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2008 International Workshop on Dynamical System and Realted Topics
(2008 / vol.10 / no.2)
CHAOS, SHADOWING AND HOMOCLINIC ORBITS
KEN PALMER
Pages. 1-26   



The subject of this article is discrete dynamical systems or, more
precisely, di®eomorphisms in $Bbb R^n$. We describe the notion of a hyperbolic set,
the most important property of which is the shadowing property. We give
a proof of the shadowing theorem. Then we show how shadowing can be
used to prove that chaos occurs near a transversal homoclinic orbit. Finally
we show that shadowing can be used to give computer-assisted proofs of the
existence of such orbits. This is a report of my own published work, alone
or in collaboration with B.A.Coomes and H.Ko»cak. However the proof of the
Shadowing Theorem (Version 2) given below is new.



1. Hyperbolic Sets and Shadowing
2. Transversal homoclinic points
3. Finding Homoclinic Orbits by Numerical Shadowing
4. Other Aspects of Numerical Shadowing



Hyperbolic sets, transversal homoclinic orbits, pseudo orbits, chaos, shadowing, Newton's method



58F22, 58F13, 65H10