Effect of bounded noise on chaotic motions of stochastically perturbed slowly varying oscillator

The effect of bounded noise on the chaotic behavior of a class of slowly varying oscillators is investigated. The stochastic Melnikov method is employed and then the criteria in both mean and mean-square sense are derived. The threshold amplitude of bounded noise given by stochastic Melnikov process is in good comparison with one determined by the numerical simulation of top Lyapunov exponents. The presence of noise scatters the chaotic domain in parameter space and the larger noise intensity results in a sparser and more irregular region. Both the simple cell mapping method and the generalized cell mapping method are applied to demonstrate the effects of noises on the attractors. Results show that the attractors are diffused and smeared by bounded noise and if the noise intensity increases, the diffusion is exacerbated.     

Measure attractors and random attractors for stochastic partial differential equations

In the theory of stochastic differential equations we can distinguish between two kinds of attractors. The first one is the attractor (measure attractor) with respect to the Markov semigroup generated by a stochastic differential equation. The second meaning of attractors (random attractors) is to be understood with respect to each trajectory of the random equation. The aim of this paper is to bring together the two meanings of attractors. In particular, we show the existence of measure attractors if random attractors exist. We can also show the uniqueness of the stationary distributions of the stochastic Navier-Stokes equation if the viscosity is large     

Dynamics of Stochastic Ginzburg-Landau Equations Driven by Colored Noise on Thin Domains

This work is concerned with the asymptotic behaviors of solutions to a class of non-autonomous stochastic Ginzburg-Landau equations driven by colored noise and deterministic non-autonomous terms defined on thin domains. The existence and uniqueness of tempered pullback random attractors are proved for the stochastic Ginzburg-Landau systems defined on $(n+1)$-dimensional narrow domain. Furthermore, the upper semicontinuity of these attractors is established, when a family of $(n+1)$-dimensional thin domains collapses onto an $n$-dimensional domain.     

Effects of random noise in a dynamical model of love

This paper aims to investigate the stochastic model of love and the effects of random noise. We first revisit the deterministic model of love and some basic properties are presented such as: symmetry, dissipation, fixed points (equilibrium), chaotic behaviors and chaotic attractors. Then we construct a stochastic love-triangle model with parametric random excitation due to the complexity and unpredictability of the psychological system, where the randomness is modeled as the standard Gaussian noise. Stochastic dynamics under different three cases of “Romeo’s romantic style”, are examined and two kinds of bifurcations versus the noise intensity parameter are observed by the criteria of changes of top Lyapunov exponent and shape of stationary probability density function (PDF) respectively. The phase portraits and time history are carried out to verify the proposed results, and the good agreement can be found. And also the dual roles of the random noise, namely suppressing and inducing chaos are revealed.     

具有白噪声的随机格点系统的随机吸引子的Kolmogorov熵

基于耗散的随机格点系统解的渐近行为理论,主要运用元素分解法与有限维空间中多面体球覆盖的拓扑性质,研究了具有白噪声的随机Klein-Gordon-Schrdinger格点动力系统的随机吸引子的Kolmogorov熵,并得到它的一个上界.     

Wong–Zakai approximations and attractors for stochastic reaction–diffusion equations on unbounded domains

In this paper, we study the Wong–Zakai approximations given by a stationary process via the Wiener shift and their associated long term behavior of the stochastic reaction–diffusion equation driven by a white noise. We first prove the existence and uniqueness of tempered pullback attractors for the Wong–Zakai approximations of stochastic reaction–diffusion equation. Then, we show that the attractors of Wong–Zakai approximations converges to the attractor of the stochastic reaction–diffusion equation for both additive and multiplicative noise.     

Asymptotic behavior of stochastic Schrödinger lattice systems driven by nonlinear noise

Abstract

We study the random dynamics of the N-dimensional stochastic Schrödinger lattice systems with locally Lipschitz diffusion terms driven by locally Lipschitz nonlinear noise. We first prove the existence and uniqueness of solutions and define a mean random dynamical system associated with the solution operators. We then establish the existence and uniqueness of weak pullback random attractors in a Bochner space. We finally prove the existence of invariant measures of the stochastic equation in the space of complex-valued square-summable sequences. The tightness of a family of probability distributions of solutions is derived by the uniform estimates on the tails of the solutions at far field.     

Dynamical behavior for stochastic lattice systems

Random attractors describe the long term behavior of the random dynamical systems. This paper is devoted to a general first order stochastic lattice dynamical systems (SLDS) with some dissipative nonlinearity. We prove the asymptotic compactness of the random dynamical system and obtain the random attractor, which is a compact random invariant set with tempered bound.     

