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

$$begin{aligned} -ddot{x} + left( 1 + varepsilon ^{-1} A(t)right) G'(x) = 0, end{aligned}$$

where A(t) is a nonnegative T-periodic function and (varepsilon > 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).