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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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