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.
Categories: nlin.CD