points
Physica D 33 (1988) 34-50 North-Holland, Amsterdam
NORMAL FORM SOLUTIONS OF DYNAMICAL SYSTEMS IN THE BASIN OF ATTRACTION OF THEIR FIXED POINTS Tassos BOUNTIS Department of Mathematics, University of Patras, 261 10 Patras, Greece
George TSAROUHAS Department of Theoretical Mechanics, University of Thessaloniki, 54006 Thessaloniki, Greece
Russell HERMAN Department of Physics, Clarkson University, Potsdam, NY 13676, USA Received 26 January 1988
Dedicated to Joe Ford, on the occasion of his 60th birthday, even though he was one of the first to point out that many of the "attracting" features of dynamical systems were anything but "normal"
The normal form theory of Poincar6, Siegel and Arnol'd is applied to an analytically solvable Lotka-Volterra system in the plane, and a periodically forced, dissipative Dufling's equation with chaotic orbits in its 3-dimensional phase space. For the planar model, we determine exactly how the convergence region of normal forms about a nodal fixed point is limited by • the presence of singularities of the solutions in the complex t-plane. Despite such limitations, however, we show, in the case of a periodically driven system, that normal forms can be used to obtain useful estimates of the bosin of attraction of a stable fixed point of the Poincar6 map, whose "boundary" is formed by the intersecting invariant manifolds of a second hyperbolic fixed point nearby.
1. Introduction
There has recently been great progress in the analysis of global aspects of the solutions of nonlinear dynamical systems of the form dxi ] j-1
j = l k=l
i = 1, 2. . . . . n. Whether these equations are derived from a Hamiltonian of N degrees of freedom (with n = 2N) [1] or, describe, in general, a non-conservative model [2], it is well-known that they can have regions in their phase space, where their solutions are extremely sensitive to the choice of initial conditions [3-5]. These sensitive, so-called chaotic regions may be found in the neighborhood of intersecting invariant manifolds of hyperbolic fixed points (or, unstable periodic orbits) of the system- and are often seen to "encircle" regions of regular behavior around elliptic, or attracting fixed points of (1.1). Moreover, for large ranges of parameters of the problem, these intersecting invariant manifolds oscillate "wildly", and form quite complicated boundaries around basins of attraction in phase space [6-8]. 0167-2789/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
T. Bountis et al./ Normal form solutions of dynamical systems
35
Since eqs. (1.1) are not explicitly solvable (for all initial conditions and all t ~ R), except in very few and special cases [9, 10], it is important to develop analytical techniques to obtain solutions, at least over some selected parts of phase space, e.g., inside these interesting basins of attraction about stable fixed points. Such techniques have already been proposed and developed by Poincar6 [11], Siegel [12] and later by Arnol'd [13], and are part of what is called the n o r m a l f o r m analysis of dynamical systems [2]. This analysis consists of using series expansions to change variables and bring a given set of equations (1.1) to their so-called normal form, in which they are often easier to solve, thereby obtaining the desired solution by transforming back to the original variables. The purpose of this paper is to apply the normal form analysis to some simple dynamical systems of the form (1.1) in regions around a fixed point of the problem. Our aim is two-fold: First, we wish to demonstrate that, in the case where the normal form series expansions converge, they do so over large regions of initial conditions, and thus yield some important information about the system, like e.g. useful estimates of the basin of attraction about a stable fixed point. Secondly, we shall attempt to show that the convergence properties (and hence, the usefulness) of normal forms can be determined by analyzing the singularities of the solutions of (1.1) in the complex time plane. An important fact to keep in mind is that normal form series expansions for dissipative systems (1.1) are known to have a finite, n o n z e r o radius of convergence [11-13]. One of the objectives of the work described here, therefore, is to investigate, on specific examples, how large can this radius of convergence be, and thus test the usefulness of the normal form analysis in practice. By contrast, normal form series expansions for H a m i l t o n i a n systems (1.1) are notoriously divergent- with zero radius of convergence- and form an important part of the well-known asymptotic methods of canonical perturbation theory [1-3]. The theory of normal forms has also been developed for dissipative, discrete dynamical in R2-i.e. mappings of the plane onto i t s e l f - b y Chenciner [14], and has been applied by Valkering [15] to the regions of attraction of Hrnon's quadratic mapping. In the case of systems of ODE's, like (1.1), normal forms have been recently applied to the study of the limit cycle of a periodically driven 2-dimensional physical model [16], and the behavior near the origin of the Lorenz system in its subcritical region [17]. Note that our original equations (1.1) have been written in such a way that x, --- 0, i = 1, 2 . . . . . n, is an equilibrium or fixed point solution of the system. Let us also assume that the coefficients A~j and B~jk are time-independent (if they were not, new dependent variables would have to be introduced in their place, as described e.g. in section 3). The first task of a normal form approach is to d i a g o n a l i z e the linear part of eqs. (1.1), bringing them to the form
~i=X,yi+ ~
Ni, k y k y t +
....
(1.2)
k,l=l
i = 1, 2 . . . . . n. Now, a transformation to new variables z i is sought: oo
yi=zi
+
~_, a i ( m ) z " ,
i = 1 , 2 . . . . . n,
(1.3)
Iml--_2
with m - ( m l , m 2 . . . . , m , ) , m i > O, for all i, and [ml-ml+m2+
"'" + m , ,
z m =- z ~ l . z~ '2 . . . z,m",
(1.4)
T. Bountis et al./ Normal form solutions of dynamical systems
36
such that the new equations Zi=~kiZi "+-
~ bi(m)z 'n,
i = 1,2 . . . . . n,
(1.5)
Iml>_2
will be easier to solve than (1.1) or (1.2) Of course, the ideal case would be bi(m) = 0 for all i and m in (1.5). This, however, only happens in the absence of all resonances, i.e. when
X i - ( m , X ) = X i - ~ mkXk¢O, i = 1 , 2 ..... n,
(1.6)
k=l
for all integer vectors m. The reason is that then one can directly evaluate all obtained by substituting (1.3) in (1.2), using (1.5), i.e.
[Xi--(m,X)]ai(m)=--~_,Ni, kt{8(m,txk+lzt)+ k,l
ai(m ) from
~_, ak(m')a,(m-rn')}+ ....
the equations
(1.7)
Ire'l>_2
since the coefficients of a~(m) would never vanish. (The delta function in (1.7) is 1 if m =/*k +/*/ and zero otherwise, /*k being the zero vector with its k t h entry replaced by 1.) Moreover, the convergence of the above series would be guaranteed (at least in a small neighborhood of the fixed point) provided the X~'s: (a) either belong to the Poincar6 domain, i.e. Re(Xi) < 0 (or Re(Xi) > 0) for all i, or (b) belong to the Siegel domain, i.e. 0 lies within the convex hull of ~k1 . . . . , Xn, in the complex X-plane and C
[Xi-(m,X)[>
[m["'
for s o m e c > 0 ,
v>_
n-2 2
'
(1.8)
for all Xi, whence no resonances are present, see [13, ch. 5]. Our first example fails within class (a) above and is free from resonances. Duffing's equation, on the other hand, when viewed as a 4-dimensional system is seen to contain an infinity of resonances. As a 2-dimensional system, however, with periodic coefficients, it has eigenvalues X1, Xe belonging to the Poincar6 domain and convergence can still be proved, see [13, section 26]. In the case of resonance, at some m, the coefficient ai(m) cannot be calculated from (1.7). Instead, we set it equal to zero, and satisfy the corresponding eq. (1.7) by an appropriate choice of hi(m), cf. (1.5), which implies that we have to include nonlinear terms in the normal form equations. Still, even if that makes it difficult to solve (1.5) exactly, we can derive from it, in many cases, useful information about the solutions in the neighborhood of the fixed point. In section 2 we apply the normal form analysis to an exactly solvable Lotka-Volterra system in the plane, near a nodal fixed point at the origin. We find that the normal form series converge over a large region around that point, with the boundary of this region determined by the condition that the normal form variables z, = D~exp(t) encounter a singularity of the solutions in the complex time plane. This connection between convergence regions of normal forms and complex t-singularities has also been found on other examples [18], whose normal form series (1.3) do not truncate after a finite number of terms. We suggest that, in cases where these singularities can be accurately located, they may offer a practical criterion by which the region of convergence of normal form solutions can be determined.
T. Bountis et al./ Normal form solutions of dynamical systems
37
In section 3, we concentrate on the higher dimensional, non-integrable Dufling's equation
5i + # y c = x - x 2 + ( x - 1 ) Q c o s t o t ,
#>o.
(1.9)
We show that normal forms indeed yield very useful estimates of the basin of attraction about a stable simple periodic orbit of (1.9), corresponding to a fixed point of the Poincar6 surface of section x(tk), :~(tk), at t k = k ( 2 ~ r / o ~ ) + t o, k = 0 , 1 , 2 . . . . . As has been recently demonstrated [6, 8], such basins of attraction can have boundaries with extremely complicated "fractal" shapes. It would, therefore, be important to develop further analytical techniques (such as those provided by normal forms) to approximate the size and extent of such basins of attraction, especially in the higher dimensional cases n > 3, of (1.1).
2. Planar Lotka-Volterra models
Before applying, however, the normal form analysis to higher dimensional systems, it is instructive to use it to solve some planar models of the form "~1 = a x l + XlX2'
3C2 =
bxz - xxx2,
ab > 0,
(2.1)
for which there is a nodal fixed point at (b, - a ) . In particular, we will solve here first the case a = b = 1, which has no resonances and whose exact solution is also available in closed form. Thus, we will be able to analyze completely the boundaries of convergence of the normal form series, and examine how they may be related to the singularities of the solutions of (2.1) in the complex t-plane. Let us therefore take a = b = 1 in (2.1) and observe that the origin (0,0) of this system is a node, about which the linear part of the equations is already diagonalized. Hence, (2.1) can be written immediately in the form
E
.9i=X,Yi +
N~,ktYkYt,
i=1,2,
(2.2)
k,l=l,2
with A1 =
~k 2 --~
1, Yi = xi, and
N,,kk=0,
NI,~t=NI,tk = ~,
N2,kl=N2,1k = _!z,
(2.3)
cf. (2.1). Note, now, that with 1~1= ~ 2 ~ 0, no resonances are present, cf. (1.6), for any choice of m I > 0 and m 2 > 0, with [ml - rnx + m2 >- 2, since we have for all m~: I)~i-- ( m , ) ~ ) 1 = I~ki- (ml)k 1 + m22~2)1> [•il = 1.
