The properties of motion close to the transition of a stable family of
periodic orbits to complex instability is investigated with two symplectic 4D
mappings, natural extensions of the standard mapping. As for the other types of
instabilities new families of periodic orbits may bifurcate at the transition;
but, more generally, families of {\sl isolated invariant curves} bifurcate,
similar to but distinct from a Hopf bifurcation. The evolution of the stable
invariant curves and their bifurcations are described.Comment: 5 pages, self-unpacking uuencoded compressed Postscript, Contribution
  at the NATO ASI Conference on "Hamiltonian Systems with Three or More Degrees
  of Freedom, Barcelona, Spain, June 19-30, 199

Ollé, Mercè

Pfenniger, Daniel

English

arXiv

In an autonomous Hamiltonian system with three or more degrees of freedom, a family of periodic orbits may become unstable when two pairs of characteristic multipliers coallesce on the unit circle at points not equal to ±1, and then move off the unit circle. This paper develops normal forms suitable for the neighbourhood of such an, instability, and, at this approximation, demonstrates the bifurcation from the periodic orbit of a family of invariant two-dimensional tori. The theory is illustrated with numerical computations of orbits of the planar general three-body problem

Ollé Torner, Mercè

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chao-dyn/9508008   31 Aug 1995BIFURCATION AT COMPLEX INSTABILITYMERCE OLLE & DANIEL PFENNIGERObservatoire de GeneveCH-1290 Sauverny, SwitzerlandAbstract. The properties of motion close to the transition of a stable family of periodicorbits to complex instability is investigated with two symplectic 4D mappings, naturalextensions of the standard mapping. As for the other types of instabilities new familiesof periodic orbits may bifurcate at the transition; but, more generally, families of isolatedinvariant curves bifurcate, similar to but distinct from a Hopf bifurcation. The evolutionof the stable invariant curves and their bifurcations are described.1. IntroductionComplex instability is a type of generic instability in Hamiltonian systems with more than2 degrees of freedom. It was described and named so by Broucke[2]in a study of periodic or-bits in the elliptic RTBP, in which he classied all the possible instability types of periodicorbits in 3-degree of freedom Hamiltonian systems. In real problems, it appears in celestialmechanics[5]and galactic dynamics[3];[4];[8];[10];[11];[12], in particle accelerator problems[7], andactually, in many engineering problem with 3 or more degrees of freedom. On the otherhand, complex instability has been studied by mathematical techniques only during aboutthe last decade[1];[6];[9].In this paper, we are concerned with the intricacies in phase space associated to thetransition from stability to complex instability. We refer to the numerical study of thistransition as well as the description of the bifurcating objects done by Pfenniger[10]. Weshall briey review those results and we shall concentrate, specially, on the evolution ofthe stable bifurcating invariant curves when varying some parameter. There appear twotypes of bifurcations resembling the bifurcations of periodic orbits: the `period-doubling'bifurcation and the bifurcation of a pair of asymmetric invariant curves.2. Symplectic mappingsAs well known, Hamiltonian systems can be transformed into symplectic mappings bymeans of the Poincare sections. In 3-degree of freedom systems, there corresponds a 4Dsection space. Thus, let us consider a symplectic mapping T , 0 a xed point, A the Jacobianmatrix around the origin, and , 1=, , and 1= the eigenvalues of A.From stability, there are only three routes to instability as a system parameter ischanged: 1) A pair of eigenvalues , on the unit circle in the complex plane bifurcates2onto the real axis at (1; 0) {tangent bifurcation{; 2) the same occurring at ( 1; 0) {perioddoubling bifurcation{; 