This is a self-contained introduction to the classical theory of homoclinic bifurcation theory, as well as its generalizations and more recent extensions to higher dimensions. It is also intended to stimulate new developments, relating the theory of fractal dimensions to bifurcations, and concerning homoclinic bifurcations as generators of chaotic dynamics. The book begins with a review chapter giving background material on hyperbolic dynamical systems. The next three chapters give a detailed treatment of a number of examples, Smale's description of the dynamical consequences of transverse homoclinic orbits, and a discussion of the subordinate bifurcations that accompany homoclinic bifurcations, including Hénon-like families. The core of the work is the investigation of the interplay between homoclinic tangencies and non-trivial basic sets. The fractal dimensions of these basic sets turn out to play an important role in determining which class of dynamics is prevalent near a bifurcation. The authors provide a new, more geometric proof of Newhouse's theorem on the co-existence of infinitely many periodic attractors, one of the deepest theorems in chaotic dynamics.
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Book Description
Editorial Reviews
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"...an interesting monograph on this collection of ideas. It begins with material often covered in graduate texts but quickly moves to the exposition of ideas only available in the original sources. It would be quite suitable for an advanced graduate level course in dynamics and bifurcation theory." John Franks, Bulletin of the American Mathematical Society
Book Description
A self-contained introduction to the classical theory of homoclinic bifurcation theory, and its generalizations and more recent extensions to higher dimensions. It is also intended to stimulate new developments, relating the theory of fractal dimensions to bifurcations, and chaotic dynamics.
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First Sentence:
In this chapter we give background information and references to the literature concerning basic notions in dynamical systems that play an important role in our study of homoclinic bifurcations. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
unstable limit capacities, bounded distortion property, persistent tangencies, hyperbolic periodic attractors, many coexisting sinks, persistent homoclinic tangencies, homoclinic tangency, linearizing coordinates, maximal invariant set, maximal invariant subset, surface diffeomorphisms, unstable separatrix, infinitely many sinks, nonwandering set, first tangency, unstable cones, unstable foliations, unstable separatrices, positive iterates, transverse homoclinic intersections, strong stable manifold, premature creation, negative limit set, expanding eigenvalue, transverse homoclinic orbits New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
In this chapter we give background information and references to the literature concerning basic notions in dynamical systems that play an important role in our study of homoclinic bifurcations. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
unstable limit capacities, bounded distortion property, persistent tangencies, hyperbolic periodic attractors, many coexisting sinks, persistent homoclinic tangencies, homoclinic tangency, linearizing coordinates, maximal invariant set, maximal invariant subset, surface diffeomorphisms, unstable separatrix, infinitely many sinks, nonwandering set, first tangency, unstable cones, unstable foliations, unstable separatrices, positive iterates, transverse homoclinic intersections, strong stable manifold, premature creation, negative limit set, expanding eigenvalue, transverse homoclinic orbits New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 5
books:
- Dynamical Systems: Valparaiso 1986 : Proceedings of a Symposium Held in Valparaiso (Lecture Notes in Mathematics) by R. Bamon in Back Matter (1), and Back Matter (2)
- Dynamical Systems (London Mathematical Society St) by Albert Fathi on page 1, and Back Matter
- Rhode Island (13 Colonies) by Roberta Wiener in Back Matter
- Ordinary Differential Equations (Classics in Applied Mathematics) by Philip Hartman in Back Matter
- Ordinary Differential Equations by Garrett Birkhoff in Back Matter
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