Chaos Near Resonance

Overview

This book offers the first systematic exposition of recent analytic results that can be used to understand and predict the global effect of resonances in phase space. The geometric methods discussed here enable one to identify complicated multi-time-scale solution sets and slow-fast chaos in physical problems. The topics include slow and partially slow manifolds, homoclinic and heteroclinic jumping, universal global bifurcations, generalized Silnikov-orbits and -manifolds, disintegration of invariant manifolds ...
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Paperback (Softcover reprint of the original 1st ed. 1999)
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Overview

This book offers the first systematic exposition of recent analytic results that can be used to understand and predict the global effect of resonances in phase space. The geometric methods discussed here enable one to identify complicated multi-time-scale solution sets and slow-fast chaos in physical problems. The topics include slow and partially slow manifolds, homoclinic and heteroclinic jumping, universal global bifurcations, generalized Silnikov-orbits and -manifolds, disintegration of invariant manifolds near resonances, and high-codimension homoclinic jumping. The main emphasis is on near-integrable dissipative systems, but a separate chapter is devoted to resonance phenomena in Hamiltonian systems. A number of applications are described from the areas of fluid mechanics, rigid body dynamics, chemistry, atmospheric science, and nonlinear optics. In addition, the theory is extended to infinite dimensions to cover resonances in certain nonlinear partial differential equations, such as single and coupled nonlinear Schrodinger equations.. "This self-contained monograph will be useful to the applied scientist who wishes to analyze resonances in complex physical problems, as well as to mathematicians interested in the geometric theory of multi- and infinite-dimensional dynamical systems.
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Editorial Reviews

From the Publisher
"An extensive bibliography and the many examples make this clearly-written book an excellent introduction to these techniques for identifying chaos in perturbations of systems with resonance."
Applied Mechanics Reviews, Vol. 53/4, April 2000

"Haller makes a point of wanting to see dynamical systems theory fulfil "its long-standing promise to solve real-life problems". His book, through a wealth of detailed examples, delivers on this promise, and is certain ti become a standard text in this area. In particular, it is an excellent introduction to this research area, and contains a wealth of bibliographical and historical detail.
Matthew Nicol, Bulletin of the LMS, No. 162, Vol. 33/3, May 2001

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Product Details

  • ISBN-13: 9781461271727
  • Publisher: Springer New York
  • Publication date: 4/30/2013
  • Series: Applied Mathematical Sciences Series , #138
  • Edition description: Softcover reprint of the original 1st ed. 1999
  • Edition number: 1
  • Pages: 430
  • Product dimensions: 6.14 (w) x 9.21 (h) x 0.91 (d)

Table of Contents

Preface
1 Concepts From Dynamical Systems 1
2 Chaotic Jumping Near Resonances: Finite-Dimensional Systems 56
3 Chaos Due to Resonances in Physical Systems 159
4 Resonances in Hamiltonian Systems 231
5 Chaotic Jumping Near Resonances: Infinite-Dimensional Systems 286
App. A Elements of Differential Geometry 371
App. B Some Facts From Analysis 390
References 401
Symbol Index 421
Index 423
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