Complex Dynamics in a ODE Model Related to Phase Transition

Abstract

Motivated by some recent studies on the Allen-Cahn phase transition model with a periodic nonautonomous term, we prove the existence of complex dynamics for the second order equationwhere A(t) is a nonnegative T -periodic function and ε > 0 is sufficiently small. More precisely, we find a full symbolic dynamics made by solutions which oscillate between any two different strict local minima x 0 and x 1 of G(x). Such solutions stay close to x 0 or x 1 in some fixed intervals, according to any prescribed coin tossing sequence. For convenience in the exposition we consider (without loss of generality) the case x 0 = 0 and x 1 = 1.