Symbolic Dynamics of Homoclinic Orbits in a Symmetric Map

Details Symbolic Dynamics of Homoclinic Orbits in a Symmetric Map Symbolic Dynamics of Homoclinic Orbits in a Symmetric Map

2002-01-17

arxiv.org/abs/nlin/0201031v1

Nonlinear Sciences

Chaotic Dynamics

11 pages 5 figures

Scientific paper

Symbolic dynamics for homoclinic orbits in the two-dimensional symmetric map, $x_{n+1}+cx_{n}+x_{n-1}=3x_{n}^3$, is discussed. Above a critical $c^{\ast}$, the system exhibits a fully-developed horse-shoe so that its global behavior is described by a complete ternary symbolic dynamics. The relative location of homoclinic orbits is determined by their sequences according to a simple rule, which can be used to numerically locate orbits in phase space. With the decrease of $c$, more and more pairs of homoclinic orbits collide and disappear. Forbidden zone in the symbolic space induced by the collision is discussed.

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