(2.4)
Thus upon changing to the z~ variables by Y i = Z i "~ 2 Iml>2
ai(m)z",
i=1,2,
(2.5)
T. Bountis et al./ Normal form solutions of dynamical systems
38
with z " = z~'lzT2 , cf. (1.3), the normal form of the equations will be purely linear:
Zi=XiZi=Zi,
i = 1,2,
(2.6)
i.e. all b, = 0 in (1.5), provided, of course, the infinite series (2.5) converges. Substituting (2.5) in (2.2), and using (2.6), we obtain the following equation for a i(m) upon equating the coefficients of z":
[hi- (m,X)]ai(m)=
--Ni,k, 8 ( m , ~ k +l'L/)-- 2Ni, ktak(m--lx/)--N~.k,
E
ak(m')at(m--m'),
Ire'l>2
(2.7) i = 1,2, where summation over all k, l = 1,2 is implied, /~1 = (1,0), ~2 is the delta function 1,
8(m'l'Lk+ttl)=
m = lUk +/~/,
=
(0,1), cf. (1.7), and 8(m,/~k +/~t)
(2.8)
O, m4:l.tk+l,L t.
Thus, from (2.7) we can now compute all the ai(m) since their coefficient on the left-hand side is bounded away from zero, due to (2.4). Having thus calculated (and stored) the values of the ai(m)'s up to a desired order, say hmL = 30, we return to (2.5), insert for z~ the solution of (2.6).
z i = D i e t,
i = 1,2,
D1, D 2 arb. const.,
(2.9)
and compute the Yi (and hence the xi) from
xi=yi=Diet+
~_, a i ( m ) D ? l D ~ 2 e (m,+"2)t,
i=1,2.
(2.10)
[m[>_2
And now we come to the question of convergence of the above series: We recall that the theorems of Poincar6 and others [11-13], in this case, guarantee this convergence over some region in the zl, z 2 (and x 1, x2) plane about (0, 0); they do not tell us, however, how large this region is. What we find here is that this region extends over a large domain of bounded (for all t < 0) motion around the origin of the z~, z 2 and xl, x 2 planes, shown here in the shaded regions of figs. l(a) and l(b) respectively. As we demonstrate below, the boundaries of these regions exactly correspond to values of t = t, at which the solutions have singularities in the complex time plane. To see this, let us note first that this system can be solved exactly and in closed form by
xl=
+ c l e x p ( c 2+ t + c l e t) l+exp(c2+qe, ) ,
cl e t
X2=l+_exp(c2+cxet),
(2.11)
where cl, c 2 are arbitrary constants specified by the initial conditions, and the choice of + depends on whether x 1 and x 2 have the same sign or not. These solutions become singular at t = t, values given by c 2 + c 1 exp ( t , ) = in,~,
n any integer.
(2.12)
T. Bountis et al./ Normalform solutions of dynamicalsystems
39
Z2
1'
A
\ -s ,(z~
1,,,, Xl
Fig. 1. Convergence regions (shown shaded) of normal form series for the planar model (2.1) with a = b = 1. (a) In the real z1, z2 (D1, D2) plane, with the boundary curves given by (2.18), and (b) the corresponding region in the xl, x 2 plane of the original variables. Thus, given initial conditions xi0 = x~(0), i = 1, 2, sufficiently close to the origin, we can invert eq. (2.10) to o b t a i n the corresponding D, = D,(xlo, X2o). Noting then that
C1 =XIO"~-X20,
c 2 = l n ( X l o l -- (XIo-I-X20), \ x20 /
(2.13)
the t , of (2.12) is determined for which the corresponding
zi* =Di(Xlo, X2o)e t°,
i = 1,2,
(2.14)
will be the " f i r s t " (i.e. the ones with the smallest magnitude) to cause the divergence of (2.5). Usually, this procedure will have to be carried out numerically since the location of the singularities and the relationship between the Di's and the initial conditions are not explicitly known, in general. In the case of o u r special example, however, we can p e r f o r m all of the above steps analytically, since, with the aid of
T. Bountiset aL/ Normalform solutionsof dynamicalsystems
40
(2.11), the normal form series (2.5) can be summed up in closed form [19]:
Z1
Z2e-z
xl= 1-z2(1-e-~)/z
'
x2= 1-z2(1-e-~)/z
'
z=-z 1+z2,
(2.15)
It is now simple to check, using (2.9), (2.13) and (2.15), at t = 0, that c I = D I + D 2,
(2.16)
c2=ln(D1/D2)
and substitute (2.16) in (2.12) to obtain In ~
n=0,+__1,+2 .....
+(Dl+D2)et*=in,~,
(2.17)
Setting the singularity at t, = 0 and using from (2.15) the expression for the singularity "surface" z ~ / z ~ = - exp [ - (z~* + z~' )1,
we can derive the intersection of this "surface" with the real D1, D 2 plane: (D 1 + D2) 2 - In2 ( D 1 / D 2 ) = ~r2,
D1D 2 > 0,
(2.18a)
D 1 + D 2 + InID1/D2J = O,
D1D 2 < 0.
(2.18b)
These expressions give precisely the two pairs of hyperbolas shown in the real z 1, z 2 (D 1,/)2) plane of fig. la. Put differently, after determining the Dx, D 2 that correspond to a given set of initial conditions, we can compute from (2.17) the value of t R = Re(t.) at which the solutions (2.10) will diverge: tR___ ½1n (
ln2 ( D1/D2) +'rr2 } (D, + D2) 2
'
D1D2 > 0,
(2.19a)
tR = l n [ l n I D 1 / D 2 { / ( D x + D2)I,
D1D2 < 0.
(2.19b)
These results have also been verified numerically by following D 1 / D 2 = a - " r a y s " in the plane of fig. l(a), and computing the radius of convergence of series (2.10) using a standard "root" test, for several values of a. Thus, the region between the hyperbolas (2.18) of fig. l(a) is transformed via (2.10) into the shaded region of fig. l(b). But then, one might ask, how can one obtain solutions of this system closer to the stable and unstable manifolds of the saddle point at (1, - 1)? Is it possible to use normal form series to construct such solutions explicitly? For simple, planar models such as this one, the answer is yes: Shift the origin of (2.1), with a = b = 1, to the point (1, - 1 ) by introducing Yl = xl + x 2,
Y2 ~--- X l
-- X2 -- 2
and note that the resulting equations for yl and Y2 have diagonalized linear parts: .Pl =Yl,
Y2 = -Y2 + ½(Yl2 - y ~ ) "
(2.20)
T. Bountis et al./ Normal form solutions of dynamical systems
41
Moreover, since the first equation is already linear, the normal form analysis of (2.10) is very simple, and is easily seen to converge over the full, unshaded area of the x 1 > 0, x 2 < 0 quadrant of fig. l(b), thus complementing nicely the region of convergence about the nodal point of the system. Still, eqs. (2.20) have eigenvalues ~1 = 1, ~2 = --1 which do not belong to the Poincar6 domain (see section 1) and give rise to an infinity of resonances:
Xi_(m,X)=o{m=(n+l,n), m = ( n , n + l),
i=1, i=2,
(2.21)
for every n = 1, 2, 3 . . . . . Since the problem is planar, however, no intersections of invariant manifolds are present and the normal form series turn out to also converge in the neighborhood of this hyperbolic fixed point. A similar situation arises in the normal form analysis of the conservative Lotka-Volterra system (2.1) with a = - b = 1, which has two fixed points: one saddle at (0, 0) and one center at ( - 1 , - 1 ) . Changing coordinates so that the center is located at the origin, we set xl = - 1 + x and x 2 = - 1 + y and write (2.1) in the form
.¢c= - y + xy, .9= x - xy.
(2.22)
This is the classic Lotka-Volterra model, with the well-known integral of the motion ( - 1 + x ) ( - 1 + y ) exp (x + y ) = const.
(2.23)
Its solutions about (0, 0) are all periodic, up to the stable and unstable manifolds of (1,1), but are not known explicitly as functions of t. We can construct these solutions, however, by normal form series expansions, which turn out to converge all the way up to the invariant manifolds of (1.1). To see how this is done, let us first diagonalize the linear part of (2.22) by changing to new variables Yl, Y2:
(2.24)
- i
i
Y2
Y2
~ 1/2
1/2i
] \ Y]
in terms of which (2.22) becomes
.~j = ~ . j j +
~_, Nj, k,YkYl,
J = 1,2,
(2.25)
k,l=1,2
with X1 = i, ~2 ~- - i and 1-i N1,11 = N2"22 = - - i i -
= -Ul,=,
Nj,12 = Nj,21 = 0,
1+i N2,11 = U l * = = - W -
(2.26)
=
where stars denote complex conjugation. Introducing now the normal form coordinates zj we write as before
yj=zj+ E aj(m) zm, Iml>-2
j=1,2,
(2.27)
T. Bountis et aL / Normal form solutions of dynamical systems
42
and note that a countable infinity of resonances occurs, as in (2.21) above, since Xj
-
( m , h)
=
(ml~kI + m2h2)
~kj -
-----0
(m 1 + m 2 >_ 2)
(with X1 = i, X2 = - i ) are satisfied for
mz=ml-1,
j=l: j=2:
mz=ml+l,
or or
rn=#l+n(/~l+/~2), m=/~2+n(/h+/~2),
(2.28a) n=l,2 .....