3) the four eigenvalues collide simultaneously by conjugate pairson the unit circle and leave it into the complex plane; this is the transition to complexinstability, also called Krein collision. Let us denote k the rotation number correspondingto the ratio between the modulus of the eigenvalues and 2. If k is rational, k = p=q,where q is the smallest positive integer, then q periodic orbits may bifurcate, becausethe eigenvalues are then integer roots of 1. But, usually, k is irrational and there are nobifurcating families of periodic orbits but of invariant curves at the transition point.In order to describe the transition from stability to complex instability we have used[10]two 4D natural generalizations of the standard map inspired by the symplectic Froeschlemap, the mappings Tsand Tt,Ts0BB@x1x2x3x41CCA=0BB@D[x1+K1sin(x1+x2) + L1sin(x1+x2+x3+x4)]x1+ x2D[x3+K2sin(x3+x4) + L2sin(x1+x2+x3+x4)]x3+ x41CCA(mod 2) (1)Tt0BB@x1x2x3x41CCA=0BB@D[x1+K1sin(x1+x2) + L1tan(x1+x2+x3+x4)]x1+ x2D[x3+K2sin(x3+x4) + L2tan(x1+x2+x3+x4)]x3+ x41CCA(mod 2) (2)The dissipation parameter D is only used for computing purposes. When D < 1, D = 1,D > 1 the mappings are respectively volume contracting, preserving and dilating. Westudy the mappings around x = 0; they have the same Jacobian A but dierent non-linearproperties as described below. We restrict the parameter space by taking L1=  L2 L,K1 K and K2= 0; the transition takes place when L =  K=4  Lcritin the interval 8 < K < 0. L is a varying parameter and K assumed not to depend on L (see [10] for adiscussion of the parameters).3. Transition stability-complex instabilityStriking phase structures appear at the transition from stability to complex instability.The main results are that, as in classical pitchfork bifurcations, the bifurcating structuresmay be \direct" (the bifurcating objects unfold on the unstable side), or they may be\inverse" (the bifurcating objects unfold on the stable side) as the parameter L is varied.For the mapping Ts(Tt), the bifurcation is direct (inverse), and the bifurcating objectsexist for L > Lcrit(L < Lcrit).When the rotation number k is rational, then we nd q-periodic orbits which bifurcate.In [10], a detailed description of such periodic orbits and their stability properties are given.When the rotation number k is irrational, stable (unstable) invariant curves bifurcatefor the mapping Ts(Tt). In order to nd the stable invariant curves for the mapping Ts,we exploit the property that they become attracting limit cycles as soon as dissipation(D < 1) occurs. Starting atD < 1, the method consists in progressively increasing D whilecontrolling the convergence of the consequents onto the invariant curves. When D = 1, wehave reached the desired invariant curve. Fig. 1 shows such an example of bifurcation.We are now interested in describing the typical bifurcations along of a family of stableinvariant curves when varying some parameter (L). Let us x a value of K such that wehave an irrational k and let us study the family of stable invariant curves of Tswhich3Figure 1. Projections of the invariant curves in mapping Ts, K =  1, for values L = 0:26 (inner one),L = 0:30, L = 0:35 (outer one) (Lcrit= 0:25)Figure 2. Evolution of the stable bifurcating invariant curves for mapping Ts, K =  1, Lcrit= 0:25bifurcates at the transition. In order to describe it, we dene, for a given invariant curve,x20the value of x2> 0 when \x1 0", that is, when jjx1jj  ,  a small positive quantity.At rst, the invariant curves are stable, but as L increases, the stability characterchanges and there appear other bifurcations (see Fig. 2, where a representation of theinvariant curves by their x20value when varying L is given). In particular, we distinguishtwo dierent regions in Fig. 2, corresponding to two dierent kinds of bifurcations. Wedescribe region 1 (Fig. 3) with some detail: we notice, rst, a deviation of the family ofstable invariant curves, for L > 0:3805; there appear slightly