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1.
In this paper, we first introduce the measure-theoretic entropy for arbitrary Borel probability measure in nonautonomous case. Then we show that there is certain variational relation for nonautonomous dynamical systems.  相似文献   

2.
The minimality of a nonautonomous dynamical system given by a compact Hausdorff space X and a sequence of continuous self-maps of X is studied. A sufficient condition for the nonminimality of such a system is formulated. Special attention is given to the particular case where X is a real compact interval I. A sequence of continuous self-maps of I forming a minimal nonautonomous system may converge uniformly. For example, the limit may be any topologically transitive map. However, if all maps in the sequence are surjective, then the limit is necessarily monotone. An example where the limit is the identity is given. As an application, in a simple way we construct a triangular map in the square I 2 with the property that every point except those in the leftmost fiber has an orbit whose -limit set coincides with the leftmost fiber.  相似文献   

3.
In this paper we present an abstract approach to inertial manifolds for nonautonomous dynamical systems. Our result on the existence of inertial manifolds requires only two geometrical assumptions, called cone invariance and squeezing property, and two additional technical assumptions, called boundedness and coercivity property. Moreover we give conditions which ensure that the global pullback attractor is contained in the inertial manifold. In the second part of the paper we consider special nonautonomous dynamical systems, namely processes (or two-parameter semi-flows). As a first application of our abstract approach and for reason of comparison with known results we verify the assumptions for semilinear nonautonomous evolution equations whose linear part satisfies an exponential dichotomy condition and whose nonlinear part is globally bounded and globally Lipschitz.  相似文献   

4.
We show two examples of systems in with such that |Zt| is strictly decreasing in time for any n but as .  相似文献   

5.
The aim of this paper is to demonstrate that topological fixed point theorems have no canonical generalization to the case of random dynamical systems. This is done by using tools from algebraic ergodic theory. We give a criterion for the existence of invariant probability measures for group valued cocycles. With that, examples of continuous random dynamical systems on a compact interval without random invariant points, which are an appropriate generalization of fixed points, are constructed.  相似文献   

6.
利用能量原理,提出一种以能量为状态参数的分析非线性非自治系统拓扑结构的定性方法。分析了能量参数确定系统状态的唯一性,揭示了能量方法的正确性和有效性以及反映系统运动状态信息的丰富性。以非自治Duffing方程为对象,采用能量状态参数,应用提出的能量方法在相空间和运动空间展开了一般的定性分析,给出了能量相图和能量时间历程、能量Poincaré截面图、能量参数分岔图,分析了其中呈现出的运动规律和捕捉到的有价值信息,并与现有的方法进行了比较。  相似文献   

7.
Dynamical spectrum is a concept in terms of exponential dichotomy. The theory of dynamical spectrum, due to Sacker and Sell, plays important roles in many fields of dynamical systems and differential equations. Noticing its significance and importance, we study in this paper the theory of dynamical spectrum for some general random dynamical systems. More precisely, after introducing a random version of the concept of exponential dichotomy, by using some methods and techniques from dynamical systems and ergodic theory, under some general integrability conditions, we establish the dynamical spectral decomposition theorem in framework of random dynamical systems, which can be regarded as a random version of the deterministic dynamical systems due to Sacker and Sell. In our result, the dynamical spectral intervals and the corresponding spectral subbundles will be given.  相似文献   

8.
In this paper we study the continuity of invariant sets for nonautonomous infinite-dimensional dynamical systems under singular perturbations. We extend the existing results on lower-semicontinuity of attractors of autonomous and nonautonomous dynamical systems. This is accomplished through a detailed analysis of the structure of the invariant sets and its behavior under perturbation. We prove that a bounded hyperbolic global solutions persists under singular perturbations and that their nonlinear unstable manifold behave continuously. To accomplish this, we need to establish results on roughness of exponential dichotomies under these singular perturbations. Our results imply that, if the limiting pullback attractor of a nonautonomous dynamical system is the closure of a countable union of unstable manifolds of global bounded hyperbolic solutions, then it behaves continuously (upper and lower) under singular perturbations.  相似文献   

9.
In this article we establish an interaction between non-smooth systems, geometric singular perturbation theory and synchronization phenomena. We find conditions for a non-smooth vector fields be locally synchronized. Moreover its regularization provide a singular perturbation problem with attracting critical manifold. We also state a result about the synchronization which occurs in the regularization of the fold-fold case. We restrict ourselves to the 3-dimensional systems ( = 3) and consider the case known as a T-singularity.  相似文献   