(2.28b)
with/~1 = (1, 0), /~2 = (0,1). Thus the normal form equations for this problem become
~,j = ~.jzj + zj ~ bj,~(zxz2) ~, j = 1,2,
(2.29)
s=l
where the coefficients bj, s will be so chosen that all resonant terms in the equations for aj(m) cancel. To this end, let us substitute (2.27) in (2.25) using (2.29) to obtain
[ ~ j - (m, h )]aj(m) = --Nj, k,~(m,~k + IXl) - 2Nj, kla,(m-- I~) + bj,,6(m, ~j + n(/q +/~z)) n-1
Im1-2
+ E aj(rn-s(btl+lx2))(m-s(Ixl+t~2),b,)-Nj,
kt E
s=l
ak(m')at(m-m'),
j=l,2,
(2.30)
Ire'l> 2
where b~ = (bx,~, b2,~), and n is the integer part of Irnl/2. Eqs. (2.30) can be solved for all aj(m), for which the m's do not satisfy conditions (2.28a, b), for j = 1 and j = 2 respectively. In the resonant cases, however, we may set: aj(m)=0,
m=lxj+n(l~l+l~2),
n=1,2 .....
(2.31)
and solve (2.30), at these values of m, for bj,,: n-1
bj,,= 2Nj, kta,(m -- I~k) -- ~_, aj(m -- s(t~l + ttz))( m -- s(l~l +It2), bs) s~l
Iml-2
+Nj, kt Y', a k ( m ' ) a , ( m - m '
).
(2.32)
Imq>_2 Before we attempt to solve the normal form equations (2.29) let us note from (2.24) that y~* = Y2, which implies z1* ---z2, and b*1 , s = b 2s, ,
s---1,2 .... •,
a~(ml, m 2 ) = a 2 ( m 2 , rnl)
'
(2.33)
cf. (2.27) and (2.29). If we now set zl = R exp (icp) ,
z2 = R exp ( - i c p ) ,
(2.34)
we can uncouple eqs. (2.29) into separate equations for R and % R=0,
~=1+
~ s~l
Im(bx,~)R2L
(2.35)
T. Bountis et al./ Normalform solutions of dynamical systems
43
y
i
"3
Fig. 2. Region of bounded (oscillatory) motion of the conservative Lotka-Volterra model (2.22). The closed curves were obtained by normal form series, which are seen to converge all the way to the straight line invariant rnattifolds of (1,1).
Since it turns out that Re(bx,,) = 0, for all s, and thus directly integrate (2.35) to obtain oo
R=R0=const.,
ep=tot+to,
to=l+
E Im(bx,~) R2s.
(2.36)
s=l
To study the convergence properties of these normal form solutions we work as follows: Pick a value of R o and use it in (2.36) to calculate the corresponding to. Substituting then in (2.27) one obtains (with t o = 0) Yl = Ro ei•t-{-
E
al(ml, mz)R~'~+"2 exp [i(m I - m2)tot],
(2.37)
Iml>2
with Y2 = Yx*, whence, from (2.24), the x, y immediately follow. These solutions are clearly oscillatory and rotate around the origin in both the Zl, z 2 and x, y planes. Taking successively larger and larger values of R 0 > 0 and 0 < cp < 2~r we plot the corresponding solution curves in the x, y plane, see fig. 2. Keeping, for example, in (2.37) terms up to Iml - 25, we find for R o = 0.5, to = 0.90442636, 6-digit agreement, and for R 0 = 0.7, to = 0.77264, 4-digit agreement, when comparing with the results of numerical integration of (2.22). In this conservative model, therefore, the normal form series solutions appear to converge over the whole domain of bounded motion around (0, 0), all the way to the invariant manifolds of the saddle point at (1.1). This may be explained by the fact that the problem is integrable, cf. (2.23), and its solutions are all periodic, given by convergent Fourier series expansions of the form (2.37), whose frequency depends on the initial conditions, cf. (2.36). It is important to note, however, that in more complicated, higher dimensional problems, if the eigenvalues of the linearized equations about a fixed point do not satisfy Re )~j < 0 (or Re )~i > 0) for all i, then, in the presence of resonances, normal form series solutions will most likely be divergent, with zero radius of convergence. If one concentrates, however, on dissipative systems, one finds that the convergence region of normal forms is large enough to provide useful estimates of the basin of attraction about a stable fixed point of the associated Poincar6 map.
44
T. Bountis et a l . / Normal form solutions o f dynamical systems
3. N o r m a l f o r m solutions of Dufling's equation
We now turn to the normal form analysis of the dissipative Duffing's equation 5/+ # 2 -
(3.1)
x + x 2 = ( x - 1)Qcos~0t,
/~ > 0. When Q = 0, (3.1) has an unstable equilibrium point at (0, 0) and a stable one at (1,0), in the x, plane. For small Q 4: 0, these turn into fixed points of the Poincar6 map of the system on its surface of section
2(tk)/tk=k2~r +,ol
~'°--=(x(tk),
(3.2)
O)
through which pass periodic orbits of period 2~r/~0 in the extended 3-dimensional phase space. At b o t h / t = 0 and Q = 0, the hyperbolic fixed point at (0,0) has stable and unstable manifolds, W ~ and W u respectively, which join smoothly, "embracing" the elliptic fixed point at (1,0). For ~t> 0, these manifolds split, and at sufficiently high Q, they intersect each other infinitely often, thus forming a very complicated boundary for the basin of attraction around the stable fixed point on Zt0 [2]. Our purpose here is to show that the region of convergence of normal forms extends sufficiently far from the stable fixed point to be able to capture most of its actual basin of attraction. Our results suggest, that this convergence region does not extend all the way to the intersecting W u and WS manifolds, due to the presence of singularities in the complex t-plane, as explained in section 2. In order to apply the normal form analysis to (3.1), we write it first as a fourth order system, shifting the stable fixed point to the origin, and get
"~1 ~ X 2 , 2 2 --= --I, l X 2 -- X 1 -- X 2 Jr- X I X 4 ,
X3 ~'~ 0"~X4,
X3(0 ) = 0,
3~4 " ~ - - 0~X3,
X4(0) = Q,
(3.3)
with x -= (x 1, x 2, x 3, x4) "r, this system can now be written as (3.4)
ic = A x + M x ,
where A, the matrix of its linear part, has eigenvalues 2tx= __~ + 2 r~__/x2 •
'
Xz =
/z 2
½¢4__/z2
'
X 3 = i~,
~4 ~
-i~,
(3.5)
and M = ( M u ) has all Mq= 0, except M21 = x 4 - x 1. Next we determine the matrix S that diagonalizes A, i.e.
SAS-I = A,
Aq=XiSij,
(3.6)
T. Bountis et al./ Normal form solutions of dynamical systems
45
and define the new variables y~ in y = (Yl, Y2, Y3, Y4)T by ~2 d
-d
0
0
Xl
-;kid
d
0
0
X2
0
0
1/2
-i/2
X3
0
0
1/2
i/2
X4
y=$x=
d-=
- Xl)
(3.7)
Substituting then x = S - l y in (3.4) and multiplying on the left by $ we obtain (Yl +Y2 - iY3 + iY4)(Yl +Y2)
p=Ay+
1
Ax_h 2
( - Y l -Y2 + iy3 + iY4)(Yl +Y:) 0 0
or, more compactly, 4
~j = Xjyj + E Nj. ktYkY,,
j=1,2,3,4,
(3.8)
k,l=l
with N I , l l -~- N1,12 = N1,22 = 1/(X 2 - Xl) , N1,14 = N1,24 = - N l , 1 3 -m- - N 1 , 2 3 = i/(2X 2 - 2Xx) ,
(3.8a)
N1,33 = N1,34 = N1,44 ~-- 0,
and
Nj, kt= Nj, tk , Nl, kt= --N2,k,,
N3,kt=N4,kt=0.
(3.8b)
Now, we are ready to introduce the normal form coordinates zj by writing
yj=zj+
Y~ aj(m)z m, j = 1 , 2 , 3 , 4 ,
(3.9)
Iml>2
cf. (1.3), (1.4), and go on to derive the equation of motion for the z/s. Before doing this, however, let us note that the hj's of this system, cf. (3.5), give rise to infinitely many resonances, ~'i - (m, X) = 0, at the
T. Bountis et al. / Normal form solutions of dynamical systems
46
following rn values:
m=(1,O,n,n), m=(O,l,n,n), m=(O,O,n+l,n), m = ( O , O , n , n + l),
j=l, j=2, j=3, j=4,
n >
1.
(3.10)
As we have explained already in sections 1 and 2, these resonances will introduce additional terms in the first two normal form equations
2 j = X j z j + z j Y'~ bj,~(z,z,)',
j=1,2,
(3.11)
s~l
while the third and fourth equations, being already linear, cf. (3.8), will remain unaffected, i.e.
~j=Xjzj,
aj(rn) = 0 ,
j=
3,4.
(3.12)
Substituting (3.9) and (3.10) above in (3.3) and equating coefficients of z m we obtain for j = 1, 2, [hi-
(rn,X)] a i ( m ) = - Nj, ktS( m, l~k + I~t) - 2Nj, ktat( m -- I~k) + bj,, 8( m, t~j + n(/~ 3 + / x , ) )
+ i aj(m -- s(/.t 3 +/.t4))(m -- ,(/tt 3 + ~4)' b,) s~l
-Nj, kt Y'~ ak(m')at(m--m'),
(3.13)
Ire'l>__2
where /~k is a 4-dimensional integer vector with 1 in the kth entry and 0 everywhere else, n = [m/2], b, = (bl, s, b2,,,0,0), and repeated indices imply summation over all k, l = 1,2,3,4. Thus all aj(m) can be computed from (3.13) except those with m = 2n + 1, cf. (3.10), for which we set aj(m)=0,
m=/~j+n(~t3+/~,),
j=1,2,
and choose the coefficient bj,, so that (3.13) is satisfied. At this point we are ready to solve the normal form equations of the problem: Starting with (3.12) we have z3(t ) = R e i°J(t-tD,
Za(t ) = R e -i'~u-',),
(3.14)
where t3, t 4 a r e constants to be determined by initial conditions at t = 0. These are given by inverting (3.7), using (3.3), y,(0)
=
y,(0)
=
x3(0 )
- -2
ix4(0 )
2
=-2
i
Q=z3(O)=Re-i~°t3'
--ix3(0) xa(0) iQ = z,(0) = R e i'°" 2 + ~ = '~
T. Bountis et al./ Normal form solutions of dynamical systems
F r o m these equations we deduce R -form
z3(t ) = (-iQ/2)e i~t, z4(t ) =
Q/2,
47
13 = ~r/2a~ and t 4 = 0 and write the solutions (3.14) in the
(3.15)
( i Q / 2 ) e -i°''.