unstable invariant curves.We also remark a gap between L = 0:387 and L = 0:388, which is due to a resonanceassociated to a family of 28-periodic orbits, which has a transition from stability to complexinstability; thus, we may expect new bifurcating families of invariant curves there, whichwe have not followed yet. Finally, there is a change of stability (by a pitchfork bifurcation)and there bifurcate \double-periodic" invariant curves (the invariant curve makes 2 turnsbefore closing in section space) apart from the central invariant curve which becomesunstable. Fig. 4 shows a `double-periodic' invariant curve. Some branches of invariantcurves in this gure remain to be continued; presumably they become unstable and ournumerical method is unable to determine unstable invariant curves.We consider now region 2 (Fig. 2). Again a change of stability occurs together with anew pitchfork bifurcation. It has, however, a dierent meaning here: for a xed value of L,4Figure 3. Bifurcations of the invariant curves in region 1Figure 4. 'Double-periodic' invariant curve in Region 2, mapping Ts, K =  1, L = 0:395L > 0:435, we have three values of x20, the external ones which correspond to two dierentasymmetric stable invariant curves and the central one which is the central invariant curvewhich has become unstable. We show two asymmetric invariant curves in Fig. 5.The eect of the bifurcation on the chaotic neighbourhood of the central periodic familycan be summarized as follows.Figure 5. Two asymmetric bifurcated invariant curves; mapping Ts, K =  1, L = 0:445Figure 6. Nekhoroshev estimation for mIn the direct case (mapping Ts), the chaotic orbits originating around the centralcomplex-unstable orbit are conned for a long time. As an example and in order to give aNekhoroshev estimation of the powerful connement, we x a number of iterates N (sayN = 103), we x the initial conditions close to the origin (say xi= 0:01, for i = 1; 2; 3; 4),we dene d = maxnNjjTns(x)k, x = (x1; x2; x3; x4) and we compute d when varying theparameter L, for a K xed. Afterwards, we compute the necessary number m of iteratesin order to have, for each value of L, a value of d, d0, such that d0 1:01d. For K =  1,we observe a huge number of iterates m between Lcrit= 0:25 < L < 0:27, and we can tm by a Nekhoroshev expression n  exp[0:08418=(L  Lcrit)1:37385], for 0:27  L  0:305(see Fig. 6). To summarize, invariant curves and conned chaotic orbits coexist.In the inverse case (mapping Tt), the transition to chaos is immediate and no con-nement occurs. The invariant tori surrounding the stable central family dissolve beforethe transition to complex instability. See [10] for the numerical details to reach the lastinvariant torus by means of the dissipation parameter.Acknowledgments M. Olle thanks Prof. G. Gomez for many useful discussions, andDr. P. Comas for his help in all computer and software issues. M. Olle is partially supportedby DGICYT grants PB90-0580 and PR95-071, and D. Pfenniger by the Swiss FNRS.M. Olle's usual address: E.T.S.E.I.B., Dept. Matematica Aplicada I, Diagonal 647,08028 Barcelona, Spain.References1. Bridges, T.J. (1990) Math. Proc. Camb. Phil. Soc. 108, 575{601; ibid: (1991) 109, 375{403.2. Broucke, R. (1969) AIAA J., 7, 1003{1009.3. Contopoulos, G., Magnenat, P. (1985) Celest. Mech., 37, 3874. Contopoulos, G., Barbanis, B. (1994) Cel. Mech. and Dyn. Astron., 59, 279{3005. Hadjidemetriou, J.D. (1975) Cel. Mech., 12, 255{2766. Heggie, D.G. (1985) Cel. Mech., 35, 357{3827. Howard, J.E., Lichtenberg, A.J., Lieberman, M.A., Cohen, R.H. (1986) Physica, 20D, 259{2848. Magnenat, P. (1982) Cel. Mech., 28, 319{343.9. Van der Meer (1985) The Hamiltonian Hopf Bifurcation. Lecture Notes in Math., 1160, Springer-V.10. Pfenniger, D. (1985) Astron. Astrophys., 150, 97{111; ibid. 150, 112{12811. Pfenniger, D., Martinet, L. (1987) Astron. Astrophys., 173, 81{8512. Pfenniger, D. (1987) Astron. Astrophys., 180, 79{93

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