Symbolic Dynamics of Homoclinic Orbits in a Symmetric Map

Abstract

Symbolic dynamics for homoclinic orbits in the two-dimensional symmetric map, $x_{n+1}+c{x}_n+x_{n-1}=3x_n^3$, 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.

11 pages 5 figures