Finally, from (3.15) we substitute in (3.11) and obtain equations for z 1 and z2:
~j---)~yz++ zj ~ by,,(a/2) 2",
j=1,2,
(3.16)
$=1
which are linear and can be readily solved, once the by,s are known, z1 = r l e x p ( ) q t ) ,
(3.17)
z2 = r2exp(-r2t),
Y1 -- Y2* = YR + iYI ------Xl + ~
(3.17a)
bl,s(Q//2) 2s,
s=l
where the relation 3'1 = ~'2" follows directly from the computation of the by,s- Moreover, since from (3.7) it is clear that Yl* = Y2, we also have zl*=z2,
.'.
q * = r 2 and with
rj=pyei°J,
j=l,2,
(3.18)
we conclude Pl = P2 and - 01 = 02. Thus, selecting small values for & and 01, at t = 0, and using (3.17) we can compute explicitly the yj's (and the xj's) at any later time t > 0, for which the series (3.9) will still converge. Let us take, for example, a uniform distribution of points in the z 1 (or z2) plane lying on a small circle of radius Pl = P 2 - 0.02, around the fixed point. They correspond, via the formulas (3.9) and (3.17), at t = 0, to an approximately circular distribution of points in the x 1, x 2 plane (see the innermost curves of figs. 3(a, b)). Now let us evaluate the series (3.9), with (3.17), for these points, at the t values t = tk =
2~k,
k = 0,5,10,15,20,23,
(3.19)
f o r / , = 0.1, ~ = 3 and Q = 0.3, whereupon we obtain the six closed curves plotted here in fig. 3a. We have stopped at k = 23 because this is approximately the t k value at which the convergence of our series, with terms up to [m I = 14, breaks down. What we can say, therefore, is that if we were to start with initial conditions chosen within the outermost curve of the x 1, x 2 plane of fig. 3(a) we would find that the normal form series accurately represents the solution of the problem, as it winds down to the fixed point, for all t > 0. To see whether this claim is justified we have taken the corresponding initial conditions of the innermost circle and integrated eq. (3.1) numerically, for t < 0, plotting the intersections of the orbits with the x 1, x 2 plane at the same t = t k values (3.19). The resulting set of curves obtained in this way are shown here in fig. 3(b). They are seen to agree very well with the normal form results of fig. 3(a), except, of course for the outermost curve, for which the normal form series did not converge, and discrepancies between the analytical and numerical approach became evident. In figs. 3(b, c) we have plotted this outermost curve, obtained by normal forms and numerical integration respectively, together with the intersecting invariant manifolds of the hyperbolic point. Thus, if
48
T. Bountis et aL/ Normal form solutions of dynamical systems
b
d
C
I
Fig. 3. Comparison between analytical (normal form) and numerical solutions of (3.1) near its stable fixed point, for g = 0.1, 0~= 3 and Q = 0.3. (a) A small "circle" of 20 initial conditions around (1,0) evaluated for t < 0 at the tk values of (3.19) via the normal form series (3.9). (c) The location of the invariant manifolds of the hyperbolic fixed point with respect to the outermost curve obtained in the above way at t23 -23 (2~r/~) where the series cease to converge. (b) and (d) are the same as (a) and (c) respectively, except that they were obtained by numerical integration of (3.1). =
we w e r e to view o u r n o r m a l f o r m result as a n e s t i m a t e of the b a s i n of a t t r a c t i o n of the stable fixed point, w e w o u l d h a v e c a p t u r e d , in this example, quite a large p o r t i o n of it. W e should remark, however, that the full b a s i n of a t t r a c t i o n is n o t c o n t a i n e d o n l y within the finite d o m a i n shown in figs. 3(c, d) b u t extends to i n f i n i t y a l o n g the stable manifold, while the u n s t a b l e m a n i f o l d of the h y p e r b o l i c fixed p o i n t has o s c i l l a t i o n s t h a t are a t t r a c t e d b y the stable fixed point. O n e c o u l d argue, of course that, h a d we i n c l u d e d m o r e terms in o u r series, their convergence region m i g h t b e s h o w n to b e larger t h a n i n d i c a t e d here, p e r h a p s even e x t e n d to the i n v a r i a n t m a n i f o l d s of the u n s t a b l e fixed point. A l t h o u g h we c a n n o t c o m p l e t e l y rule o u t this possibility, we e m p h a s i z e that this
T. Bountis et al./ Normalform solutions of dynamical systems
49
certainly does not happen with the integrable example (2.1) with a = b = 1 and other similar models we have studied elsewhere [18]. As was explained earlier, all these dynamical systems have singularities in the complex t-plane, which may prevent convergence long before the solution reaches a basin boundary as t increases in the negative direction. An obvious exception, of course, occurs if the normal form series happen to truncate, exactly, after a finite number of terms, in which case complex t-singularities would be irrelevant. However, even in cases where we expect these singularities to be relevant, if the problem is higher than two-dimensional and non-integrable, it will typically have an infinity of singularities, which often accumulate on complicated patterns in the complex t-plane [20]. This would make it difficult to tell a priori, which singularity (or singularities) will be responsible for the divergence in each case. Still, since the convergence of series in more than one complex variable is difficult to study in general, looking at singularities in the complex t-plane offers an alternative approach, especially in problems where these singularities can be accurately located and are distributed sparsely enough, so that their individual effect on the normal form series can be determined.
4. Concluding remarks Our main purpose in this paper was to demonstrate on simple systems of nonlinear ODE's the applicability and usefulness of normal forms in obtaining solutions about fixed points of planar models, or, around stable, simple closed orbits of periodically driven systems, with 3-dimensional phase space. Along the way, we discovered some interesting connections between the convergence properties of normal forms and the singularities of the solutions in the complex t-plane. In our planar examples, we were able to make this connection precise, and obtained large regions of convergence around different fixed points of one non-conservative model. These regions of convergence were seen to complement each other, while the boundary between them was determined by the location of the complex t-singularities of the solution. In the case of a conservative (and integrable) planar system, on the other hand, the corresponding "arrangement" of the singularities permitted the convergence of normal forms over a full, (arbitrarily long) period of the motion and thus yielded all bounded solutions around the origin up to the invariant manifolds of a nearby saddle point. Applying then these methods to the neighborhood of a stable fixed point of the Poincar6 map of a periodically driven Duffing's equation, we obtained useful estimates of the basin of attraction around that point. These estimates did not extend all the way to the actual basin boundary, because the normal form series were numerically seen to diverge, before the solutions reached the invariant manifolds in the neighborhood of a nearby hyperbolic fixed point. We suggest that the convergence of normal form series is connected with the location of movable singularities of the solutions in the complex t-plane. After all, these singularities limit the convergence of an expansion of the solution in powers of t, while the normal form series are expanded in powers of exponentials of t. The predictive value of this observation, however, is questionable, if the location of the singularities is not accurately known, or their number is too large to permit, in practice, an investigation of which one(s) of them cause divergence. In any case, we expect the application of normal forms to the basins of attraction of dynamical systems to prove quite useful, especially in problems with several fixed points, which are separated by complicated basin boundaries. Such problems are currently under study and more results concerning the convergence properties of normal forms will be given in future publications.
50
T. Bountis et al./ Normal form solutions of dynamical systems
Acknowledgements T w o of u s (T.B. a n d G . T . ) wish to t h a n k A. R a u h a n d J. P a d e for several h e l p f u l c o n v e r s a t i o n s c o n c e r n i n g n o r m a l forms, a n d a c k n o w l e d g e the h o s p i t a h t y of the Physics D e p a r t m e n t of the U n i v e r s i t y of O l d e n b u r g , w h e r e p a r t of this w o r k was c a r r i e d out.
References [1] M. Lieberman and A. Lichtenberg, Regular and Stochastic Motion, Appl. Math. Sci., vol. 38 (Springer, Berhn, 1983). [2] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Appl. Math. Sci., vol. 42 (Springer, Berlin, 1983). [3] R.H.G. Helleman, in: Fundamental Studies in Statistical Mechanics, vol. 5, E.G.D. Cohen, ed. (North-Holland, Amsterdam 1981). [4] Chaotic Behaviour of Deterministic Systems, Proc. of Les Houches 1981 Summer School, G. Iooss, R.H.G. Helleman and R. Stora, eds. (North-Holland, Amsterdam, 1984). [5] J. Ford, Physics Today (April 1983), p. 40. [6] C. Grebogi, S.W. McDonald, E. Ott and J.A. Yorke, Phys. Lett. A99 (1983) 415; see also Physica D 17 (1985) 125. [7] C. Grebogi, E. Ott and J.A. Yorke, Phys. Rev. Lett. 56 (10) (1986) 1011; see also refs. listed therein. [8] F.C. Moon and G.-X. Li, Phys. Rev. Lett. 5 (1985) 1439. [9] T. Bountis, V. Papageorgiou and P. Wintemitz, J. Math. Phys. 27 (1986) 1215. [10] A. Ramani, B. Dorizzi, B. Grammaticos and T. Bountis, J. Math. Phys. 25 (1984) 878; see also Physica A 128 (1984) 268. [11] A.D. Brjuno, Analytical form of differential equations, Trans. Moscow Math. Soc. 25 (1971) 131. [12] V.M. Starzhinski, Applied Methods in the Theory of Nonlinear Oscillations (MIR, Moscow, 1980) English transl. [13] V.I. Arnol'd, Geometrical Methods in the Theory of Ordinary Differential Equations (Springer, Berlin, 1983). [14] A. Chenciner, in ref. [4] above. [15] T. Valkering, Physica D 18 (1986) 483. [16] G. Tsarouhas, Normal form of time dependent chemical rate equations for irradiation produced point defects, Phys. Lett. All6 (6) (1986) 264. [17] J. Pade, A. Rauh and G. Tsarouhas, Application of normal form theory to the Lorenz model in the subcritical region, Physica D 29 (1987) 236. [18] T. Bountis and G. Tsarouhas, On the apphcation of normal forms near attracting fixed points of dynamical systems, Physica A 153 (1988) 160. [19] A. Rauh, private communication. [20] T. Bountis, V. Papageorglou and M. Bier, On the singularity analysis of intersecting separatices in near-integrable dynamical systems, Physica D 24 (1987) 292.
