Shadowing in Dynamical Systems: Theory and Applications

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Developing the theory of hyperbolic sets both for diffeomorphisms and flows, with an emphasis on shadowing, this text shows that hyperbolic sets are expansive and have the shadowing property. It then uses shadowing to prove that hyperbolic sets are robust under perturbation, that they have an asymptotic phase property and also that the dynamics near a transversal homoclinic orbit is chaotic. It turns out that chaotic dynamical systems arising in practice are not quite hyperbolic. However, they possess enough hyperbolicity to enable us to use shadowing ideas to give computer-assisted proofs that computed orbits of such systems can be shadowed by true orbits for long periods of time, that they possess periodic orbits of long periods and that it is really true that they are chaotic.