Rotation numbers for random dynamical systems on the circle

In this paper, we study rotation numbers of random dynamical systems on the circle. We prove the existence of rotation numbers and the continuous dependence of rotation numbers on the systems. As an application, we prove a theorem on analytic conjugacy to a circle rotation.

     

Two notes on measure-theoretic entropy of random dynamical systems

In this paper, Brin-Katok local entropy formula and Katok＇s definition of the measuretheoretic entropy using spanning set are established for the random dynamical system over an invertible ergodic system.     

Note on the Markus-Yamabe conjecture for gradient dynamical systems

Let be a C¹ vector field which has a singular point O and its linearization is asymptotically stable at every point of Rn. We say that the vector field v satisfies the Markus-Yamabe conjecture if the critical point O is a global attractor of the dynamical system . In this note we prove that if v is a gradient vector field, i.e. v=∇f (f∈C²), then the basin of attraction of the critical point O is the whole Rn, thus implying the Markus-Yamabe conjecture for this class of vector fields. An analogous result for discrete dynamical systems of the form xm+1=∇f(xm) is proved.     

Leibniz代数胚上动力系统的轨道范例与图示

首先陈述Leibniz代数胚上的动力系统和在局部坐标系下该动力系统的方程,在此基础上,给出了动力系统轨道的范例和图示.     

Exponential attractors for first-order lattice dynamical systems

In l², we investigate the existence of an exponential attractor for the solution semigroup of a first-order lattice dynamical system acting on a closed bounded positively invariant set which needs not to be compact since l² is infinite dimensional. Up to our knowledge, this is the first time to examine the existence of exponential attractors for lattice dynamical systems.     

关于局部随机吸引子若干性质的注记(英文)

In this note, we study some properties of local random pull-back attractors on compact metric spaces. We obtain some relations between attractors and their fundamental neighborhoods and basins of attraction. We also obtain some properties of omega-limit sets, as well as connectedness of random attractors. A simple deterministic example is given to illustrate some confusing problems.     

Stability of Hausdorff dimension of Axiom A attractors under random perturbations

This paper proves that the Hausdorff dimension of an Axiom A attractor is stable under random perturbations.     

Random periodic solutions of random dynamical systems

In this paper, we give the definition of the random periodic solutions of random dynamical systems. We prove the existence of such periodic solutions for a C¹ perfect cocycle on a cylinder using a random invariant set, the Lyapunov exponents and the pullback of the cocycle.     

Universal quantification for deterministic chaos in dynamical systems

A cell dynamical system model for deterministic chaos enables precise quantification of the round-off error growth, i.e., deterministic chaos in digital computer realizations of mathematical models of continuum dynamical systems. The model predicts the following: (a) The phase space trajectory (strange attractor) when resolved as a function of the computer accuracy has intrinsic logarithmic spiral curvature with the quasiperiodic Penrose tiling pattern for the internal structure. (b) The universal constant for deterministic chaos is identified as the steady-state fractional round-off error k for each computational step and is equal to 1/τ² ( = 0.382) where τ is the golden mean. k being less than half accounts for the fractal (broken) Euclidean geometry of the strange attractor. (c) The Feigenbaum's universal constantsa and d are functions of k and, further, the expression 2a² = πd quantifies the steady-state ordered emergence of the fractal geometry of the strange attractor. (d) The power spectra of chaotic dynamical systems follow the universal and unique inverse power law form of the statistical normal distribution. The model prediction of (d) is verified for the Lorenz attractor and for the computable chaotic orbits of Bernoulli shifts, pseudorandom number generators, and cat maps.     

关于非自治动力系统中的h-极小覆盖

设（X,d1,f1∞）与（Y ,d2,g1,∞）为两个非自治动力系统,h是从（X,d1,f．∞）到（Y,d2,g1∞）的拓扑半共轭．通过对自治动力系统中的h一极小覆盖的研究,本文得到了以下结论：1）对于任意的Y∈Y及X∈h-1（y）,orb（x,f1∞）被h映射为orb（y,g1∞）,w（x,f1∞）被h映射为w（y,g1∞）;2）在（X,d1,f1∞）中引入关于拓扑半共轭的h-极小覆盖的定义,证明了h一极小覆盖的存在性;3）对于任意的XEX和Y∈Y,在（w（z,f1∞）,f1∞。（x,f1,∞）与（w（y,g∞）,g1,∞（y,g1∞））均构成原系统的子系统的前提下,R（f1∞）被h映射为R（g1∞）．这些结论丰富了非自治动力系统的内容．     

Some characteristics of complex behavior of orbits in dynamical systems

We present a brief review of mathematical notions of complexity based on instability of orbits. We show that the complexity as a function of time may grow exponentially in chaotic situations or polynomially for systems with zero topological entropy. At the end we discuss the class of nonchaotic systems for which all orbits are stable but nevertheless behavior of orbits is complex. We introduce a new notion of complexity for such a kind of systems.     

A nonrandom lyapunov spectrum for nonlinear stochastic dynamical systems

We present a nonrandom version of the Multiplicative Ergodic (Oseledec) Theorem for a nonlinear stochastic dynamical system on a smooth compact Riemannian Manifold M. This theorem characterises the a.s. asymptotic behaviour of the derivative system. Our approach (based on work of Furstenberg and Kifer, who deal with a linear system) is to consider an associated system on the projective bundle over M and to relate the behaviour of the theorem to the ergodic behaviour of this system. When the system has no random element, our work reduces to an alternative approach to the Multiplicative Ergodic Theorem for a diffeomorphism of M.     

Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications

In this paper, we first present some sufficient conditions for the existence of a global random attractor for general stochastic lattice dynamical systems. These sufficient conditions provide a convenient approach to obtain an upper bound of Kolmogorov ε-entropy for the global random attractor. Then we apply the abstract result to the stochastic lattice sine-Gordon equation.     

Sharp eigenvalue asymptotics for fourth order operators on the circle

We determine high energy asymptotics of eigenvalues of fourth order operator on the circle.     

随机环境中马氏链的最小闭集

主要研究了随机环境中马氏链的最小闭集的一些性质,并就保守集C在何种情况下存在最小闭子集的开问题结合Foguel的L1-理论进行了讨论,得到了一些结果.     

On the dynamics of nonautonomous periodic general dynamical systems and differential inclusions

We establish the existence and stability results for periodic nonautonomous uniform forward attractors of periodic general dynamical systems (set-valued dynamical systems). We also investigate the dynamical behavior of nonautonomous periodic differential inclusion x^′(t)∈f(t,x(t)) on R^m with only upper semi-continuous right-hand side by applying the abstract results. Firstly, we show that if the system has a compact uniformly attracting set, then it has a periodic nonautonomous uniform forward attractor A. Secondly, we prove that A is robust with respect to both internal and external perturbations. Finally, we apply the robustness result to discuss the effects of small time delays to asymptotic stability of the system.     

Some problems on fractal geometry and topological dynamical systems

This paper is a continuation of [14], some new problems on fractal geometry and topological dynamical systems have been posed.     

Integrals polynomial in velocity for two-degrees-of-freedom dynamical systems whose configuration space is a torus

We consider dynamical systems with two degrees of freedom whose configuration space is a torus and which admit first integrals polynomial in velocity. We obtain constructive criteria for the existence of conditional linear and quadratic integrals on the two-dimensional torus. Moreover, we show that under some additional conditions the degree of an irreducible integral of the geodesic flow on the torus does not exceed 2.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 37–44, July, 1998.The author wishes to express his thanks to V. V. Kozlov for his interest and his help in this work.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00747.