NORMAL FORM SOLUTIONS OF DYNAMICAL SYSTEMS IN THE BASIN OF ATTRACTION OF THEIR FIXED POINTS Tassos BOUNTIS Department of Mathematics, University of Patras, 261 10 Patras, Greece
George TSAROUHAS Department of Theoretical Mechanics, University of Thessaloniki, 54006 Thessaloniki, Greece
Russell HERMAN Department of Physics, Clarkson University, Potsdam, NY 13676, USA Received 26 January 1988
Dedicated to Joe Ford, on the occasion of his 60th birthday, even though he was one of the first to point out that many of the "attracting" features of dynamical systems were anything but "normal"
The normal form theory of Poincar6, Siegel and Arnol'd is applied to an analytically solvable Lotka-Volterra system in the plane, and a periodically forced, dissipative Dufling's equation with chaotic orbits in its 3-dimensional phase space. For the planar model, we determine exactly how the convergence region of normal forms about a nodal fixed point is limited by • the presence of singularities of the solutions in the complex t-plane. Despite such limitations, however, we show, in the case of a periodically driven system, that normal forms can be used to obtain useful estimates of the bosin of attraction of a stable fixed point of the Poincar6 map, whose "boundary" is formed by the intersecting invariant manifolds of a second hyperbolic fixed point nearby.
1. Introduction
There has recently been great progress in the analysis of global aspects of the solutions of nonlinear dynamical systems of the form dxi ] j-1
j = l k=l
i = 1, 2. . . . . n. Whether these equations are derived from a Hamiltonian of N degrees of freedom (with n = 2N) [1] or, describe, in general, a non-conservative model [2], it is well-known that they can have regions in their phase space, where their solutions are extremely sensitive to the choice of initial conditions [3-5]. These sensitive, so-called chaotic regions may be found in the neighborhood of intersecting invariant manifolds of hyperbolic fixed points (or, unstable periodic orbits) of the system- and are often seen to "encircle" regions of regular behavior around elliptic, or attracting fixed points of (1.1). Moreover, for large ranges of parameters of the problem, these intersecting invariant manifolds oscillate "wildly", and form quite complicated boundaries around basins of attraction in phase space [6-8]. 0167-2789/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
T. Bountis et al./ Normal form solutions of dynamical systems
35
Since eqs. (1.1) are not explicitly solvable (for all initial conditions and all t ~ R), except in very few and special cases [9, 10], it is important to develop analytical techniques to obtain solutions, at least over some selected parts of phase space, e.g., inside these interesting basins of attraction about stable fixed points. Such techniques have already been proposed and developed by Poincar6 [11], Siegel [12] and later by Arnol'd [13], and are part of what is called the n o r m a l f o r m analysis of dynamical systems [2]. This analysis consists of using series expansions to change variables and bring a given set of equations (1.1) to their so-called normal form, in which they are often easier to solve, thereby obtaining the desired solution by transforming back to the original variables. The purpose of this paper is to apply the normal form analysis to some simple dynamical systems of the form (1.1) in regions around a fixed point of the problem. Our aim is two-fold: First, we wish to demonstrate that, in the case where the normal form series expansions converge, they do so over large regions of initial conditions, and thus yield some important information about the system, like e.g. useful estimates of the basin of attraction about a stable fixed point. Secondly, we shall attempt to show that the convergence properties (and hence, the usefulness) of normal forms can be determined by analyzing the singularities of the solutions of (1.1) in the complex time plane. An important fact to keep in mind is that normal form series expansions for dissipative systems (1.1) are known to have a finite, n o n z e r o radius of convergence [11-13]. One of the objectives of the work described here, therefore, is to investigate, on specific examples, how large can this radius of convergence be, and thus test the usefulness of the normal form analysis in practice. By contrast, normal form series expansions for H a m i l t o n i a n systems (1.1) are notoriously divergent- with zero radius of convergence- and form an important part of the well-known asymptotic methods of canonical perturbation theory [1-3]. The theory of normal forms has also been developed for dissipative, discrete dynamical in R2-i.e. mappings of the plane onto i t s e l f - b y Chenciner [14], and has been applied by Valkering [15] to the regions of attraction of Hrnon's quadratic mapping. In the case of systems of ODE's, like (1.1), normal forms have been recently applied to the study of the limit cycle of a periodically driven 2-dimensional physical model [16], and the behavior near the origin of the Lorenz system in its subcritical region [17]. Note that our original equations (1.1) have been written in such a way that x, --- 0, i = 1, 2 . . . . . n, is an equilibrium or fixed point solution of the system. Let us also assume that the coefficients A~j and B~jk are time-independent (if they were not, new dependent variables would have to be introduced in their place, as described e.g. in section 3). The first task of a normal form approach is to d i a g o n a l i z e the linear part of eqs. (1.1), bringing them to the form
~i=X,yi+ ~
Ni, k y k y t +
....
(1.2)
k,l=l
i = 1, 2 . . . . . n. Now, a transformation to new variables z i is sought: oo
yi=zi
+
~_, a i ( m ) z " ,
i = 1 , 2 . . . . . n,
(1.3)
Iml--_2
with m - ( m l , m 2 . . . . , m , ) , m i > O, for all i, and [ml-ml+m2+
"'" + m , ,
z m =- z ~ l . z~ '2 . . . z,m",
(1.4)
T. Bountis et al./ Normal form solutions of dynamical systems
36
such that the new equations Zi=~kiZi "+-
~ bi(m)z 'n,
i = 1,2 . . . . . n,
(1.5)
Iml>_2
will be easier to solve than (1.1) or (1.2) Of course, the ideal case would be bi(m) = 0 for all i and m in (1.5). This, however, only happens in the absence of all resonances, i.e. when
X i - ( m , X ) = X i - ~ mkXk¢O, i = 1 , 2 ..... n,
(1.6)
k=l
for all integer vectors m. The reason is that then one can directly evaluate all obtained by substituting (1.3) in (1.2), using (1.5), i.e.
[Xi--(m,X)]ai(m)=--~_,Ni, kt{8(m,txk+lzt)+ k,l
ai(m ) from
~_, ak(m')a,(m-rn')}+ ....
the equations
(1.7)
Ire'l>_2
since the coefficients of a~(m) would never vanish. (The delta function in (1.7) is 1 if m =/*k +/*/ and zero otherwise, /*k being the zero vector with its k t h entry replaced by 1.) Moreover, the convergence of the above series would be guaranteed (at least in a small neighborhood of the fixed point) provided the X~'s: (a) either belong to the Poincar6 domain, i.e. Re(Xi) < 0 (or Re(Xi) > 0) for all i, or (b) belong to the Siegel domain, i.e. 0 lies within the convex hull of ~k1 . . . . , Xn, in the complex X-plane and C
[Xi-(m,X)[>
[m["'
for s o m e c > 0 ,
v>_
n-2 2
'
(1.8)
for all Xi, whence no resonances are present, see [13, ch. 5]. Our first example fails within class (a) above and is free from resonances. Duffing's equation, on the other hand, when viewed as a 4-dimensional system is seen to contain an infinity of resonances. As a 2-dimensional system, however, with periodic coefficients, it has eigenvalues X1, Xe belonging to the Poincar6 domain and convergence can still be proved, see [13, section 26]. In the case of resonance, at some m, the coefficient ai(m) cannot be calculated from (1.7). Instead, we set it equal to zero, and satisfy the corresponding eq. (1.7) by an appropriate choice of hi(m), cf. (1.5), which implies that we have to include nonlinear terms in the normal form equations. Still, even if that makes it difficult to solve (1.5) exactly, we can derive from it, in many cases, useful information about the solutions in the neighborhood of the fixed point. In section 2 we apply the normal form analysis to an exactly solvable Lotka-Volterra system in the plane, near a nodal fixed point at the origin. We find that the normal form series converge over a large region around that point, with the boundary of this region determined by the condition that the normal form variables z, = D~exp(t) encounter a singularity of the solutions in the complex time plane. This connection between convergence regions of normal forms and complex t-singularities has also been found on other examples [18], whose normal form series (1.3) do not truncate after a finite number of terms. We suggest that, in cases where these singularities can be accurately located, they may offer a practical criterion by which the region of convergence of normal form solutions can be determined.
T. Bountis et al./ Normal form solutions of dynamical systems
37
In section 3, we concentrate on the higher dimensional, non-integrable Dufling's equation
5i + # y c = x - x 2 + ( x - 1 ) Q c o s t o t ,
#>o.
(1.9)
We show that normal forms indeed yield very useful estimates of the basin of attraction about a stable simple periodic orbit of (1.9), corresponding to a fixed point of the Poincar6 surface of section x(tk), :~(tk), at t k = k ( 2 ~ r / o ~ ) + t o, k = 0 , 1 , 2 . . . . . As has been recently demonstrated [6, 8], such basins of attraction can have boundaries with extremely complicated "fractal" shapes. It would, therefore, be important to develop further analytical techniques (such as those provided by normal forms) to approximate the size and extent of such basins of attraction, especially in the higher dimensional cases n > 3, of (1.1).
2. Planar Lotka-Volterra models
Before applying, however, the normal form analysis to higher dimensional systems, it is instructive to use it to solve some planar models of the form "~1 = a x l + XlX2'
3C2 =
bxz - xxx2,
ab > 0,
(2.1)
for which there is a nodal fixed point at (b, - a ) . In particular, we will solve here first the case a = b = 1, which has no resonances and whose exact solution is also available in closed form. Thus, we will be able to analyze completely the boundaries of convergence of the normal form series, and examine how they may be related to the singularities of the solutions of (2.1) in the complex t-plane. Let us therefore take a = b = 1 in (2.1) and observe that the origin (0,0) of this system is a node, about which the linear part of the equations is already diagonalized. Hence, (2.1) can be written immediately in the form
E
.9i=X,Yi +
N~,ktYkYt,
i=1,2,
(2.2)
k,l=l,2
with A1 =
~k 2 --~
1, Yi = xi, and
N,,kk=0,
NI,~t=NI,tk = ~,
N2,kl=N2,1k = _!z,
(2.3)
cf. (2.1). Note, now, that with 1~1= ~ 2 ~ 0, no resonances are present, cf. (1.6), for any choice of m I > 0 and m 2 > 0, with [ml - rnx + m2 >- 2, since we have for all m~: I)~i-- ( m , ) ~ ) 1 = I~ki- (ml)k 1 + m22~2)1> [•il = 1.
