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April  2005, 12(3): 465-480. doi: 10.3934/dcds.2005.12.465

Homoclinic tangencies in $R^n$

Victoria Rayskin 1,

1. 

Department of Mathematics, 520 Portola Plaza, Box 951555, University of California, Los Angeles, CA 90095-1555, United States

Received  October 2003 Revised  September 2004 Published  December 2004

Let $f: M \rightarrow M$ denote a diffeomorphism of a smooth manifold $M$. Let $p \in M$ be its hyperbolic fixed point with stable and unstable manifolds $W_S$ and $W_U$ respectively. Assume that $W_S$ is a curve. Suppose that $W_U$ and $W_S$ have a degenerate homoclinic crossing at a point $B\ne p$, i.e., they cross at $B$ tangentially with a finite order of contact.
It is shown that, subject to $C^1$-linearizability and certain conditions on the invariant manifolds, a transverse homoclinic crossing will arise arbitrarily close to $B$. This proves the existence of a horseshoe structure arbitrarily close to $B$, and extends a similar planar result of Homburg and Weiss [10].
Keywords: Homoclinic tangency, horseshoe structure., order of contact, $\lambda$-Lemma, invariant manifolds.
Mathematics Subject Classification: 37B10, 37C05, 37C15, 37D1.
Citation: Victoria Rayskin. Homoclinic tangencies in $R^n$. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 465-480. doi: 10.3934/dcds.2005.12.465
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