广义符号动力系统的混沌性

本文给出了符号动力系统的一般数学模型,它是离散时空系统的一种特殊情形.在现有离散时空系统的混沌概念和研究方法的基础上.研究了这类广义符号动力系统的混沌性,得到了一类在Devaney意义下新的广义符号混沌动力系统,从而推广了现有符号动力系统混沌性的研究范围.     

广义控制系统的动态阶配置

本文考虑广义控制系统。通过使用输出导数反馈配置系统的动态阶。     

Attractors for Second Order Lattice Dynamical Systems

We consider the existence and the approximation of the global attractor for second order damped lattice dynamical systems.     

A Continuous Archetype of Nonuniform Chaos in Area-Preserving Dynamical Systems

We propose a piecewise linear, area-preserving, continuous map of the two-torus as a prototype of nonlinear two-dimensional mixing transformations that preserve a smooth measure (e.g., the Lebesgue measure). The model lends itself to a closed-form analysis of both statistical and geometric properties. We show that the proposed model shares typical features that characterize chaotic dynamics associated with area-preserving nonlinear maps, namely, strict inequality between the line-stretching exponent and the Lyapunov exponent, the dispersive behavior of stretch-factor statistics, the singular spatial distribution of expanding and contracting fibers, and the sign-alternating property of cocycle dynamics. The closed-form knowledge of statistical and geometric properties (in particular of the invariant contracting and dilating directions) makes the proposed model a useful tool for investigating the relationship between stretching and folding in bounded chaotic systems, with potential applications in the fields of chaotic advection, fast dynamo, and quantum chaos theory.     

Decidability of Chaos for Some Families of Dynamical Systems

We show that the existence of positive Lyapounov exponents and/or SRB measures are undecidable (in the algorithmic sense) properties within some parametrized families of interesting dynamical systems: the quadratic family and Hénon maps. Because the existence of positive exponents (or SRB measures) is, in a natural way, a manifestation of chaos, these results may be understood as saying that the chaotic character of a dynamical system is undecidable. Our investigation is directly motivated by questions asked by Carleson and Smale in this direction.     

Periodic Solutions of Second Order Superlinear Singular Dynamical Systems

We study the existence of periodic solutions of second order superlinear dynamical systems with a singularity of repulsive type. The proof is based on a well-known fixed point theorem for completely continuous operators. We do not need to consider so-called strong force conditions. Recent results in the literature are generalized and significantly improved.     

Multiplicity of Positive Periodic Solutions to Second Order Singular Dynamical Systems

The existence and multiplicity of non-collision periodic solutions for second order singular dynamical systems are discussed in this paper. Using the Green’s function of linear differential equation, we consider general singularity and do not need any kind of strong force condition. The proof relies on a nonlinear alternative principle of Leray–Schauder and a fixed point theorem in cones.     

Entropy,Chaos, and Weak Horseshoe for Infinite‐Dimensional Random Dynamical Systems

In this paper, we study the complicated dynamics of infinite‐dimensional random dynamical systems that include deterministic dynamical systems as their special cases in a Polish space. Without assuming any hyperbolicity, we prove if a continuous random map has a positive topological entropy, then it contains a topological horseshoe. We also show that the positive topological entropy implies the chaos in the sense of Li‐Yorke. The complicated behavior exhibited here is induced by the positive entropy but not the randomness of the system.© 2017 Wiley Periodicals, Inc.     

A Numerical Method for Establishing Liapunov Stability of Linear Second Order Non-autonomous Dynamical Systems

A numerical method which basically utilizes the Liapunov DirectMethod is presented which establishes sufficient conditionsfor the stability of the null solution of the nonautonomousdifferential equation x+b(t)x+a(t)x = 0. This is accomplishedwithout the unstructured task of searching for given Liapunovfunctions for the particular problem being analysed. The methodis simple and straightforward to apply, and can be used to obtainstability results as long as b is not identically zero. Thenumerical approach is demonstrated by applying it to the dampedMathieu equation, the Euler equation, and an equation with quasiperiodiccoefficients.     

动力系统中的Lipschitz稳定性

本文讨论动力系统的Ｌｉｐｓｃｈｉｔｚ稳定性与吸引性之间的关系．给出动力系统中弱吸引性、吸引性和强吸引性这三个概念相互等价的条件，在一定的条件下证明了（弱、强）吸引子与全局（弱、强）吸引子是一致的．本文还讨论了度量Ｌｙａｐｕｎｏｖ稳定性和拓扑Ｌｙａｐｕｎｏｖ稳定性及它们与Ｌｉｐｓｃｈｉｔｚ稳定性之间的关系．     

通过同步实现从混沌到有序的转变

本文旨在研究连续的混沌系统是否存在“混沌＋混沌=有序”的现象．证明了两个双向耦合的连续混沌系统在一些情况下可产生有序的动力学行为．作为例子,通过选取适当的耦合参数使Lorenz系统以及Chen和Lee引入的混沌系统同步,进而对同步系统的动力学行为进行了理论分析和数值模拟．结果表明,逐渐改变参数,系统实现了从混沌到有序的过渡．     

高阶非线性动力系统全局分析——胞胞映射法应用

本文阐述高阶非线性动力系统全局分析和应用胞胞映射进行分析的一般特点,以及胞胞映射方法对于高阶系统全局分析的有效性;并具体进行了一个弱耦合van der Pol振荡系统的全局分析,确定系统具有两个稳定的极限环,并确定了整个四维空间被分为两个部分,这两部分分别是沿两个极限环运动的渐近稳定域(吸引域).     

非自治离散系统的分布混沌性

该文在非自治离散系统中定义了分布混沌,研究了映射序列f_n,∞=(f_n,f_n+t,…),_n∈N（N为自然数集）的混沌行为,讨论了f_n,∞的分布混沌性是否意味着乘积系统f_n,∞^[m]（m为正整数）的分布混沌性,或者后者的分布混沌性是否意味着前者的分布混沌性.     

Affinely Prime Dynamical Systems

This paper deals with representations of groups by "affine" automorphisms of compact,convex spaces,with special focus on "irreducible" representations:equivalently "minimal" actions.When the group in question is PSL(2,R),the authors exhibit a oneone correspondence between bounded harmonic functions on the upper half-plane and a certain class of irreducible representations.This analysis shows that,surprisingly,all these representations are equivalent.In fact,it is found that all irreducible affine representations of this group are equivalent.The key to this is a property called "linear Stone-Weierstrass"for group actions on compact spaces.If it holds for the "universal strongly proximal space"of the group (to be defined),then the induced action on the space of probability measures on this space is the unique irreducible affine representation of the group.     

随机动力系统的进展和问题

本文主要介绍随机动力系统的主要成果和进一步研究的问题。     

Banach Algebra Dynamical Systems

This paper generalizes the C*-dynamical system to the Banach algebra dynamical system(A,G,α)and define the crossed product A α G of a given Banach algebra dynamical system(A,G,α).Then the representation of A α G is described when A admits a bounded left approximate identity.In a natural way,the authors define the reduced crossed product A α,r G and discuss the question when A α G coincides with A α,r G.     

Boolean Monomial Dynamical Systems

An important problem in the theory of finite dynamical systems is to link the structure of a system with its dynamics. This paper contains such a link for a family of nonlinear systems over the field with two elements. For systems that can be described by monomials (including Boolean AND systems), one can obtain information about the limit cycle structure from the structure of the monomials. In particular, the paper contains a sufficient condition for a monomial system to have only fixed points as limit cycles. This condition depends on the cycle structure of the dependency graph of the system and can be verified in polynomial time.Received February 2, 2004