Selection of a stochastic Landau-Lifshitz equation and the stochastic persistence problem

Abstract

In this article, we study the persistence of properties of a given classical deterministic differential equation under a stochastic perturbation of two distinct forms: external and internal. The first case corresponds to add a noise term to a given equation using the framework of Itô or Stratonovich stochastic differential equations. The second case corresponds to consider a parameters dependent differential equations and to add a stochastic dynamics on the parameters using the framework of random ordinary differential equations. Our main concerns for the preservation of properties is stability/instability of equilibrium points and symplectic/Poisson Hamiltonian structures. We formulate persistence theorem in these two cases and prove that the cases of external and internal stochastic perturbations are drastically different. We then apply our results to develop a stochastic version of the Landau-Lifshitz equation. We discuss in particular previous results obtain by Etore and al. in [28] and we finally propose a new family of stochastic Landau-Lifshitz equations.

Hybrid quantum-classical chaotic NEMS

“…In such hybrid systems any small perturbation destroys the regular motion and leads to unpredictable evolution of the system. In the first case, the stochasticity appears to be externally, and in the second case, it is an intrinsic property of the system [3]. For instance, the kicked rotator in the classical regime, when the strength of kicking is increased, the regular periodic motion is destroyed and chaotic motion is observed.…”