High-frequency chaotic solutions for a slowly varying dynamical system

Abstract

In this article we study the asymptotic dynamics of highly oscillatory solutions for the unbalanced Allen-Cahn equation with a slowly varying coefficient. We describe the underlying structure of these solutions through a function we call the adiabatic profile, which accounts for the asymptotic area covered by the Solutions in the phase space. In finite intervals, we construct solutions given any adiabatic profile. In the case of a periodic coefficient we show that the system has chaotic behavior by constructing high-frequency complex solutions which can be characterized by a bi-infinite sequence of real numbers in [c(1), c(2)] boolean OR {0} (0 < c(1) < c(2)).