Special Session 17: 

Complex dynamics in a ODE model related to phase transition

Fabio Zanolin
University of Udine
Italy
Co-Author(s):    Duccio Papini
Abstract:
Motivated by some recent studies on the Allen--Cahn phase transition model with a periodic non-autonomous term, we prove the existence of complex dynamics for the second order equation $$-\ddot{x} + (1 + \varepsilon^{-1} A(t)) G`(x) = 0,$$ where $A(t)$ is a non-negative $T$-periodic function and $\varepsilon > 0$ is sufficiently small. In particular, we find a full symbolic dynamics made by solutions which oscillate between any two different strict local minima $x_0$ and $x_1$ of $G(x).$ Our approach is based on a variant of the theory of topological horseshoes.