homoclinic_2005 - Chaotic Dynamics Near Transverse...

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Chaotic Dynamics Near Transverse Homoclinic Points CDS140B Lecturer: Wang Sang Koon winter, 2005 The Study of Deterministic Chaos. Despite the fact that the system is deterministic, it has the property that imprecise knowledge of the intial condition may lead to unpredictability after some finite time. We will cover Symbolic Dynamics which is the paradigm for deterministic chaos. Conley-Moser Conditions which allow one to verify the existence of Smale Horseshoe - like dynamics and chaos. Homoclinic Orbits and Heteroclinic Cycles where horseshoe-like dynamics exists and where the whole machinery of symbolic dynamics can make this chaotic behavior more precise. Conley-Moser Conditions for a general two-dimensional map to have an invariant set on which it has the shift dynamics. These conditions are a combination of geometric and analytical condi- tions. The geometrical part consists of generalizing the notion of horizontal and vertical rectangles by allowing the boundaries to be Lipschitz curves, rather than straight lines. horizontal “rectangles” map to “vertical” rectangles,
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This note was uploaded on 01/04/2012 for the course CDS 140b taught by Professor List during the Fall '10 term at Caltech.

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homoclinic_2005 - Chaotic Dynamics Near Transverse...

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