DIFFERENTIATING INVARIANT MANIFOLDS OF DYNAMICAL SYSTEMS WITH APPLICATIONS TO MELNIKOV THEORY.

Title: DIFFERENTIATING INVARIANT MANIFOLDS OF DYNAMICAL SYSTEMS WITH APPLICATIONS TO MELNIKOV THEORY.
Identifier: AAI8629714
identifier: 8629714
Creator: MILLER, WALTER MCPHERSON.
Contributor: Michael Shub
Date: 1986
Language: English
Publisher: City University of New York.
Subject: Mathematics
Abstract: Parameterized versions of the stable-unstable manifold theorem assert that if a family of dynamical systems depend smoothly on a parameter then a corresponding subfamily of invariant manifolds of these dynamical systems also vary smoothly with respect to the parameter. In applications Melnikov functions are constructed to detect dynamically significant geometric interaction of these manifolds, especially those leading to chaotic behaviour.;Here we formulate the derivative of variation of persistent invariant manifolds of dynamical systems with respect to parameters. In general these formulations are valid in Banach spaces.;We show how these derivative formulations lead naturally to Melnikov functions. Applications are given. For example we use these functions to give necessary and sufficient conditions for global bifurcation of degenerate intersection of stable and unstable manifolds to transversality.
Type: dissertation
Source: PQT Legacy CUNY.xlsx
degree: Ph.D.
Program: Mathematics

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