Local and semi-local bifurcations in Hamiltonian dynamical systems

Heinz Hanβmann (auth.)9783540388944, 354038894X

Once again KAM theory is committed in the context of nearly integrable Hamiltonian systems. While elliptic and hyperbolic tori determine the distribution of maximal invariant tori, they themselves form n-parameter families. Hence, without the need for untypical conditions or external parameters, torus bifurcations of high co-dimension may be found in a single given Hamiltonian system. The text moves gradually from the integrable case, in which symmetries allow for reduction to bifurcating equilibria, to non-integrability, where smooth parametrisations have to be replaced by Cantor sets. Planar singularities and their versal unfoldings are an important ingredient that helps to explain the underlying dynamics in a transparent way.

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Table of contents :
Front Matter….Pages I-XV
Introduction….Pages 1-15
Bifurcations of Equilibria….Pages 17-89
Bifurcations of Periodic Orbits….Pages 91-107
Bifurcations of Invariant Tori….Pages 109-142
Perturbations of Ramified Torus Bundles….Pages 143-159
Planar Singularities….Pages 161-165
Stratifications….Pages 167-171
Normal Form Theory….Pages 173-184
Proof of the Main KAM Theorem….Pages 185-200
Proofs of the Necessary Lemmata….Pages 201-206
Back Matter….Pages 207-241