10.
In this work, we introduce the notion of conductance in the context of Cuntz–Krieger C-algebras. These algebras can be seen as a noncommutative version of topological Markov chains. Conductance is a useful notion in the theory of Markov chains to study the approach of a system to the equilibrium state. Our goal is twofold. On one hand, conductance can be used to measure the complexity of dynamical systems, complementing topological entropy. On the other hand, using C-algebras, we can give a natural framework to analyze the path space of a finite graph associated to a Markov shift.  相似文献   

11.
In this paper new estimates on the C 0-norm of solutions are shown for first order convex Hamiltonian systems possessing super-quadratic potentials. Applying these estimates, some new results on the existence of subharmonics are obtained, which generalize the main results in Ekeland and Hofer [5], and a question about a priori estimates on subharmonics raised by Ekeland and Hofer [5] is answered when the convex Hamiltonian systems have globally super-quadratic potentials. Using the uniform estimates on the subharmonics, the behavior of convergence of subharmonics is studied too.  相似文献   

12.
International Applied Mechanics - For non-autonomous polynomial equations, a method for designing a stabilizing control based on the direct Lyapunov method and solving an algebraic equation of high...  相似文献   

13.
14.
In the present paper forward nonautonomous competitive systems of two parabolic second order partial differential equations are studied. The concept of forward uniform persistence for such systems is introduced. Sufficient conditions, expressed in terms of the principal spectrum, are given for those systems to be forward uniformly persistent.  相似文献   

15.
Journal of Dynamics and Differential Equations - We study the stability of general n-dimensional nonautonomous linear differential equations with infinite delays. Delay independent criteria, as...  相似文献   

16.
For a topological dynamical system $(X,T)$ ( X , T ) and $d\in \mathbb N $ d ∈ N , the associated dynamical parallelepiped $\mathbf{Q}^{[d]}$ Q [ d ] was defined by Host–Kra–Maass. For a minimal distal system it was shown by them that the relation $\sim _{d-1}$ ~ d ? 1 defined on $\mathbf{Q}^{[d-1]}$ Q [ d ? 1 ] is an equivalence relation; the closing parallelepiped property holds, and for each $x\in X$ x ∈ X the collection of points in $\mathbf{Q}^{[d]}$ Q [ d ] with first coordinate $x$ x is a minimal subset under the face transformations. We give examples showing that the results do not extend to general minimal systems.  相似文献   

17.
The scaling function and the linear model for a circle endomorphism are two important smooth invariants under conjugacy. We discuss these two invariants and some relations between them. Furthermore, we use these relations to discuss some realization results in this direction. The discussion in this paper avoids quasiconformal mapping theory and Gibbs theory and g-measure theory, which are used in our previous discussions, therefore, is straightforward and simple.This paper is dedicated to Professor Shui-Nee Chow on the occasion of his 60th Birthday.Mathematics Subject Classification 2000: Primary 37E10, Secondary 34C14  相似文献   

18.
We investigate problems of practical stability of set dynamical systems and prove the main properties of sets of initial conditions optimal with respect to inclusion. For systems with linear dynamical component and specific types of set components, we obtain optimal deformation functions and optimal estimates for practical stability.  相似文献   

19.
In this paper we study the existence of invariant manifolds for a special type of nonautonomous systems which arise in the study of discretization methods. According to [10], a one-step scheme of step-size for an autonomous system can be interpreted as the -flow of a perturbed nonautonomous system. The perturbation is rapidly forced in the sense that it is periodic with respect to time with period . Assuming a saddle node for the autonomous system, we prove that these rapidly forced perturbations have center manifolds which exist in a uniform neighborhood and which converge to a center manifold of the autonomous system as tends to zero. Our results are applied to obtain a smooth continuation as well as estimates of the well known center manifolds for one-step schemes. They also form the basis for studying saddle-node homoclinic orbits under discretization.  相似文献   

20.
Li  Li 《Nonlinear dynamics》1999,19(3):237-260
In this paper, based upon the concept of the conservation of average energy, a theorem for the periodic solution of strongly nonlinear nonautonomous systems is proved. By using this theorem not only the necessary and sufficient conditions for the existence and stability of the periodic solutions can be obtained but, at the same time, the approximate expressions for those periodic solutions can also be obtained.  相似文献   

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