Random Invariant Manifold Research Articles

Stability and constructions of the Poincaré maps for a class of stochastic partial differential equations

This work is concerned with a class of stochastic partial differential equations with a multiplicative noise. The Poincaré map, as a main tool to study the periodic orbits, plays a crucially important role in the invariant sets of the system, which captures the dominated dynamical behavior of the trajectories. It derives out the existence and the stability of the stochastic Poincaré maps of the system, which return to a small neighborhood of a fixed point of a deterministic Poincaré map one time or even infinite times with the small noise disturbing. It further expresses the constructions of the stochastic Poincaré maps through the random invariant manifold and the moving orthonormal system.

Mean-square invariant manifolds for stochastic weak-damping wave equations with nonlinear noise

We study mean-square invariant manifolds for stochastic wave equations with nonlinear infinite-dimensional noise and weak damping, where all nonlinear terms satisfy the nonhomogeneous Lipschitz conditions. First, we consider the existence of a mean-square random dynamical system generated from the mild solution. Second, we give a careful analysis for the spectrum of the wave operator and show that the wave operator satisfies the pseudo exponential dichotomy. Finally, by using the Lyapunov-Perron method and analyzing the conditional expectation solution, we show the existence of a mean-square random invariant unstable manifold as well as a mean-square random invariant stable set.

Effective filtering for slow-fast systems via Wong-Zakai approximation

Nonlinear filtering problem has been widely investigated in diverse fields. One important topic related to it is how to formulate an effective approximation scheme for a given observation. By using Wong-Zakai approximation and random invariant manifold theory, we will propose an effective approximation result for a class of slow-fast systems with respect to filtering. We will firstly establish the smooth reduced system via random invariant manifold theory, and then show exponential attractive property of it. At last, we will prove the filtering of the smooth reduced system can well approximate that of the original system.

Random Perturbation of Invariant Manifolds for Non-Autonomous Dynamical Systems

Random invariant manifolds are geometric objects useful for understanding dynamics near the random fixed point under stochastic influences. Under the framework of a dynamical system, we compared perturbed random non-autonomous partial differential equations with original stochastic non-autonomous partial differential equations. Mainly, we derived some pathwise approximation results of random invariant manifolds when the Gaussian white noise was replaced by colored noise, which is a type of Wong–Zakai approximation.

Mean-square invariant manifolds for ill-posed stochastic evolution equations driven by nonlinear noise

This paper discerns the invariant manifold of a class of ill-posed stochastic evolution equations driven by a nonlinear multiplicative noise. To be more precise, we establish the existence of mean-square random unstable invariant manifold and only mean-square stable invariant set. Due to the lack of the Hille-Yosida condition, we construct a modified variation of constants formula by the resolvent operator. With the price of imposing an unusual condition involving a non-decreasing map, we set up the Lyapunov-Perron method and derive the required estimates. We also emphasize that the Lyapunov-Perron map in the forward time loses the invariant due to the adaptedness, we alternatively establish the existence of mean-square random stable sets.

Random invariant manifolds and foliations for slow-fast PDEs with strong multiplicative noise

This article is devoted to the dynamical behaviors of a class of slow-fast PDEs perturbed by strong multiplicative noise. We will accomplish the existence of random invariant manifolds and foliations, and show exponential tracking property of them. Moreover, the asymptotic approximation for both objects will be presented.

Random invariant manifolds of stochastic evolution equations driven by Gaussian and non-Gaussian noises

The goal of this work is to compare the invariant manifold of the stochastic evolution equation driven by an α-stable process with the invariant manifold of the stochastic evolution equation forced by Brownian motion. First, we show that the solution of the Marcus stochastic evolution equation driven by a type of α-stable process converges to the solution of the related Stratonovich stochastic evolution equation forced by Brownian motion. Then, we study the invariant stable manifold of the stochastic evolution equation driven by an α-stable process. Finally, we prove that the invariant stable manifold of the Marcus stochastic evolution equation driven by an α-stable process converges in probability to the invariant stable manifold of the Stratonovich stochastic evolution equation forced by Brownian motion. The connection between the random dynamical system driven by non-Gaussian noise and the random dynamical system driven by Gaussian noise is established.

Effective Filtering for Multiscale Stochastic Dynamical Systems Driven by Lévy Processes*

The work is about multiscale stochastic dynamical systems driven by Lévy processes. We prove that such a system can be approximated by a low-dimensional system on a random invariant manifold, and the original filter can be also approximated by the reduced low-dimensional filter. Finally, we investigate the reduction for $$\varepsilon =0$$ and obtain that these reduced systems does not approximate these multiscale stochastic dynamical systems.

Mean-square random invariant manifolds for stochastic differential equations

We develop a theory of mean-square random invariant manifolds for mean-square random dynamical systems generated by stochastic differential equations. This theory is applicable to stochastic partial differential equations driven by nonlinear noise. The existence of mean-square random invariant unstable manifolds is proved by the Lyapunov-Perron method based on a backward stochastic differential equation involving the conditional expectation with respect to a filtration. The existence of mean-square random stable invariant sets is also established but the existence of mean-square random stable invariant manifolds remains open.