(2.4)
Thus upon changing to the z~ variables by Y i = Z i "~ 2 Iml>2
ai(m)z",
i=1,2,
(2.5)
T. Bountis et al./ Normal form solutions of dynamical systems
38
with z " = z~'lzT2 , cf. (1.3), the normal form of the equations will be purely linear:
Zi=XiZi=Zi,
i = 1,2,
(2.6)
i.e. all b, = 0 in (1.5), provided, of course, the infinite series (2.5) converges. Substituting (2.5) in (2.2), and using (2.6), we obtain the following equation for a i(m) upon equating the coefficients of z":
[hi- (m,X)]ai(m)=
--Ni,k, 8 ( m , ~ k +l'L/)-- 2Ni, ktak(m--lx/)--N~.k,
E
ak(m')at(m--m'),
Ire'l>2
(2.7) i = 1,2, where summation over all k, l = 1,2 is implied, /~1 = (1,0), ~2 is the delta function 1,
8(m'l'Lk+ttl)=
m = lUk +/~/,
=
(0,1), cf. (1.7), and 8(m,/~k +/~t)
(2.8)
O, m4:l.tk+l,L t.
Thus, from (2.7) we can now compute all the ai(m) since their coefficient on the left-hand side is bounded away from zero, due to (2.4). Having thus calculated (and stored) the values of the ai(m)'s up to a desired order, say hmL = 30, we return to (2.5), insert for z~ the solution of (2.6).
z i = D i e t,
i = 1,2,
D1, D 2 arb. const.,
(2.9)
and compute the Yi (and hence the xi) from
xi=yi=Diet+
~_, a i ( m ) D ? l D ~ 2 e (m,+"2)t,
i=1,2.
(2.10)
[m[>_2
And now we come to the question of convergence of the above series: We recall that the theorems of Poincar6 and others [11-13], in this case, guarantee this convergence over some region in the zl, z 2 (and x 1, x2) plane about (0, 0); they do not tell us, however, how large this region is. What we find here is that this region extends over a large domain of bounded (for all t < 0) motion around the origin of the z~, z 2 and xl, x 2 planes, shown here in the shaded regions of figs. l(a) and l(b) respectively. As we demonstrate below, the boundaries of these regions exactly correspond to values of t = t, at which the solutions have singularities in the complex time plane. To see this, let us note first that this system can be solved exactly and in closed form by
xl=
+ c l e x p ( c 2+ t + c l e t) l+exp(c2+qe, ) ,
cl e t
X2=l+_exp(c2+cxet),
(2.11)
where cl, c 2 are arbitrary constants specified by the initial conditions, and the choice of + depends on whether x 1 and x 2 have the same sign or not. These solutions become singular at t = t, values given by c 2 + c 1 exp ( t , ) = in,~,
n any integer.
(2.12)
T. Bountis et al./ Normalform solutions of dynamicalsystems
39
Z2
1'
A
\ -s ,(z~
1,,,, Xl
Fig. 1. Convergence regions (shown shaded) of normal form series for the planar model (2.1) with a = b = 1. (a) In the real z1, z2 (D1, D2) plane, with the boundary curves given by (2.18), and (b) the corresponding region in the xl, x 2 plane of the original variables. Thus, given initial conditions xi0 = x~(0), i = 1, 2, sufficiently close to the origin, we can invert eq. (2.10) to o b t a i n the corresponding D, = D,(xlo, X2o). Noting then that
C1 =XIO"~-X20,
c 2 = l n ( X l o l -- (XIo-I-X20), \ x20 /
(2.13)
the t , of (2.12) is determined for which the corresponding
zi* =Di(Xlo, X2o)e t°,
i = 1,2,
(2.14)
will be the " f i r s t " (i.e. the ones with the smallest magnitude) to cause the divergence of (2.5). Usually, this procedure will have to be carried out numerically since the location of the singularities and the relationship between the Di's and the initial conditions are not explicitly known, in general. In the case of o u r special example, however, we can p e r f o r m all of the above steps analytically, since, with the aid of
T. Bountiset aL/ Normalform solutionsof dynamicalsystems
40
(2.11), the normal form series (2.5) can be summed up in closed form [19]:
Z1
Z2e-z
xl= 1-z2(1-e-~)/z
'
x2= 1-z2(1-e-~)/z
'
z=-z 1+z2,
(2.15)
It is now simple to check, using (2.9), (2.13) and (2.15), at t = 0, that c I = D I + D 2,
(2.16)
c2=ln(D1/D2)
and substitute (2.16) in (2.12) to obtain In ~
n=0,+__1,+2 .....
+(Dl+D2)et*=in,~,
(2.17)
Setting the singularity at t, = 0 and using from (2.15) the expression for the singularity "surface" z ~ / z ~ = - exp [ - (z~* + z~' )1,
we can derive the intersection of this "surface" with the real D1, D 2 plane: (D 1 + D2) 2 - In2 ( D 1 / D 2 ) = ~r2,
D1D 2 > 0,
(2.18a)
D 1 + D 2 + InID1/D2J = O,
D1D 2 < 0.
(2.18b)
These expressions give precisely the two pairs of hyperbolas shown in the real z 1, z 2 (D 1,/)2) plane of fig. la. Put differently, after determining the Dx, D 2 that correspond to a given set of initial conditions, we can compute from (2.17) the value of t R = Re(t.) at which the solutions (2.10) will diverge: tR___ ½1n (
ln2 ( D1/D2) +'rr2 } (D, + D2) 2
'
D1D2 > 0,
(2.19a)
tR = l n [ l n I D 1 / D 2 { / ( D x + D2)I,
D1D2 < 0.
(2.19b)
These results have also been verified numerically by following D 1 / D 2 = a - " r a y s " in the plane of fig. l(a), and computing the radius of convergence of series (2.10) using a standard "root" test, for several values of a. Thus, the region between the hyperbolas (2.18) of fig. l(a) is transformed via (2.10) into the shaded region of fig. l(b). But then, one might ask, how can one obtain solutions of this system closer to the stable and unstable manifolds of the saddle point at (1, - 1)? Is it possible to use normal form series to construct such solutions explicitly? For simple, planar models such as this one, the answer is yes: Shift the origin of (2.1), with a = b = 1, to the point (1, - 1 ) by introducing Yl = xl + x 2,
Y2 ~--- X l
-- X2 -- 2
and note that the resulting equations for yl and Y2 have diagonalized linear parts: .Pl =Yl,
Y2 = -Y2 + ½(Yl2 - y ~ ) "
(2.20)
T. Bountis et al./ Normal form solutions of dynamical systems
41
Moreover, since the first equation is already linear, the normal form analysis of (2.10) is very simple, and is easily seen to converge over the full, unshaded area of the x 1 > 0, x 2 < 0 quadrant of fig. l(b), thus complementing nicely the region of convergence about the nodal point of the system. Still, eqs. (2.20) have eigenvalues ~1 = 1, ~2 = --1 which do not belong to the Poincar6 domain (see section 1) and give rise to an infinity of resonances:
Xi_(m,X)=o{m=(n+l,n), m = ( n , n + l),
i=1, i=2,
(2.21)
for every n = 1, 2, 3 . . . . . Since the problem is planar, however, no intersections of invariant manifolds are present and the normal form series turn out to also converge in the neighborhood of this hyperbolic fixed point. A similar situation arises in the normal form analysis of the conservative Lotka-Volterra system (2.1) with a = - b = 1, which has two fixed points: one saddle at (0, 0) and one center at ( - 1 , - 1 ) . Changing coordinates so that the center is located at the origin, we set xl = - 1 + x and x 2 = - 1 + y and write (2.1) in the form
.¢c= - y + xy, .9= x - xy.
(2.22)
This is the classic Lotka-Volterra model, with the well-known integral of the motion ( - 1 + x ) ( - 1 + y ) exp (x + y ) = const.
(2.23)
Its solutions about (0, 0) are all periodic, up to the stable and unstable manifolds of (1,1), but are not known explicitly as functions of t. We can construct these solutions, however, by normal form series expansions, which turn out to converge all the way up to the invariant manifolds of (1.1). To see how this is done, let us first diagonalize the linear part of (2.22) by changing to new variables Yl, Y2:
(2.24)
- i
i
Y2
Y2
~ 1/2
1/2i
] \ Y]
in terms of which (2.22) becomes
.~j = ~ . j j +
~_, Nj, k,YkYl,
J = 1,2,
(2.25)
k,l=1,2
with X1 = i, ~2 ~- - i and 1-i N1,11 = N2"22 = - - i i -
= -Ul,=,
Nj,12 = Nj,21 = 0,
1+i N2,11 = U l * = = - W -
(2.26)
=
where stars denote complex conjugation. Introducing now the normal form coordinates zj we write as before
yj=zj+ E aj(m) zm, Iml>-2
j=1,2,
(2.27)
T. Bountis et aL / Normal form solutions of dynamical systems
42
and note that a countable infinity of resonances occurs, as in (2.21) above, since Xj
-
( m , h)
=
(ml~kI + m2h2)
~kj -
-----0
(m 1 + m 2 >_ 2)
(with X1 = i, X2 = - i ) are satisfied for
mz=ml-1,
j=l: j=2:
mz=ml+l,
or or
rn=#l+n(/~l+/~2), m=/~2+n(/h+/~2),
(2.28a) n=l,2 .....
(2.28b)
with/~1 = (1, 0), /~2 = (0,1). Thus the normal form equations for this problem become
~,j = ~.jzj + zj ~ bj,~(zxz2) ~, j = 1,2,
(2.29)
s=l
where the coefficients bj, s will be so chosen that all resonant terms in the equations for aj(m) cancel. To this end, let us substitute (2.27) in (2.25) using (2.29) to obtain
[ ~ j - (m, h )]aj(m) = --Nj, k,~(m,~k + IXl) - 2Nj, kla,(m-- I~) + bj,,6(m, ~j + n(/q +/~z)) n-1
Im1-2
+ E aj(rn-s(btl+lx2))(m-s(Ixl+t~2),b,)-Nj,
kt E
s=l
ak(m')at(m-m'),
j=l,2,
(2.30)
Ire'l> 2
where b~ = (bx,~, b2,~), and n is the integer part of Irnl/2. Eqs. (2.30) can be solved for all aj(m), for which the m's do not satisfy conditions (2.28a, b), for j = 1 and j = 2 respectively. In the resonant cases, however, we may set: aj(m)=0,
m=lxj+n(l~l+l~2),
n=1,2 .....
