January  2003, 9(1): 69-95. doi: 10.3934/dcds.2003.9.69

Heteroclinic orbits and chaotic invariant sets for monotone twist maps

1. 

Department of Mathematics, Northwestern University Evanston, IL 60208, United States

2. 

Department of Mathematics, Northwestern University, Evanston, Illinois 60208, United States

Received  September 2002 Published  November 2002

We consider the monotone twist map $\bar f$ on $(\mathbb R/\mathbb Z)\times R$, itslift $f$ on $R^2$ and its associated variational principle $h:\mathbb R^2\to\mathbb R$ through its generating function. By working with the variationalprinciple $h$, we first show that for an adjacent minimal chain$\{(u^k, v^k)\}_{k=s}^t$ of fixed points of $f$, if there exists abarrier $B_k$ for each adjacent minimal pair $u^k < u^{k+1}$, $ s \le k \le {t-1} $, then there exists a heteroclinic orbit between $(u^s, v^s)$ and$(u^t, v^t)$, then by assuming that there is a barrier for any twoneighboring globally minimal critical points and $m$ is sufficientlylarge, we construct an invariant set $\Lambda^m\subset (\mathbb R/\mathbb Z)\times\mathbb R$ such that the shift map of the $n$-symbol space is a factor of$\bar f^m|_{\Lambda^m}$, where $n$ is the total number of the globallyminimal fixed points of $\bar f$.
Citation: Tifei Qian, Zhihong Xia. Heteroclinic orbits and chaotic invariant sets for monotone twist maps. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 69-95. doi: 10.3934/dcds.2003.9.69
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