Synchronization of stochastic lattice equations

In this paper we consider a system of two coupled nonlinear lattice stochastic equations driven by additive white noise processes. We prove the master slave synchronization of the components of the coupled system, namely, for $$t\rightarrow \infty $$ the solution of one of the subsystems (the slave component) converges to the values of a Lipschitz continuous function of the other component, the master component. To establish this kind of synchronization we will prove the existence of an exponentially attracting random invariant manifold for the coupled system.

Effective filtering for multiscale stochastic dynamical systems in Hilbert spaces

The work concerns effective filtering for a type of slow-fast data assimilation system in Hilbert spaces. First, we prove that the original system reduces to a system on a random invariant manifold. Second, we show that the nonlinear filtering of the original system can approximate that of the reduced system. Finally, we apply the result to an example.

Invariant manifolds and foliations for random differential equations driven by colored noise

In this paper, we prove the existence of local stable and unstable invariant manifolds for a class of random differential equations driven by nonlinear colored noise defined in a fractional power of a separable Banach space. In the case of linear noise, we show the pathwise convergence of these random invariant manifolds as well as invariant foliations as the correlation time of the colored noise approaches zero.

Finite dimensional reducing and smooth approximating for a class of stochastic partial differential equations

This work provides a finite dimensional reducing and a smooth approximating for a class of stochastic partial differential equations with an additive white noise. Using the invariant random cone to show the asymptotical completion, this stochastic partial differential equation is reduced to a stochastic ordinary differential equation on a random invariant manifold. Furthermore, after deriving the finite dimensional reducing for another stochastic partial differential equation driven by a Wong-Zakai scheme via a smooth colored noise, it is proved that when the smooth colored noise tends to the white noise, the solution and the finite dimensional reducing of the approximate system converge pathwisely to those of the original system.

Random invariant manifolds for ill-posed stochastic evolution equations

We establish the existence of random stable and unstable manifolds for ill-posed stochastic partial differential equations (SPDEs). Namely, we assume that the linear part does not generate a [Formula: see text]-semigroup. Using the theory of integrated semigroups, we are able to analyze the long-time behavior of random dynamical systems generated by such SPDEs.

Effective Filtering Analysis for Non-Gaussian Dynamic Systems

This work is about a slow-fast data assimilation system under non-Gaussian noisy fluctuations. Firstly, we show the existence of a random invariant manifold for a stochastic dynamical system with non-Gaussian noise and two-time scales. Secondly, we obtain a low dimensional reduction of this system via a random invariant manifold. Thirdly, we prove that the low dimensional filter on the random invariant manifold approximates the original filter, in a probabilistic sense.

A Wong-Zakai approximation for random invariant manifolds

Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic influences. But these random objects are difficult to be visualized geometrically or computed numerically. The current work provides a perturbation approach to approximate these random invariant manifolds. We first discuss the existence of a random invariant manifold for a class of stochastic evolutionary equations. Then, we approximate the random invariant manifold by the invariant manifold of a new system with smooth colored noise (i.e., integrated Ornstein-Uhlenbeck processes). The convergence in a pathwise Wong-Zakai sense is shown.

Approximate the dynamical behavior for stochastic systems by Wong–Zakai approaching

Random invariant manifolds and foliations play an important role in the study of the qualitative dynamical behaviors for nonlinear stochastic partial differential equations. In a general way, these random objects are difficult to be visualized geometrically or computed numerically. The current work provides a perturbation approach to approximate these random invariant manifolds and foliations. After briefly discussing the existence of random invariant manifolds and foliations for a class of stochastic systems driven by additive noises, the corresponding Wong–Zakai type of convergence result in path-wise sense is established.

Approximation of random invariant manifolds for a stochastic Swift-Hohenberg equation

Random invariant manifolds are considered for a stochastic Swift-Hohenberg equation with multiplicative noise in the Stratonovich sense. Using a stochastic transformation and a technique of cut-off function, existence of random invariant manifolds and attracting property of the corresponding random dynamical system are obtained by Lyaponov-Perron method. Then in the sense of large probability, an approximation of invariant manifolds has been investigated and this is further used to describe the geometric shape of the invariant manifolds.

Spatial Homogenization of Stochastic Wave Equation with Large Interaction

AbstractA dynamical approximation of a stochastic wave equation with large interaction is derived. A random invariant manifold is discussed. By a key linear transformation, the random invariant manifold is shown to be close to the random invariant manifold of a second-order stochastic ordinary differential equation.

Singular fold with real noise

We study the effect of small real noise on the jump behavior near a singular fold point, which is an important step in understanding the burst-spike behavior in many biological models. We show by the theory of center manifolds and random invariant manifolds that if the order of the noise is high enough, trajectories essentially pass the fold point in the manner as though there is no noise.