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1.
This paper focuses on chaos induced by snap-back repellers in non-autonomous discrete systems. A new concept of snap-back repeller for non-autonomous discrete systems is introduced and several new criteria of chaos induced by snap-back repellers in non-autonomous discrete systems are established. In addition, it is proved that a regular and nondegenerate snap-back repeller in non-autonomous discrete systems implies chaos in the (strong) sense of Li–Yorke. Two illustrative examples are proved.  相似文献   

2.
This paper is concerned with chaos induced by strictly turbulent maps in noncompact sets of complete metric spaces. Two criteria of chaos for such types of maps are established, and then a criterion of chaos, characterized by snap-back repellers in complete metric spaces, is obtained. All the maps presented in this paper are proved to be chaotic either in the sense of both Li–Yorke and Wiggins or in the sense of both Li–Yorke and Devaney. The results weaken the assumptions in some existing criteria of chaos. Several illustrative examples are provided with computer simulation.  相似文献   

3.
Li and Yorke not only introduced the term “chaos” along with a mathematically rigorous definition of what they meant by it, but also gave a condition for chaos in scalar difference equations, their equally famous “period three implies chaos” result. Generalizations of the Li and Yorke definition of chaos to difference equations in ? n are reviewed here as well as higher dimensional conditions ensuring its existence, specifically the “snap-back repeller” condition of Marotto and its counterpart for saddle points. In addition, further generalizations to mappings in Banach spaces and complete metric spaces are considered. These will be illustrated with various simple examples including an application to chaotic dynamics on the metric space (?  n , D) of fuzzy sets on the base space ? n .  相似文献   

4.
This paper is concerned with chaos induced by regular snap-back repellers. One new criterion of chaos induced by strictly coupled-expanding maps in compact sets of metric spaces is established. By employing this criterion, the nondegenerateness assumption in the Marotto theorem established in 1978 is weakened. In addition, it is proved that a regular snap-back repeller and a regular homoclinic orbit to a regular expanding fixed point in finite-dimensional spaces imply chaos in the sense of Li-Yorke. An illustrative example is provided with computer simulations.  相似文献   

5.
In this paper, we study small perturbations of a class of chaotic discrete systems in Banach spaces induced by snap-back repellers. If a map has a regular and non-degenerate snap-back repeller, then it still has a regular and non-degenerate snap-back repeller under a sufficiently small perturbation. Consequently, the perturbed system is still chaotic in the sense of both Devaney and Li–Yorke as the original one. Furthermore, in order to study structural stability of maps with regular and non-degenerate snap-back repellers, we first discuss structural stability of strictly A-coupled-expanding maps in Banach spaces. Applying this result, we show that a map with a regular and non-degenerate snap-back repeller in a Banach space is C 1 structurally stable on its chaotic invariant set.  相似文献   

6.
The chaotic behavior of one-dimensional, 2-segment and 3-segment piecewise-linear maps is examined by using the concept of snap-back repellers introduced by Marotto and the parameters conditions of existence for snap-back repeller are obtained. Simulation results are presented to show the snap-back repeller,some periodic points and attracting interval cycles with chaotic intervals.  相似文献   

7.
A nonlinear discrete time Cournot duopoly game is investigated in this paper. The conditions of existence for saddle-node bifurcation, transcritical bifurcation and flip bifurcation are derived using the center manifold theorem and the bifurcation theory. We prove that there exists chaotic behavior in the sense of Marotto's definition of chaos. The numerical simulations not only show the consistence with our theoretical analysis, but also exhibit the complex but interesting dynamical behaviors of the model. The computation of maximum Lyapunov exponents confirms the theoretical analysis of the dynamical behaviors of the system.  相似文献   

8.
Marotto extended Li–Yorke?s theorem on chaos from one-dimension to multi-dimension through introducing the notion of snapback repeller in 1978. Due to a technical flaw, he redefined snapback repeller in 2005 to validate this theorem. This presentation provides two methodologies to facilitate the application of Marotto?s theorem. The first one is to estimate the radius of repelling neighborhood for a repelling fixed point. This estimation is of essential and practical significance as combined with numerical computations of snapback points. Secondly, we propose a sequential graphic-iteration scheme to construct homoclinic orbit for a repeller. This construction allows us to track the homoclinic orbit. Applications of the present methodologies with numerical computation to a chaotic neural network and a predator–prey model are demonstrated.  相似文献   

9.
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.  相似文献   

10.
This paper is concerned with chaos of time-varying (i.e. non-autonomous) discrete systems in metric spaces. Some basic concepts are introduced for general time-varying systems, including periodic point, coupled-expansion for transitive matrix, uniformly topological equiconjugacy, and three definitions of chaos, i.e. chaos in the sense of Devaney and Wiggins, respectively, and in a strong sense of Li–Yorke. An interesting observation is that a finite-dimensional linear time-varying system can be chaotic in the original sense of Li–Yorke, but cannot have chaos in the strong sense of Li–Yorke, nor in the sense of Devaney in a set containing infinitely many points, and nor in the sense of Wiggins in a set starting from which all the orbits are bounded. A criterion of chaos in the original sense of Li–Yorke is established for finite-dimensional linear time-varying systems. Some basic properties of topological conjugacy are discussed. In particular, it is shown that topological conjugacy alone cannot guarantee two topologically conjugate time-varying systems to have the same topological properties in general. In addition, a criterion of chaos induced by strict coupled-expansion for a certain irreducible transitive matrix is established, under which the corresponding nonlinear system is proved chaotic in the strong sense of Li–Yorke. Two illustrative examples are finally provided with computer simulations for illustration.  相似文献   

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