Local and Semi-Local Bifurcations in Hamiltonian Dynamical Systems: Results and Examples

ISBN-13: 9783540388944 / Angielski / Miękka / 2006 / 264 str.

Heinz Hanbmann

Opis

This book demonstrates that while elliptic and hyperbolic tori determine the distribution of maximal invariant tori, they themselves form n-parameter families. Therefore, torus bifurcations of high co-dimension may be found in a single given Hamiltonian system, absent untypical conditions or external parameters. The text moves logically from the integrable case, in which symmetries allow for reduction to bifurcating equilibria, to non-integrability, where smooth parametrisations must be replaced by Cantor sets.

Recenzje

From the reviews:

"This book deals with bifurcations of invariant tori in Hamiltonian systems, in particular in near-integrable systems. ... The book closes with a series of appendices, in which technical or more fundamental details are summarised ... . This well-written monograph represents a welcome contribution to the literature in this field-I am not aware of a similarly comprehensive and systematic treatment of bifurcations of invariant tori. I recommend the book to any researchers or graduate students working in this field." (Thomas Wagenknecht, Mathematical Reviews, Issue 2008 a)

Spis treści

Bifurcations of Equilibria.- Bifurcations of Periodic Orbits.- Bifurcations of Invariant Tori.- Perturbations of Ramified Torus Bundles.- Planar Singularities.- Stratifications.- Normal Form Theory.- Proof of the Main KAM Theorem.- Proofs of the Necessary Lemmata.

Blurb

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.