(2.31)
and solve (2.30), at these values of m, for bj,,: n-1
bj,,= 2Nj, kta,(m -- I~k) -- ~_, aj(m -- s(t~l + ttz))( m -- s(l~l +It2), bs) s~l
Iml-2
+Nj, kt Y', a k ( m ' ) a , ( m - m '
).
(2.32)
Imq>_2 Before we attempt to solve the normal form equations (2.29) let us note from (2.24) that y~* = Y2, which implies z1* ---z2, and b*1 , s = b 2s, ,
s---1,2 .... •,
a~(ml, m 2 ) = a 2 ( m 2 , rnl)
'
(2.33)
cf. (2.27) and (2.29). If we now set zl = R exp (icp) ,
z2 = R exp ( - i c p ) ,
(2.34)
we can uncouple eqs. (2.29) into separate equations for R and % R=0,
~=1+
~ s~l
Im(bx,~)R2L
(2.35)
T. Bountis et al./ Normalform solutions of dynamical systems
43
y
i
"3
Fig. 2. Region of bounded (oscillatory) motion of the conservative Lotka-Volterra model (2.22). The closed curves were obtained by normal form series, which are seen to converge all the way to the straight line invariant rnattifolds of (1,1).
Since it turns out that Re(bx,,) = 0, for all s, and thus directly integrate (2.35) to obtain oo
R=R0=const.,
ep=tot+to,
to=l+
E Im(bx,~) R2s.
(2.36)
s=l
To study the convergence properties of these normal form solutions we work as follows: Pick a value of R o and use it in (2.36) to calculate the corresponding to. Substituting then in (2.27) one obtains (with t o = 0) Yl = Ro ei•t-{-
E
al(ml, mz)R~'~+"2 exp [i(m I - m2)tot],
(2.37)
Iml>2
with Y2 = Yx*, whence, from (2.24), the x, y immediately follow. These solutions are clearly oscillatory and rotate around the origin in both the Zl, z 2 and x, y planes. Taking successively larger and larger values of R 0 > 0 and 0 < cp < 2~r we plot the corresponding solution curves in the x, y plane, see fig. 2. Keeping, for example, in (2.37) terms up to Iml - 25, we find for R o = 0.5, to = 0.90442636, 6-digit agreement, and for R 0 = 0.7, to = 0.77264, 4-digit agreement, when comparing with the results of numerical integration of (2.22). In this conservative model, therefore, the normal form series solutions appear to converge over the whole domain of bounded motion around (0, 0), all the way to the invariant manifolds of the saddle point at (1.1). This may be explained by the fact that the problem is integrable, cf. (2.23), and its solutions are all periodic, given by convergent Fourier series expansions of the form (2.37), whose frequency depends on the initial conditions, cf. (2.36). It is important to note, however, that in more complicated, higher dimensional problems, if the eigenvalues of the linearized equations about a fixed point do not satisfy Re )~j < 0 (or Re )~i > 0) for all i, then, in the presence of resonances, normal form series solutions will most likely be divergent, with zero radius of convergence. If one concentrates, however, on dissipative systems, one finds that the convergence region of normal forms is large enough to provide useful estimates of the basin of attraction about a stable fixed point of the associated Poincar6 map.
44
T. Bountis et a l . / Normal form solutions o f dynamical systems
3. N o r m a l f o r m solutions of Dufling's equation
We now turn to the normal form analysis of the dissipative Duffing's equation 5/+ # 2 -
(3.1)
x + x 2 = ( x - 1)Qcos~0t,
/~ > 0. When Q = 0, (3.1) has an unstable equilibrium point at (0, 0) and a stable one at (1,0), in the x, plane. For small Q 4: 0, these turn into fixed points of the Poincar6 map of the system on its surface of section
2(tk)/tk=k2~r +,ol
~'°--=(x(tk),
(3.2)
O)
through which pass periodic orbits of period 2~r/~0 in the extended 3-dimensional phase space. At b o t h / t = 0 and Q = 0, the hyperbolic fixed point at (0,0) has stable and unstable manifolds, W ~ and W u respectively, which join smoothly, "embracing" the elliptic fixed point at (1,0). For ~t> 0, these manifolds split, and at sufficiently high Q, they intersect each other infinitely often, thus forming a very complicated boundary for the basin of attraction around the stable fixed point on Zt0 [2]. Our purpose here is to show that the region of convergence of normal forms extends sufficiently far from the stable fixed point to be able to capture most of its actual basin of attraction. Our results suggest, that this convergence region does not extend all the way to the intersecting W u and WS manifolds, due to the presence of singularities in the complex t-plane, as explained in section 2. In order to apply the normal form analysis to (3.1), we write it first as a fourth order system, shifting the stable fixed point to the origin, and get
"~1 ~ X 2 , 2 2 --= --I, l X 2 -- X 1 -- X 2 Jr- X I X 4 ,
X3 ~'~ 0"~X4,
X3(0 ) = 0,
3~4 " ~ - - 0~X3,
X4(0) = Q,
(3.3)
with x -= (x 1, x 2, x 3, x4) "r, this system can now be written as (3.4)
ic = A x + M x ,
where A, the matrix of its linear part, has eigenvalues 2tx= __~ + 2 r~__/x2 •
'
Xz =
/z 2
½¢4__/z2
'
X 3 = i~,
~4 ~
-i~,
(3.5)
and M = ( M u ) has all Mq= 0, except M21 = x 4 - x 1. Next we determine the matrix S that diagonalizes A, i.e.
SAS-I = A,
Aq=XiSij,
(3.6)
T. Bountis et al./ Normal form solutions of dynamical systems
45
and define the new variables y~ in y = (Yl, Y2, Y3, Y4)T by ~2 d
-d
0
0
Xl
-;kid
d
0
0
X2
0
0
1/2
-i/2
X3
0
0
1/2
i/2
X4
y=$x=
d-=
- Xl)
(3.7)
Substituting then x = S - l y in (3.4) and multiplying on the left by $ we obtain (Yl +Y2 - iY3 + iY4)(Yl +Y2)
p=Ay+
1
Ax_h 2
( - Y l -Y2 + iy3 + iY4)(Yl +Y:) 0 0
or, more compactly, 4
~j = Xjyj + E Nj. ktYkY,,
j=1,2,3,4,
(3.8)
k,l=l
with N I , l l -~- N1,12 = N1,22 = 1/(X 2 - Xl) , N1,14 = N1,24 = - N l , 1 3 -m- - N 1 , 2 3 = i/(2X 2 - 2Xx) ,
(3.8a)
N1,33 = N1,34 = N1,44 ~-- 0,
and
Nj, kt= Nj, tk , Nl, kt= --N2,k,,
N3,kt=N4,kt=0.
(3.8b)
Now, we are ready to introduce the normal form coordinates zj by writing
yj=zj+
Y~ aj(m)z m, j = 1 , 2 , 3 , 4 ,
(3.9)
Iml>2
cf. (1.3), (1.4), and go on to derive the equation of motion for the z/s. Before doing this, however, let us note that the hj's of this system, cf. (3.5), give rise to infinitely many resonances, ~'i - (m, X) = 0, at the
T. Bountis et al. / Normal form solutions of dynamical systems
46
following rn values:
m=(1,O,n,n), m=(O,l,n,n), m=(O,O,n+l,n), m = ( O , O , n , n + l),
j=l, j=2, j=3, j=4,
n >
1.
(3.10)
As we have explained already in sections 1 and 2, these resonances will introduce additional terms in the first two normal form equations
2 j = X j z j + z j Y'~ bj,~(z,z,)',
j=1,2,
(3.11)
s~l
while the third and fourth equations, being already linear, cf. (3.8), will remain unaffected, i.e.
~j=Xjzj,
aj(rn) = 0 ,
j=
3,4.
(3.12)
Substituting (3.9) and (3.10) above in (3.3) and equating coefficients of z m we obtain for j = 1, 2, [hi-
(rn,X)] a i ( m ) = - Nj, ktS( m, l~k + I~t) - 2Nj, ktat( m -- I~k) + bj,, 8( m, t~j + n(/~ 3 + / x , ) )
+ i aj(m -- s(/.t 3 +/.t4))(m -- ,(/tt 3 + ~4)' b,) s~l
-Nj, kt Y'~ ak(m')at(m--m'),
(3.13)
Ire'l>__2
where /~k is a 4-dimensional integer vector with 1 in the kth entry and 0 everywhere else, n = [m/2], b, = (bl, s, b2,,,0,0), and repeated indices imply summation over all k, l = 1,2,3,4. Thus all aj(m) can be computed from (3.13) except those with m = 2n + 1, cf. (3.10), for which we set aj(m)=0,
m=/~j+n(~t3+/~,),
j=1,2,
and choose the coefficient bj,, so that (3.13) is satisfied. At this point we are ready to solve the normal form equations of the problem: Starting with (3.12) we have z3(t ) = R e i°J(t-tD,
Za(t ) = R e -i'~u-',),
(3.14)
where t3, t 4 a r e constants to be determined by initial conditions at t = 0. These are given by inverting (3.7), using (3.3), y,(0)
=
y,(0)
=
x3(0 )
- -2
ix4(0 )
2
=-2
i
Q=z3(O)=Re-i~°t3'
--ix3(0) xa(0) iQ = z,(0) = R e i'°" 2 + ~ = '~
T. Bountis et al./ Normal form solutions of dynamical systems
F r o m these equations we deduce R -form
z3(t ) = (-iQ/2)e i~t, z4(t ) =
Q/2,
47
13 = ~r/2a~ and t 4 = 0 and write the solutions (3.14) in the
(3.15)
( i Q / 2 ) e -i°''.
