symbolic_2005

symbolic_2005 - Introduction to Chaos and Symbol Dynamics...

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Introduction to Chaos and Symbol Dynamics CDS140B Lecturer: Wang Sang Koon winter, 2005 1 Introduction to Chaos. 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. Example: Look at the phase space of a pendulum and a pendulum with perodical forcing. Notice the complexity of the homoclinic tangle. Overview: 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. 1
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2 Symbolic Dynamics and the Shift Map Phase Space. The phase space for the shift map Σ is the space of “bi-infinite” sequences of 0’s and 1’s, with a specific metric. Two sequences are “near each other” if they are identical on a long
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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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symbolic_2005 - Introduction to Chaos and Symbol Dynamics...

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