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.
Bibliographical noteFunding Information:
This work is supported in part by NSF Grants DMS-0803287 and DMS-0854982.
- Averaging principle
- Landau-Lifshitz equation
- Magnetization dynamics
- Stochasticity in deterministic systems