Symbolic dynamics of the cubic map

Abstract

A symmetric cubic map on the interval describing isolated stable or bistable orbits and different kind of bifurcation was analyzed. Stable orbits were presented by itineraries, permutation parentheses and matrices. Itineraries were used to order the cycles with respect to the parameter and also to order the coordinates of the cycle in the interval. Linear symbolic dynamics was introduced to study the periodic unstable points coexisting with stable orbits by means of a matrix of intervals. The non-zero entries of these matrices have an N-shape corresponding to the N-shape of the cubic. The characteristic polynomial of the matrix of intervals was in one-to-one correspondence with the itinerary. Equations to calculate the number of different kind of orbits were explicitly written and used to calculate those numbers for a small number of periods.