Poincare-Birkhoff theorem

If the resonant tori, which are the invariant tori whose rotational numbers are rational, are broken under perturbations, the pairs of elliptic and hyperbolic cycles are created in the resonance zone. This fact is known as a result of the Poincare-Birkhoff theorem [4], which holds only if the twist condition, Eq. (2), is satisfied. Around elliptic cycles thus created, new types of tori, which are... 

The hierarchy of tori is theoretically predicted by the Poincare-Birkhoff theorem [10] in nearly integrable systems with two degrees of freedom. For instance, the hierarchy in the Henon-Heiles system represented by the Hamiltonian is shown in Fig. 1. 

Self-similarity is expected to be one of the important concepts to understand statistics and motion in Hamiltonian systems. However, the Poincare-Birkhoff theorem and the two models introduced by Aizawa et al. and Meiss et al. are based on the two-dimensionality of the phase spaces, and they cannot be directly applied to high-dimensional systems. As far as I know, existence of the self-similarity has not been clearly exhibited, since visualizing the selfsimilarity is not easy due to the high-dimensionality of Poincare sections, which has 2N — 2 dimension for systems with N degrees of freedom. 

But what is the nature of motions that do not lie on KAM tori This question is answered by the Poincare-Birkhoff theorem. This theorem tells us that motions that do not lie on KAM tori pinch off into periodic orbits which generally alternate between two general types. Some periodic orbits are stable (motion near the orbit tends to stay close to it), whereas others are unstable (motion near the orbit tends to diverge away from it). Unstable peri-... 

On a resonant invariant curve, out of the infinite set of r-multiple fixed points, only a finite (even) number survive, half stable and half unstable, as a consequence of the Poincare-Birkhoff fixed point theorem (Arnold and Avez, 1968 Lichtenberg and Lieberman, 1983), as shown schematically in Figure 14b. 

According to the Poincare-Birkhoff fixed point theorem all resonant tori break up for arbitrarily small perturbations. If the rotation number is p/q the perturbation leaves q pairs of hyperbolic and elliptic periodic orbits. The unstable hyperbolic orbits are embedded in a layer filled by aperiodic, chaotic orbits that do not stay on an invariant surface, but cover a finite non-zero volume of the phase space in a chaotic layer around the original resonant torus. The elliptic points, however, are wrapped around by new concentric tori that form islands of regular orbits within the chaotic band (Fig. 2.5). 

The classical and most common attempt, already discussed by Poincare, consists in trying to remove all dependencies on the angles from the Hamiltonian. This is usually called the normal form of Birkhoff. Such a normal form turns out to be particularly useful when the unperturbed Hamiltonian is linear, i.e., in (1) we have Ho(p) = (u),p) with some lo R". Therefore we illustrate the theory in the latter case. However, with some caveat, the method is useful also in the general case see Section 4.3 on Nekhoroshev s theorem. 

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