Entropy and Lyapunov Exponents Relationships in Stochastic Dynamical Systems

Stochastic differential equations and classical techniques related to the Fokker-Planck equation are standard bases for the analysis of nonlinear systems perturbed by noise, such as seismic wave propagation in random media and response of structures to turbulent wind. In this paper, a complementary approach based on entropy production is proposed to analyse the stochastic stability of dynamical systems. For a large class of stochastic dynamical systems, it is shown that the entropy information production is equal to the negative sum of Lyapunov exponents as the noise strength tends to zero. This result is correlated to the topological entropy property, which is in some cases such as the hyperbolic case, equal the sum of Lyapunov exponents. Several examples are given to illustrate the proposed procedure.