Complex Dynamics in a ODE Model Related to Phase Transition

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 equation −x ̈+(1+ε−1A(t))G′(x)=0, where 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 x0 and x1 of G(x). Such solutions stay close to x0 or x1 in some fixed intervals, according to any prescribed coin tossing sequence. For convenience in the exposition we consider (without loss of generality) the case x0=0 and x1=1.