Hyperbolic Periodic Orbits of Ordinary Differential Equations, Stable and Unstable Manifolds and Asymptotic Phase
We wish to develop a theory for autonomous systems of ordinary differential equations analogous to the theory we have developed for diflfeomorphisms in Chapters 1 through 5. It turns out that the object analogous to the fixed point of a diflfeomorphism is a periodic solution rather than an equilibrium point. To some extent, we can reduce the study of a periodic solution to that of the fixed point of a diflfeomorphism by using the Poincaré map. However, first we begin by recalling a few elementary facts from the theory of ordinary differential equations.
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