Infinitely many co-existing sinks from degenerate homoclinic tangencies

Author:
Gregory J. Davis

Journal:
Trans. Amer. Math. Soc. **323** (1991), 727-748

MSC:
Primary 58F15

DOI:
https://doi.org/10.1090/S0002-9947-1991-0982238-7

MathSciNet review:
982238

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Abstract: The evolution of a horseshoe is an interesting and important phenomenon in Dynamical Systems as it represents a change from a nonchaotic state to a state of chaos. As we are interested in determining how this transition takes place, we are studying certain families of diffeomorphisms. We restrict our attention to certain one-parameter families of diffeomorphisms in two dimensions. It is assumed that each family has a curve of dissipative periodic saddle points, , and . We also require the stable and unstable manifolds of to form homoclinic tangencies as the parameter varies through . Our emphasis is the exploration of the behavior of families of diffeomorphisms for parameter values near . We show that there are parameter values near at which has infinitely many co-existing periodic sinks.

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DOI:
https://doi.org/10.1090/S0002-9947-1991-0982238-7

Article copyright:
© Copyright 1991
American Mathematical Society