# American Institute of Mathematical Sciences

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October  2010, 14(3): 1181-1197. doi: 10.3934/dcdsb.2010.14.1181

## Existence of transversal homoclinic orbits in higher dimensional discrete dynamical systems

 1 Department of Applied Mathematics, National Chiayi University, Chiayi, Taiwan 2 Department of Applied Science, Naval Academy, Zuoying District, Kaohsiung City, 813, Taiwan

Received  August 2009 Revised  April 2010 Published  July 2010

A rigorous numerical proof for establishing existence of a transversal homoclinic orbit for a saddle fixed point with higher dimensional unstable eigenspaces is presented. As the first component of this method, a shadowing theorem that guarantees the existence of such a homoclinic orbit near a suitable pseudo orbit given the invertibility of a certain Jacobian is proved. The second component consists of a refinement procedure for numerically computing a pseudo homoclinic orbit with sufficiently small local errors so as to satisfy the hypothesis of the theorem. The third component verifies that the homoclinic orbit is transversal. In [6], they proved the existence of transversal homoclinic orbits near anti-integrable limits and near singularities for the Arneodo-Coullet-Tresser maps. In this paper, the existence of transversal homoclinic orbits were proved far away from anti-integrable limits and singularities for these maps.
Citation: Chen-Chang Peng, Kuan-Ju Chen. Existence of transversal homoclinic orbits in higher dimensional discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1181-1197. doi: 10.3934/dcdsb.2010.14.1181
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