Order and Chaos in Dynamical Systems

We consider the characteristics of order and chaos in dynamical systems, with emphasis on the orbits in astronomical systems. Celestial mechanics deals with orbits in the solar system, which are mainly ordered. On the other hand the orbits of stars in galaxies were considered to be chaotic. However numerical experiments have shown that in general a system contains both ordered and chaotic orbits. Thus a new classification of dynamical systems has been established. We describe ordered and chaotic orbits in galaxies and in mappings. Some ordered orbits appear even in strongly perturbed systems. The transition from order to chaos is due to resonance overlapping. Then we describe some recent developments concerning order and chaos in the solar system and in galaxies. The outer spiral arms in strong barred galaxies are composed mainly of sticky chaotic orbits. Ordered and chaotic orbits appear also in Bohmian quantum mechanics. If the initial probability p is not equal to the square of the wave function |ψ|², then in the case of ordered orbits p never approaches |ψ|², while in the case of chaotic orbits p → |ψ|² after a time interval called “quantum Nekhoroshev time”.     

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.     

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.     

Discontinuous Dynamical Systems in Optimal Control Theory

We develop foundations of the theory of discontinuous Hamiltonian systems appearing in the problems of optimal control. We consider analogs of the classical Poisson and Liouville theorems for discontinuous Hamiltonian systems. We study the local geometry of discontinuous dynamical systems and describe singularities in general position and the behavior of integral trajectories near an elliptical submanifold (sliding mode).     

Optimal Control Formulation for Complementarity Dynamical Systems

In this paper, we show that the complementarity dynamical systems can be reformulated as optimal control problems. By using this reformulation, we present a pseudospectral scheme to discretize the complementarity dynamical systems. Applying this discretization, the complementarity dynamical system is reduced to a sequence of nonlinear programming problems. Numerical examples and comparison with two other methods are included to demonstrate the capability of the proposed method.     

金融动力系统的同步脉冲控制

针对一类金融三维动力系统,提出了一种同步控制法.基于微分脉冲系统理论,设计了带状态反馈的脉冲控制策略来实现系统的同步控制.理论分析和数值模拟的结果都验证了控制器的有效性.这一控制策略简单便于实现,更可以用于一般非线性混沌系统的控制.     

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.     

Pseudospectral Chebyshev Optimal Control of Constrained Nonlinear Dynamical Systems

A pseudospectral method for generating optimal trajectories of linear and nonlinear constrained dynamic systems is proposed. The method consists of representing the solution of the optimal control problem by an mth degree interpolating polynomial, using Chebyshev nodes, and then discretizing the problem using a cell-averaging technique. The optimal control problem is thereby transformed into an algebraic nonlinear programming problem. Due to its dynamic nature, the proposed method avoids many of the numerical difficulties typically encountered in solving standard optimal control problems. Furthermore, for discontinuous optimal control problems, we develop and implement a Chebyshev smoothing procedure which extracts the piecewise smooth solution from the oscillatory solution near the points of discontinuities. Numerical examples are provided, which confirm the convergence of the proposed method. Moreover, a comparison is made with optimal solutions obtained by closed-form analysis and/or other numerical methods in the literature.     

一类非线性金融系统数学模型的混沌控制

研究一类非线性金融系统数学模型的混沌控制.首先,运用动力学方法研究位移反馈控制、加速反馈控制、增强反馈控制、延迟反馈控制在该系统的应用条件和反馈增益系数.其次,运用数值模拟的方法验证各种控制方法的可行性,比较各种控制方法的应用效果.最后选择较好的控制方法,为金融危机的宏观调控提供建议.     

Solving Optimal Control Problems by Exploiting Inherent Dynamical Systems Structures

Computing globally efficient solutions is a major challenge in optimal control of nonlinear dynamical systems. This work proposes a method combining local optimization and motion planning techniques based on exploiting inherent dynamical systems structures, such as symmetries and invariant manifolds. Prior to the optimal control, the dynamical system is analyzed for structural properties that can be used to compute pieces of trajectories that are stored in a motion planning library. In the context of mechanical systems, these motion planning candidates, termed primitives, are given by relative equilibria induced by symmetries and motions on stable or unstable manifolds of e.g. fixed points in the natural dynamics. The existence of controlled relative equilibria is studied through Lagrangian mechanics and symmetry reduction techniques. The proposed framework can be used to solve boundary value problems by performing a search in the space of sequences of motion primitives connected using optimized maneuvers. The optimal sequence can be used as an admissible initial guess for a post-optimization. The approach is illustrated by two numerical examples, the single and the double spherical pendula, which demonstrates its benefit compared to standard local optimization techniques.     

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．     

On Nonlinear Perturbations of Nonlinear Dynamical Systems, and Applications to Control

The nonlinear variation-of-constants formula is generalizedto infinite-dimensional systems and applied to the stabilityproblem and used to obtain bounds on the states of a nonlinearcontrol system when the controls are bounded. The theory isillustrated by a simple model from nuclear reactor dynamics.     

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.     

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.     

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     

Random Conformal Dynamical Systems

We consider random dynamical systems such as groups of conformal transformations with a probability measure, or transversally conformal foliations with a Laplace operator along the leaves, in which case we consider the holonomy pseudo-group. We prove that either there exists a measure invariant under all the elements of the group (or the pseudo-group), or almost surely a long composition of maps contracts a ball exponentially. We deduce some results about the unique ergodicity. Received: June 2005, Revision: January 2006, Accepted: March 2006