Infinitely many co-existing sinks from degenerate homoclinic tangencies
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- by Gregory J. Davis PDF
- Trans. Amer. Math. Soc. 323 (1991), 727-748 Request permission
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 $\{ {F_t}\}$ of diffeomorphisms in two dimensions. It is assumed that each family has a curve of dissipative periodic saddle points, ${P_t};\;F_t^n({P_t}) = {P_t}$, and $|\det DF_t^n({P_t})| < 1$. We also require the stable and unstable manifolds of ${P_t}$ to form homoclinic tangencies as the parameter $t$ varies through ${t_0}$. Our emphasis is the exploration of the behavior of families of diffeomorphisms for parameter values $t$ near ${t_0}$. We show that there are parameter values $t$ near ${t_0}$ at which ${F_t}$ has infinitely many co-existing periodic sinks.References
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Additional Information
- © Copyright 1991 American Mathematical Society
- 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