Hyperbolic Sets of Ordinary Differential Equations
In this chapter we develop the theory of hyperbolic sets for flows. First we show that the continuity of the splitting into stable and unstable bundles follows from the other items in the definition. Next we develop the theory of exponential dichotomies for linear differential equations, paying special attention to the roughness theorem. We use the latter to prove that hyperbolic sets are expansive both in a “continuous” way and a “discrete” way. Finally we show that hyperbolic sets are robust under perturbation, our major tool here being Lemma 2.17.
KeywordsNormal Bundle Implicit Function Theorem Fundamental Matrix Exponential Dichotomy Satisfying Inequality
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