Stochastic Chaos with Its Control and Synchronization

The discovery of chaos in the sixties of last century was a breakthrough in concept,revealing the truth that some disorder behavior,called chaos,could happen even in a deterministic nonlinear system under barely deterministic disturbance.After a series of serious studies,people begin to acknowledge that chaos is a specific type of steady state motion other than the conventional periodic and quasi-periodic ones,featuring a sensitive dependence on initial conditions,resulting from the intrinsic randomness of a nonlinear system itself.In fact,chaos is a collective phenomenon consisting of massive individual chaotic responses,corresponding to different initial conditions in phase space.Any two adjacent individual chaotic responses repel each other,thus causing not only the sensitive dependence on initial conditions but also the existence of at least one positive top Lyapunov exponent(TLE) for chaos.Meanwhile,all the sample responses share one common invariant set on the Poincaré map,called chaotic attractor,which every sample response visits from time to time ergodically.So far,the existence of at least one positive TLE is a commonly acknowledged remarkable feature of chaos.We know that there are various forms of uncertainties in the real world.In theoretical studies,people often use stochastic models to describe these uncertainties,such as random variables or random processes.Systems with random variables as their parameters or with random processes as their excitations are often called stochastic systems.No doubt,chaotic phenomena also exist in stochastic systems,which we call stochastic chaos to distinguish it from deterministic chaos in the deterministic system.Stochastic chaos reflects not only the intrinsic randomness of the nonlinear system but also the external random effects of the random parameter or the random excitation.Hence,stochastic chaos is also a collective massive phenomenon,corresponding not only to different initial conditions but also to different samples of the random parameter or the random excitation.Thus,the unique common feature of deterministic chaos and stochastic chaos is that they all have at least one positive top Lyapunov exponent for their chaotic motion.For analysis of random phenomena,one used to look for the PDFs(Probability Density Functions) of the ensemble random responses.However,it is a pity that PDF information is not favorable to studying repellency of the neighboring chaotic responses nor to calculating the related TLE,so we would rather study stochastic chaos through its sample responses.Moreover,since any sample of stochastic chaos is a deterministic one,we need not supplement any additional definition on stochastic chaos,just mentioning that every sample of stochastic chaos should be deterministic chaos.We are mainly concerned with the following two basic kinds of nonlinear stochastic systems,i.e.one with random variables as its parameters and one with ergodical random processes as its excitations.To solve the stochastic chaos problems of these two kinds of systems,we first transform the original stochastic system into their equivalent deterministic ones.Namely,we can transform the former stochastic system into an equivalent deterministic system in the sense of mean square approximation with respect to the random parameter space by the orthogonal polynomial approximation,and transform the latter one simply through replacing its ergodical random excitations by their representative deterministic samples.Having transformed the original stochastic chaos problem into the deterministic chaos problem of equivalent systems,we can use all the available effective methods for further chaos analysis.In this paper,we aim to review the state of art of studying stochastic chaos with its control and synchronization by the above-mentioned strategy.

ON EDGE-HAMILTONIAN PROPERTY OF BI-CAYLEY GRAPHS

Let G be a finite group, and S be a subset of G. The bi-Cayley graph BCay（G, S） of G with respect to S is defined as the bipartite graph with vertex set G x {0,1} and edge set {（g,0）, （gs, 1）1 g ε G, s εS}. In this paper, we first provide two interesting results for edge-hamiltonian property of Cayley graphs and bi-Cayley graphs. Next, we investigate the edge^hamiltonian property of F = BCay（G, S）, and prove that F is hamiltonian if and only if F is edge-hamiltonian when F is a connected bi-Cayley graph.

HOPF BIFURCATION OF AN INFECTED PREDATOR-PREY SYSTEM WITH PREY REFUGE

In this paper, an infected predator-prey model with prey refuge is investigated. The effects of refuge on the stability of the equilibria of the system are analyzed. Moreover, using the criterion introduced by Liu, we derive the Hopf bifurcation conditions of the system with respect to the refuge value.

PERTURBATION EQUATION OF DLW HIERARCHY AND ITS HAMILTONIAN STRUCTURE

The first-order perturbation equation of DLW hierarchy is derived by virtue of zero-curvature equation. Then, Hamiltonian structure of the obtained system is given by means of component-trace identity.

