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Local and Semi-Local Bifurcations in Hamiltonian Dynamical Systems

Results and Examples

  • Book
  • © 2007

Overview

Authors:
  1. Heinz HanĪ²mann

    1. Mathematisch Instituut, Universiteit Utrecht, TA Utrecht, The Netherlands

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Part of the book series: Lecture Notes in Mathematics (LNM, volume 1893)

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Table of contents (10 chapters)

  1. Front Matter

    Pages I-XV
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  2. Introduction

    Pages 1-15

  3. Bifurcations of Equilibria

    Pages 17-89

  4. Bifurcations of Periodic Orbits

    Pages 91-107

  5. Bifurcations of Invariant Tori

    Pages 109-142

  6. Perturbations of Ramified Torus Bundles

    Pages 143-159

  7. Planar Singularities

    Pages 161-165

  8. Stratifications

    Pages 167-171

  9. Normal Form Theory

    Pages 173-184

  10. Proof of the Main KAM Theorem

    Pages 185-200

  11. Proofs of the Necessary Lemmata

    Pages 201-206

  12. Back Matter

    Pages 207-241
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Keywords

About this book

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.

Reviews

From the reviews:

"This book deals with bifurcations of invariant tori in Hamiltonian systems, in particular in near-integrable systems. ā€¦ The book closes with a series of appendices, in which technical or more fundamental details are summarised ā€¦ . This well-written monograph represents a welcome contribution to the literature in this fieldā€”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)

Authors and Affiliations

  • Mathematisch Instituut, Universiteit Utrecht, TA Utrecht, The Netherlands

    Heinz HanĪ²mann

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