arXiv subjects

Zhenxin Liu

Publications and source records attributed to Zhenxin Liu.

48 records · Page 3Linked to original sources

arXiv2016

Concentration and limit behaviors of stationary measures

In this paper, we study limit behaviors of stationary measures of the Fokker-Planck equations associated with a system of ordinary differential equations perturbed by a class of multiplicative including additive white noises. As the noises are vanishing, various results on the invariance and concentration of the limit measures are obtained. In particular, we show that if the noise perturbed systems admit a uniform Lyapunov function, then the stationary measures form a relatively sequentially compact set whose weak$^*$-limits are invariant measures of the unperturbed system concentrated on its global attractor. In the case that the global attractor contains a strong local attractor, we further show that there exists a family of admissible multiplicative noises with respect to which all limit measures are actually concentrated on the local attractor; and on the contrary, in the presence of a strong local repeller in the global attractor, there exists a family of admissible multiplicative noises with respect to which no limit measure can be concentrated on the local repeller. Moreover, we show that if there is a strongly repelling equilibrium in the global attractor, then limit measures with respect to typical families of multiplicative noises are always concentrated away from the equilibrium. As applications of these results, an example of stochastic Hopf bifurcation is provided. Our study is closely related to the problem of noise stability of compact invariant sets and invariant measures of the unperturbed system.

math.DS↗

arXiv2015

Favard separation method for almost periodic stochastic differential equations

Favard separation method is an important means to study almost periodic solutions to linear differential equations; later, Amerio applied Favard's idea to nonlinear differential equations. In this paper, by appropriate choosing separation and almost periodicity in distribution sense, we obtain the Favard and Amerio type theorems for stochastic differential equations.

math.PR↗

arXiv2015

Integral identity and measure estimates for stationary Fokker-Planck equations

We consider a Fokker-Planck equation in a general domain in ${\mathbb{R}}^n$ with $L^p_{\mathrm{loc}}$ drift term and $W^{1,p}_{\mathrm{loc}}$ diffusion term for any $p>n$. By deriving an integral identity, we give several measure estimates of regular stationary measures in an exterior domain with respect to diffusion and Lyapunov-like or anti-Lyapunov-like functions. These estimates will be useful to problems such as the existence and nonexistence of stationary measures in a general domain as well as the concentration and limit behaviors of stationary measures as diffusion vanishes.

math.AP↗

arXiv2010

Square-mean almost automorphic solutions for some stochastic differential equations

The concept of square-mean almost automorphy for stochastic processes is introduced. The existence and uniqueness of square-mean almost automorphic solutions to some linear and non-linear stochastic differential equations are established provided the coefficients satisfy some conditions. The asymptotic stability of the unique square-mean almost automorphic solution in square-mean sense is discussed.

math.DS↗

arXiv2007

The random case of Conley's theorem: III. Random semiflow case and Morse decomposition

In the first part of this paper, we generalize the results of the author \cite{Liu,Liu2} from the random flow case to the random semiflow case, i.e. we obtain Conley decomposition theorem for infinite dimensional random dynamical systems. In the second part, by introducing the backward orbit for random semiflow, we are able to decompose invariant random compact set (e.g. global random attractor) into random Morse sets and connecting orbits between them, which generalizes the Morse decomposition of invariant sets originated from Conley \cite{Con} to the random semiflow setting and gives the positive answer to an open problem put forward by Caraballo and Langa \cite{CL}.

math.DS↗

arXiv2007

The random case of Conley's theorem: II. The complete Lyapunov function

Conley in \cite{Con} constructed a complete Lyapunov function for a flow on compact metric space which is constant on orbits in the chain recurrent set and is strictly decreasing on orbits outside the chain recurrent set. This indicates that the dynamical complexity focuses on the chain recurrent set and the dynamical behavior outside the chain recurrent set is quite simple. In this paper, a similar result is obtained for random dynamical systems under the assumption that the base space $(Ω,\mathcal F,\mathbb P)$ is a separable metric space endowed with a probability measure. By constructing a complete Lyapunov function, which is constant on orbits in the random chain recurrent set and is strictly decreasing on orbits outside the random chain recurrent set, the random case of Conley's fundamental theorem of dynamical systems is obtained. Furthermore, this result for random dynamical systems is generalized to noncompact state spaces.

math.DS↗

arXiv2006

Stochastic inertial manifolds for damped wave equations

In this paper, stochastic inertial manifold for damped wave equations subjected to additive white noise is constructed by the Lyapunov-Perron method. It is proved that when the intensity of noise tends to zero the stochastic inertial manifold converges to its deterministic counterpart almost surely.

math.DS↗

arXiv2006

Persistence of Hyperbolic Tori in Generalized Hamiltonian Systems

In this paper we prove the persistence of hyperbolic invariant tori in generalized Hamiltonian systems, which may admit a distinct number of action and angle variables. The systems under consideration can be odd dimensional in tangent direction. Our results generalize the well-known results of Graff and Zehnder in standard Hamiltonians. In our case the unperturbed Hamiltonian systems may be degenerate. We also consider the persistence problem of hyperbolic tori on sub-manifolds.

math.DS↗

arXiv2006

Conley index for random dynamical systems

Conley index theory is a very powerful tool in the study of dynamical systems, differential equations and bifurcation theory. In this paper, we make an attempt to generalize the Conley index to discrete random dynamical systems. And we mainly follow the Conley index for maps given by Franks and Richeson in [6]. Furthermore, we simply discuss the relations of isolated invariant sets between time-continuous random dynamical systems and the corresponding time-$h$ maps. For applications we give several examples to illustrate our results.

math.DS↗

arXiv2006

Attractor-repeller pair, Morse decomposition and Lyapunov function for random dynamical systems

In the stability theory of dynamical systems, Lyapunov functions play a fundamental role. In this paper, we study the attractor-repeller pair decomposition and Morse decomposition for compact metric space in the random setting. In contrast to [8], by introducing slightly stronger definitions of random attractor and repeller, we characterize attractor-repeller pair decompositions and Morse decompositions for random dynamical systems through the existence of Lyapunov functions. These characterizations, we think, deserve to be known widely.

math.DS↗

arXiv2005

The random case of Conley's theorem

The well-known Conley's theorem states that the complement of chain recurrent set equals the union of all connecting orbits of the flow $ϕ$ on the compact metric space $X$, i.e. $X-\mathcal{CR}(ϕ)=\bigcup [B(A)-A]$, where $\mathcal{CR}(ϕ)$ denotes the chain recurrent set of $ϕ$, $A$ stands for an attractor and $B(A)$ is the basin determined by $A$. In this paper we show that by appropriately selecting the definition of random attractor, in fact we define a random local attractor to be the $ω$-limit set of some random pre-attractor surrounding it, and by considering appropriate measurability, in fact we also consider the universal $σ$-algebra $\mathcal F^u$-measurability besides $\mathcal F$-measurability, we are able to obtain the random case of Conley's theorem.

math.DS↗

arXiv2005

Persistence of lower dimensional invariant tori on sub-manifolds in Hamiltonian systems

Chow, Li and Yi in [2] proved that the majority of the unperturbed tori {\it on sub-manifolds} will persist for standard Hamiltonian systems. Motivated by their work, in this paper, we study the persistence and tangent frequencies preservation of lower dimensional invariant tori on smooth sub-manifolds for real analytic, nearly integrable Hamiltonian systems. The surviving tori might be elliptic, hyperbolic, or of mixed type.

math.DS↗