EXPONENTIAL STABILITY FOR A CLASS OF SWITCHED STOCHASTIC SYSTEMS

In this paper, the stability properties for a class of switched stochastic systems with commutative componentwise subsystem matrices are studied. Under some switching law, the trivial solutions of the above systems are proved to be exponentially stable in mean square and almost sure exponentially stable if the random perturbations are sufficiently “small”.

EXISTENCE AND MOMENT ESTIMATES FOR SOLUTIONS TO NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper, we show the existence and uniqueness of solutions to a large class of SFDEs with the generalized local Lipschitzian coefficients. Some moment estima- tes of the solutions are given by establishing new Ito operator inequalities based on the Razumikhin technique. These estimates improve, extend and unify some related results including exponential stability of Mao （1997） [20], decay stability of Wu et al. （2010,2011） [32,33], Pavlovic et al. （2012） [24], asymptotic behavior of Luo et al. （2011） [18] and Song et al. （2013） [26]. Moreover, stochastic version of Wintner theorem in continuous space is established by the comparison principle, which improve and extend the main results of Xu et al. （2008 [39], 2013 [36]）. When the methods presented are applied to the SFDEs with impulses and SFDEs in Hilbert spaces, we extend the related results of Govindana et al. （2013） [7], Liu et al. （2007） [15], Vinod- kumar （2010） [29] and Xu et al. （2012） [35]. Two examples are provided to illustrate the effectiveness of our results.

A SURVEY ON THE PERIODIC SOLUTIONS TO KAPLAN-YORKE TYPE DELAY DIFFERENTIAL EQUATION-I

Since 1974, studying the original delay differential equation given by Kaplan and Yorke is about the problem on the existence of its periodic solutions, there have been a series of interesting and significant results in the previous literature. In this paper, we present a survey of some basic results. Some interesting open problems are also

(2+1)-DIMENSIONAL Tu HIERARCHY AND ITS INTEGRABLE COUPLINGS AS WELL AS THE MULTI-COMPONENT INTEGRABLE HIERARCHY

Under the frame of the （2＋1）-dimensional zero curvature equation and Tu model, （2＋1）-dimensional Tu hierarchy is obtained. Again by employing a subalgebra of the loop algebra ↑-A2 the integrable coupling system of the above hierarchy is presented. Finally, A multi-component integrable hierarchy is obtained by employing a multi-component loop algebra ↑-GM.

ALTERING CONNECTIVITY WITH LARGE DEFORMATION MESH FOR LAGRANGIAN METHOD AND ITS APPLICATION IN MULTIPLE MATERIAL SIMULATION

A new approach for treating the mesh with Lagrangian scheme of finite volume method is presented. It has been proved that classical Lagrangian method is difficult to cope with large deformation in tracking material particles due to severe distortion of cells, and the changing connectivity of the mesh seems especially attractive for solving such issues. The mesh with large deformation based on computational geometry is optimized by using new method. This paper develops a processing system for arbitrary polygonal unstructured grid,the intelligent variable grid neighborhood technologies is utilized to improve the quality of mesh in calculation process, and arbitrary polygonal mesh is used in the Lagrangian finite volume scheme. The performance of the new method is demonstrated through series of numerical examples, and the simulation capability is efficiently presented in coping with the systems with large deformations.

HYPERCYCLIC MULTIPLICATION COMPOSITION OPERATORS ON WEIGHTED BANACH SPACE

This paper characterizes some sufficient and necessary conditions for the hypercyclicity of multiples of composition operators on Hlog,0∞.

SOME GENERALIZED GRONWALL-LIKE INEQUALITIES WITH DELAY ON TIME SCALES

In this paper, we establish some new Gronwall-like inequalities which can be used as tools in the theory of integral equations with delay on time scales.

ASYMPTOTICALLY ALMOST PERIODIC FUNCTIONS IN PROBABILITY

In this paper, we first study the properties of asymptotically almost periodic functions in probability and then prove the existence of almost periodic solutions in probability to some differential equations with random terms.