On perturbations of generalized Landau-Lifshitz dynamics

TL;DR

This paper studies how small deterministic and stochastic perturbations affect systems with conservation laws, including the Landau-Lifshitz equation, revealing that bifurcations can induce stochastic behavior in the slow dynamics.

Contribution

It introduces a framework for analyzing perturbations of conservation law systems, especially addressing bifurcations and their impact on the limiting slow motion, including stochastic effects.

Findings

Bifurcations lead to stochastic limiting processes.

The Landau-Lifshitz equation is a special case within this framework.

Effective description of the slow motion in perturbed systems.

Abstract

We consider deterministic and stochastic perturbations of dynamical systems with conservation laws in $\R^3$. The Landau-Lifshitz equation for the magnetization dynamics in ferromagnetics is a special case of our system. The averaging principle is a natural tool in such problems. But bifurcations in the set of invariant measures lead to essential modification in classical averaging. The limiting slow motion in this case, in general, is a stochastic process even if pure deterministic perturbations of a deterministic system are considered. The stochasticity is a result of instabilities in the non-perturbed system as well as of existence of ergodic sets of a positive measure. We effectively describe the limiting slow motion.