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MiS Preprint
79/2000
Gradient systems with wiggly energies and related averaging problems
Govind Menon
Abstract
Gradient systems with wiggly energies of the form $$\dot{x} =-\nabla (F(x)+\epsilon A(\frac{x}{\epsilon})),\ \ \ x \in R^d $$ and $A:T^d \to R$ were proposed by Abeyaratne, Chu and James to study the kinetics of martensitic phase transitions. Their model may be recast in the framework of the theory of averaging as a dynamical system on $R^d \times T^d$, with the slow variable $x \in R^d$ and fast variable $\theta \in T^d$. However, this problem lies completely outside the classical theory of averaging, since the vertical flow on $T^d$ is not ergodic for sets of positive measure, and we must interpret averages to mean weak limits. We obtain rigorous averaging results for d=2. We use Schwartz's generalisation of the Poincaré-Bendixson theorem to heuristically derive homogenized equations for the weak limits. These equations depend on the $\omega$-limit sets for the vertical flow on fibres. When the vertical flow is structurally stable, we use the persistence of hyperbolic structures to prove that these are the correct equations. We combine these theorems with a study of two parameter bifurcations of flows on $T^2$ to characterize the weak limits. Our results may be interpreted as follows. $R^2$ breaks into: (1) a bounded open set surrounding $\{ \nabla F^{-1} (0)\}$ where there is only sticking, (2) a transition region outside this set, where the dynamics is a combination of sticking and slipping, and (3) the rest of the plane, which contains a countable number of resonance zones, with nonempty interior, and their nowhere dense complement. Inside a resonance zone the direction of the weak limits is given by the rotation number $\rho \in Q$. The Cantor set structure of the resonance zones is described by well-known results of Arnol'd and Herman in the theory of circle diffeomorphisms. Consequently, the homogenized equations vary on all scales. We also study the linear transport equation associated to the wiggly gradient flow, and show that its homogenization limit is not well-posed. V.P. Smyshlyaev has studied this problem independently, and some of our results are similar.
