Existence and Uniqueness of Solution for a Class of Stochastic Differential Equations

A class of stochastic differential equations given by dx(t)=f(x(t))dt+g(x(t))dW(t),  x(t0)=x0,  t0≤t≤T<+∞, are investigated. Upon making some suitable assumptions, the existence and uniqueness of solution for the equations are obtained. Moreover, the existence and uniqueness of solution for stochastic Lorenz system, which is illustrated by example, are in good agreement with the theoretical analysis

Dynamics of Stochastic Coral Reefs Model with Multiplicative Nonlinear Noise

Little seems to be known about the ergodicity of random dynamical systems with multiplicative nonlinear noise. This paper is devoted to discern asymptotic behavior dynamics through the stochastic coral reefs model with multiplicative nonlinear noise. By support theorem and Hörmander theorem, the Markov semigroup corresponding to the solutions is to prove the Foguel alternative. Based on boundary distributions theory, the required conservative operators related to the solutions are further established to ensure the existence a stationary distribution. Meanwhile, the density of the distribution of the solutions either converges to a stationary density or weakly converges to some probability measure

Noise and Delay Induced on Dynamics of the Aeroelasticity Model

The purpose of this paper is to investigate the stochastic bifurcation and stability problem of the Aeroelasticit-y of two-dimensional supersonic lifting surfaces with delay term. Applying Hopf bifurcation theory, Lyapunov exponent and invariant measure theory, we analyze the D- and P-bifurcation of the stochastic system. The analysis is based on the reduction of the infinite-dimensional problem to one described on a two-dimensional stochastic center manifol

Asymptotic Behavior of the Stochastic Rayleigh-van der Pol Equations with Jumps

We study the stability, attractors, and bifurcation of stochastic Rayleigh-van der Pol equations with jumps. We first established the stochastic stability and the large deviations results for the stochastic Rayleigh-van der Pol equations. We then examine the existence limit circle and obtain some new random attractors. We further establish stochastic bifurcation of random attractors. Interestingly, this shows the effect of the Poisson noise which can stabilize or unstabilize the system which is significantly different from the classical Brownian motion process

Deterministic and Stochastic Bifurcations of the Catalytic CO Oxidation on Ir(111) Surfaces with Multiple Delays

The main purpose is to investigate both deterministic and stochastic bifurcations of the catalytic CO oxidation. Firstly, super-and subcritical bifurcations are determined by the signs of the Poincaré-Lyapunov coefficients of the center manifold scalar bifurcation equations. Secondly, we explore the stochastic bifurcation of the catalytic CO oxidation on Ir(111) surfaces with multiple delays according to the qualitative changes in the invariant measure, the Lyapunov exponent, and the stationary probability density of system response. Some new criteria ensuring stability and stochastic bifurcation are obtained