Symbolic Dynamics of Homoclinic Orbits in a Symmetric Map
Abstract
Symbolic dynamics for homoclinic orbits in the two-dimensional symmetric map, xn+1+cxn+xn-1=3xn3, is discussed. Above a critical c, 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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