Instability studies on the stochastic dynamical system

Abstract

Colored noise effects on the stochastic dynamical system were studied. We have find the method to solve the Fokker-Planck equation and the Langevin equation driven by colored noise. Our formalism can be applied to the Parametric oscillator with noise squeezing phenomena. Correlation of noise expanded the region in which the system dynamics would be bounded. But it does not change the critical behavior of the system. Dynamical complexity measurement indicator for the stochastic dynamical system was defined through the current velocity. Traditional definition of the stochastic Lyapunov exponent can not explain the contribution of noise. Because of the non-differentiable manifold of the stochastic dynamical system, we need to smooth out the stochastic manifold. And, to make precise definition of the fine-grained entropy, we used the Reimmanian geometry. In the consideration of the geometry, stochastic system can have a vanishing entropy production rate. Our indicator also can used for the phase space compressibility of the Reimannian space. Finally we have calculated the information entropy production rate of the molecular motor system. Production of the information entropy is essential for the non equilibrium transport phenomena.