Dynamical Systems IX: Dynamical Systems with Hyperbolic Behaviour: 66 (Encyclopaedia of Mathematical Sciences) - Hardcover

 
9783540570431: Dynamical Systems IX: Dynamical Systems with Hyperbolic Behaviour: 66 (Encyclopaedia of Mathematical Sciences)
Hardcover
ISBN 10: 3540570438 ISBN 13: 9783540570431
Verlag: Springer, 1995

Inhaltsangabe

This volume is devoted to the "hyperbolic theory" of dynamical systems (DS), that is, the theory of smooth DS’s with hyperbolic behaviour of the tra­ jectories (generally speaking, not the individual trajectories, but trajectories filling out more or less "significant" subsets in the phase space. Hyperbolicity the property that under a small displacement of any of a trajectory consists in point of it to one side of the trajectory, the change with time of the relative positions of the original and displaced points resulting from the action of the DS is reminiscent of the mot ion next to a saddle. If there are "sufficiently many" such trajectories and the phase space is compact, then although they "tend to diverge from one another" as it were, they "have nowhere to go" and their behaviour acquires a complicated intricate character. (In the physical literature one often talks about "chaos" in such situations. ) This type of be­ haviour would appear to be the opposite of the more customary and simple type of behaviour characterized by its own kind of stability and regularity of the motions (these words are for the moment not being used as a strict ter­ 1 minology but rather as descriptive informal terms). The ergodic properties of DS’s with hyperbolic behaviour of trajectories (Bunimovich et al. 1985) have already been considered in Volume 2 of this series. In this volume we therefore consider mainly the properties of a topological character (see below 2 for further details).

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The book deals with smooth dynamical systems with hyperbolic behaviour of trajectories filling out "large subsets" of the phase space. Such systems lead to complicated motion (so-called "chaos"). The book begins with a discussion of the topological manifestations of uniform and total hyperbolicity: hyperbolic sets, Smale's Axiom A, structurally stable systems, Anosov systems, and hyperbolic attractors of dimension or codimension one. There are various modifications of hyperbolicity and in this connection the properties of Lorenz attractors, pseudo-analytic Thurston diffeomorphisms, and homogeneous flows with expanding and contracting foliations are investigated. These last two questions are discussed in the general context of the theory of homeomorphisms of surfaces and of homogeneous flows.

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Verlag
Springer
Erscheinungsdatum
1995
Sprache
Englisch
ISBN 10 
3540570438
ISBN 13 
9783540570431
Einband
Tapa dura
Anzahl der Seiten
252
Herausgeber
Anosov D.V.
Kontakt zum Hersteller
Nicht verfügbar
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Weitere beliebte Ausgaben desselben Titels

9783642081682: Dynamical Systems IX: Dynamical Systems with Hyperbolic Behaviour: 66 (Encyclopaedia of Mathematical Sciences)

Vorgestellte Ausgabe

ISBN 10:  3642081681 ISBN 13:  9783642081682
Verlag: Springer Berlin Heidelberg, 2010
Softcover