The Quasi-Periodic Centre-Saddle Bifurcation

Abstract

Nearly integrable families of Hamiltonian systems are considered in the neighbourhood of normally parabolic invariant tori. In the integrable case such tori bifurcate into normally elliptic and normally hyperbolic invariant tori. With a KAM-theoretic approach it is shown that both the normally parabolic tori and the bifurcation scenario survive a non-integrable perturbation, parametrised by pertinent large Cantor sets. These results are applied to rigid body dynamics.1998 Academic Press INTRODUCTIONA simple example of a centre-saddle bifurcation (of equilibria) is the nonlinear oscillator x +x 2 =*; Fig. 2.1 below shows how the phase portrait changes as the parameter * varies. How does this bifurcation behave in Hamiltonian systems with several degrees of freedom? As additional degrees of freedom lead to the superposition with a (quasi)-periodic motion, the equilibria in Fig. 2.1 get replaced by lower dimensional tori. Already in two degrees of freedom the (periodic) centre-saddle bifurcation becomes a generic phenomenon, the ro^le of the parameter * being played by the value of the energy. Correspondingly, examples are abundant, e.g., the He non Heiles system, the Kovalevskaya top and the second fundamental model of resonance to name but a few. While equilibria and periodic orbits can be addressed with the implicit mapping theorem, the bifurcating tori in three or more degrees of freedom involve small denominators. Let us put this problem into context. Given a non-degenerate integrable Hamiltonian system, we know from KAM-theory that most maximal invariant tori survive a small (Hamiltonian) perturbation, cf. [2,36,12] or references therein. These maximal tori are the regular fibres of the ramified torus bundle defined by the integrable system. The singular fibres of the ramified torus bundle, i.e. the lower article no. DE973365 305 0022-0396Â98 25.00

Perturbations of Superintegrable Systems

“…Indeed, the same ratio and the whole bifurcation scenario may fall into a resonance gap. However, for most values of parameters s 3 , I 1 , I 3 the quasi-periodic centre-saddle bifurcations turn out to be present in the original dynamics [28].…”

“…This makes the phase space a ramified torus bundle, the regular fibres of n-tori forming families that are organized by families of invariant (n − 1)-tori, which in turn are organized by bifurcating (n − 1)-tori and by (n − 2)-tori and so on, down to the equilibria of the Hamiltonian system. In the case n = d of Lagrangean tori one can apply KAM theory directly to the integrable system, yielding persistence of the Lagrangean tori [2,8,20], of elliptic/hyperbolic lower dimensional tori [7,8,35] and of invariant tori 1 Mathematisch Instituut, Universiteit Utrecht, 3508 TA Utrecht, The Netherlands that undergo bifurcations [10,28,29]. Such results are valid under non-degeneracy conditions [30], the best-known being Kolmogorov's non-degeneracy condition which expresses that the frequency mapping from the families of Lagrangean tori to R d is a submersion.…”

On Energy Surfaces and the Resonance Web

“…For the parabolic case we have mainly numerical indications for the behavior of the perturbed orbits; initial steps of a rigorous analysis of instabilities associated with PR are presented in [39] (a longer detailed version is in preparation). The behavior near a nonresonant parabolic torus does not yield instability-the lower dimensional parabolic torus persists [24]-and it appears that the behavior near it is indistinguishable from that appearing near the lower dimensional normally elliptic torus. However, numerical simulations indicate that the behavior near PR is dramatically different; orbits which appear to be chaotic and of a different nature than the homoclinic chaos are abundant.…”

“…Indeed, when λ p f is real and nonvanishing the corresponding family of tori is said to be normally hyperbolic, when it vanishes it is called normally parabolic, and when it is pure imaginary it is normally elliptic. For more details on the above see [33,2,5,12,13,14,15,21,22,23,24,26,44,3] and references therein. 4 Notice that a single torus belonging to this family has neutral stability in the actions directions.…”

Quasi-periodic bifurcations in reversible systems

“…We still have to show that the tori (15) persist under a small perturbation of (14). As only the relative sign between a(y 0 ) and b(y 0 ) is relevant we fix sgn b(y 0 ) = +1.…”