Attractors for Multi-valued Non-autonomous Dynamical Systems: Relationship,Characterization and Robustness

In this paper we study cocycle attractors, pullback attractors and uniform attractors for multi-valued non-autonomous dynamical systems. We first consider the relationship between the three attractors and find that, under suitable conditions, they imply each other. Then, for generalized dynamical systems, we find that these attractors can be characterized by complete trajectories, which implies that the uniform attractor is lifted invariant, though it has no standard invariance by definition. Finally, we study both upper and lower semi-continuity of these attractors. A weak equi-attraction method is introduced to study the lower semi-continuity, and we show with an example the advantages of this method. A reaction-diffusion system and a scalar ordinary differential inclusion are studied as applications.     

Long time behavior of stochastic {MHD} equations perturbed by multiplicative noises

In this paper, 2-dimensional (2D) magnetohydrodynamics (MHD) equations perturbed by multiplicative noises in both the velocity and the magnetic field is studied. We first considered the stability, or the upper semi-continuity, for equivalent random dynamical systems (RDS), and then applying the abstract result we established the existence and the upper semi-continuity of tempered random attractors for the stochastic MHD equations. This result shows that the asymptotic behavior of MHD equations is stable under stochastic perturbations.     

Regularity of pullback attractors for non-autonomous stochastic coupled reaction-diffusion systems

We consider the dynamical behavior of the typical non-autonomous autocatalytic stochastic coupled reaction-diffusion systems on the entire space $\mathbb{R}^n$. Some new uniform asymptotic estimates are implemented to investigate the existence of pullback attractors in the Sobolev space $H^1(\mathbb{R}^n)^3$ for the three-component reversible Gray-Scott system.     

Exponentially Stable Stationary Solutions for Stochastic Evolution Equations and Their Perturbation

We consider the exponential stability of stochastic evolution equations with Lipschitz continuous non-linearities when zero is not a solution for these equations. We prove the existence of a non-trivial stationary solution which is exponentially stable, where the stationary solution is generated by the composition of a random variable and the Wiener shift. We also construct stationary solutions with the stronger property of attracting bounded sets uniformly. The existence of these stationary solutions follows from the theory of random dynamical systems and their attractors. In addition, we prove some perturbation results and formulate conditions for the existence of stationary solutions for semilinear stochastic partial differential equations with Lipschitz continuous non-linearities.     

Attractors of randomly forced logistic model with delay: stochastic sensitivity and noise-induced transitions

We consider the time-delayed logistic model under the influence of random perturbations. A parametric analysis of stochastically forced regular attractors (equilibria, closed invariant curves, discrete cycles) of this model is performed using the stochastic sensitivity functions technique. A spatial arrangement of random states in stochastic attractors is described by confidence domains. The phenomenon of noise-induced transitions in a zone of discrete cycles is discussed.     

Maximal Unstable Dissipative Interval to Preserve Multi-scroll Attractors via Multi-saturated Functions

In this paper, we present families of piecewise linear systems which are controlled by a continuous piecewise monoparametric control function for the generation of monoparametric families of multi-scroll attractors. Thus, the maximum range of values that the parameter set can take in order to preserve the useful dynamics for generating of multi-scroll attractors is found and it will be called maximal robust dynamics interval. This class of dynamical systems is the result of combining two or more unstable “one-spiral” trajectories. We give necessary and sufficient conditions in order to preserve multi-scroll attractors in terms of a parameter, i.e., a family of multi-scroll attractors is generated by means of a family of switching systems with multiple monoparametric companion matrices. Lastly, we provide an example to show how the developed theory works.     

ON EXPONENTIAL STABILITY OF NON-AUTONOMOUS STOCHASTIC SEMILINEAR EVOLUTION EQUATIONS

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Stochastic perturbations of fat solenoidal attractors

In this article stochastic perturbations of a class of fat solenoidal attractors are considered. We show the robustness of their invariant densities and rates of mixing under the stochastic perturbations by investigating the properties of their transfer operators.     

Dynamics of the Stochastic Chemostat Model with Monod-Haldane Response Function

This paper is devoted to the asymptotic dynamics of stochastic chemostat model with Monod-Haldane response function. We first prove the existence of random attractors by means of the conjugacy method and further construct a general condition for internal structure of the random attractor, implying extinction of the species even with small noise. Moreover, we show that the attractors of Wong-Zakai approximations converges to the attractor of the stochastic chemostat model in an appropriate sense.