Finally, from (3.15) we substitute in (3.11) and obtain equations for z 1 and z2:
~j---)~yz++ zj ~ by,,(a/2) 2",
j=1,2,
(3.16)
$=1
which are linear and can be readily solved, once the by,s are known, z1 = r l e x p ( ) q t ) ,
(3.17)
z2 = r2exp(-r2t),
Y1 -- Y2* = YR + iYI ------Xl + ~
(3.17a)
bl,s(Q//2) 2s,
s=l
where the relation 3'1 = ~'2" follows directly from the computation of the by,s- Moreover, since from (3.7) it is clear that Yl* = Y2, we also have zl*=z2,
.'.
q * = r 2 and with
rj=pyei°J,
j=l,2,
(3.18)
we conclude Pl = P2 and - 01 = 02. Thus, selecting small values for & and 01, at t = 0, and using (3.17) we can compute explicitly the yj's (and the xj's) at any later time t > 0, for which the series (3.9) will still converge. Let us take, for example, a uniform distribution of points in the z 1 (or z2) plane lying on a small circle of radius Pl = P 2 - 0.02, around the fixed point. They correspond, via the formulas (3.9) and (3.17), at t = 0, to an approximately circular distribution of points in the x 1, x 2 plane (see the innermost curves of figs. 3(a, b)). Now let us evaluate the series (3.9), with (3.17), for these points, at the t values t = tk =
2~k,
k = 0,5,10,15,20,23,
(3.19)
f o r / , = 0.1, ~ = 3 and Q = 0.3, whereupon we obtain the six closed curves plotted here in fig. 3a. We have stopped at k = 23 because this is approximately the t k value at which the convergence of our series, with terms up to [m I = 14, breaks down. What we can say, therefore, is that if we were to start with initial conditions chosen within the outermost curve of the x 1, x 2 plane of fig. 3(a) we would find that the normal form series accurately represents the solution of the problem, as it winds down to the fixed point, for all t > 0. To see whether this claim is justified we have taken the corresponding initial conditions of the innermost circle and integrated eq. (3.1) numerically, for t < 0, plotting the intersections of the orbits with the x 1, x 2 plane at the same t = t k values (3.19). The resulting set of curves obtained in this way are shown here in fig. 3(b). They are seen to agree very well with the normal form results of fig. 3(a), except, of course for the outermost curve, for which the normal form series did not converge, and discrepancies between the analytical and numerical approach became evident. In figs. 3(b, c) we have plotted this outermost curve, obtained by normal forms and numerical integration respectively, together with the intersecting invariant manifolds of the hyperbolic point. Thus, if
48
T. Bountis et aL/ Normal form solutions of dynamical systems
b
d
C
I
Fig. 3. Comparison between analytical (normal form) and numerical solutions of (3.1) near its stable fixed point, for g = 0.1, 0~= 3 and Q = 0.3. (a) A small "circle" of 20 initial conditions around (1,0) evaluated for t < 0 at the tk values of (3.19) via the normal form series (3.9). (c) The location of the invariant manifolds of the hyperbolic fixed point with respect to the outermost curve obtained in the above way at t23 -23 (2~r/~) where the series cease to converge. (b) and (d) are the same as (a) and (c) respectively, except that they were obtained by numerical integration of (3.1). =
we w e r e to view o u r n o r m a l f o r m result as a n e s t i m a t e of the b a s i n of a t t r a c t i o n of the stable fixed point, w e w o u l d h a v e c a p t u r e d , in this example, quite a large p o r t i o n of it. W e should remark, however, that the full b a s i n of a t t r a c t i o n is n o t c o n t a i n e d o n l y within the finite d o m a i n shown in figs. 3(c, d) b u t extends to i n f i n i t y a l o n g the stable manifold, while the u n s t a b l e m a n i f o l d of the h y p e r b o l i c fixed p o i n t has o s c i l l a t i o n s t h a t are a t t r a c t e d b y the stable fixed point. O n e c o u l d argue, of course that, h a d we i n c l u d e d m o r e terms in o u r series, their convergence region m i g h t b e s h o w n to b e larger t h a n i n d i c a t e d here, p e r h a p s even e x t e n d to the i n v a r i a n t m a n i f o l d s of the u n s t a b l e fixed point. A l t h o u g h we c a n n o t c o m p l e t e l y rule o u t this possibility, we e m p h a s i z e that this
T. Bountis et al./ Normalform solutions of dynamical systems
49
certainly does not happen with the integrable example (2.1) with a = b = 1 and other similar models we have studied elsewhere [18]. As was explained earlier, all these dynamical systems have singularities in the complex t-plane, which may prevent convergence long before the solution reaches a basin boundary as t increases in the negative direction. An obvious exception, of course, occurs if the normal form series happen to truncate, exactly, after a finite number of terms, in which case complex t-singularities would be irrelevant. However, even in cases where we expect these singularities to be relevant, if the problem is higher than two-dimensional and non-integrable, it will typically have an infinity of singularities, which often accumulate on complicated patterns in the complex t-plane [20]. This would make it difficult to tell a priori, which singularity (or singularities) will be responsible for the divergence in each case. Still, since the convergence of series in more than one complex variable is difficult to study in general, looking at singularities in the complex t-plane offers an alternative approach, especially in problems where these singularities can be accurately located and are distributed sparsely enough, so that their individual effect on the normal form series can be determined.
4. Concluding remarks Our main purpose in this paper was to demonstrate on simple systems of nonlinear ODE's the applicability and usefulness of normal forms in obtaining solutions about fixed points of planar models, or, around stable, simple closed orbits of periodically driven systems, with 3-dimensional phase space. Along the way, we discovered some interesting connections between the convergence properties of normal forms and the singularities of the solutions in the complex t-plane. In our planar examples, we were able to make this connection precise, and obtained large regions of convergence around different fixed points of one non-conservative model. These regions of convergence were seen to complement each other, while the boundary between them was determined by the location of the complex t-singularities of the solution. In the case of a conservative (and integrable) planar system, on the other hand, the corresponding "arrangement" of the singularities permitted the convergence of normal forms over a full, (arbitrarily long) period of the motion and thus yielded all bounded solutions around the origin up to the invariant manifolds of a nearby saddle point. Applying then these methods to the neighborhood of a stable fixed point of the Poincar6 map of a periodically driven Duffing's equation, we obtained useful estimates of the basin of attraction around that point. These estimates did not extend all the way to the actual basin boundary, because the normal form series were numerically seen to diverge, before the solutions reached the invariant manifolds in the neighborhood of a nearby hyperbolic fixed point. We suggest that the convergence of normal form series is connected with the location of movable singularities of the solutions in the complex t-plane. After all, these singularities limit the convergence of an expansion of the solution in powers of t, while the normal form series are expanded in powers of exponentials of t. The predictive value of this observation, however, is questionable, if the location of the singularities is not accurately known, or their number is too large to permit, in practice, an investigation of which one(s) of them cause divergence. In any case, we expect the application of normal forms to the basins of attraction of dynamical systems to prove quite useful, especially in problems with several fixed points, which are separated by complicated basin boundaries. Such problems are currently under study and more results concerning the convergence properties of normal forms will be given in future publications.
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T. Bountis et al./ Normal form solutions of dynamical systems
Acknowledgements T w o of u s (T.B. a n d G . T . ) wish to t h a n k A. R a u h a n d J. P a d e for several h e l p f u l c o n v e r s a t i o n s c o n c e r n i n g n o r m a l forms, a n d a c k n o w l e d g e the h o s p i t a h t y of the Physics D e p a r t m e n t of the U n i v e r s i t y of O l d e n b u r g , w h e r e p a r t of this w o r k was c a r r i e d out.
References [1] M. Lieberman and A. Lichtenberg, Regular and Stochastic Motion, Appl. Math. Sci., vol. 38 (Springer, Berhn, 1983). [2] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Appl. Math. Sci., vol. 42 (Springer, Berlin, 1983). [3] R.H.G. Helleman, in: Fundamental Studies in Statistical Mechanics, vol. 5, E.G.D. Cohen, ed. (North-Holland, Amsterdam 1981). [4] Chaotic Behaviour of Deterministic Systems, Proc. of Les Houches 1981 Summer School, G. Iooss, R.H.G. Helleman and R. Stora, eds. (North-Holland, Amsterdam, 1984). [5] J. Ford, Physics Today (April 1983), p. 40. [6] C. Grebogi, S.W. McDonald, E. Ott and J.A. Yorke, Phys. Lett. A99 (1983) 415; see also Physica D 17 (1985) 125. [7] C. Grebogi, E. Ott and J.A. Yorke, Phys. Rev. Lett. 56 (10) (1986) 1011; see also refs. listed therein. [8] F.C. Moon and G.-X. Li, Phys. Rev. Lett. 5 (1985) 1439. [9] T. Bountis, V. Papageorgiou and P. Wintemitz, J. Math. Phys. 27 (1986) 1215. [10] A. Ramani, B. Dorizzi, B. Grammaticos and T. Bountis, J. Math. Phys. 25 (1984) 878; see also Physica A 128 (1984) 268. [11] A.D. Brjuno, Analytical form of differential equations, Trans. Moscow Math. Soc. 25 (1971) 131. [12] V.M. Starzhinski, Applied Methods in the Theory of Nonlinear Oscillations (MIR, Moscow, 1980) English transl. [13] V.I. Arnol'd, Geometrical Methods in the Theory of Ordinary Differential Equations (Springer, Berlin, 1983). [14] A. Chenciner, in ref. [4] above. [15] T. Valkering, Physica D 18 (1986) 483. [16] G. Tsarouhas, Normal form of time dependent chemical rate equations for irradiation produced point defects, Phys. Lett. All6 (6) (1986) 264. [17] J. Pade, A. Rauh and G. Tsarouhas, Application of normal form theory to the Lorenz model in the subcritical region, Physica D 29 (1987) 236. [18] T. Bountis and G. Tsarouhas, On the apphcation of normal forms near attracting fixed points of dynamical systems, Physica A 153 (1988) 160. [19] A. Rauh, private communication. [20] T. Bountis, V. Papageorglou and M. Bier, On the singularity analysis of intersecting separatices in near-integrable dynamical systems, Physica D 24 (1987) 292.