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A dynamical system is called a null system, if the topological sequence entropy along any strictly increasing sequence of non-negative integers is 0. Let 0≦p≦q≦1. A dynamical system is Dqp chaotic, if there is an uncountable subset in which any two different points have trajectory approaching time set with lower density p and upper density q. In this paper, we show that there is a null system which is also D3/41/4 chaotic. 相似文献
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树映射有异状点的一个充要条件 总被引:8,自引:0,他引:8
讨论了树上连续自映射的拓扑熵与非稳定流形之间的关系. 证明了:树上连续自映射有异状点的充要条件是其拓扑熵大于零. 因而推广了区间上连续自映射的一个结果. 相似文献
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Tatsuya Arai 《Topology and its Applications》2007,154(7):1254-1262
Let f be a continuous map from a compact metric space X to itself. The map f is called to be P-chaotic if it has the pseudo-orbit-tracing property and the closure of the set of all periodic points for f is equal to X. We show that every P-chaotic map from a continuum to itself is chaotic in the sense of Devaney and exhibits distributional chaos of type 1 with positive topological entropy. 相似文献
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J. S. Cánovas 《Acta Mathematica Hungarica》2000,88(1-2):123-131
We provide an example showing that, in general, the topological sequence entropy of a continuous interval map cannot be attained on the set of non-wandering points. This proves that there is no connection between the topological sequence entropy of an interval map and its behavior on sets of special dynamical meaning. 相似文献
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In this paper we prove a sufficient condition for the continuous map of a compact metric space for being distributively chaotic in a sequence. As an application, it is proved that a continuous map of an interval is chaotic in the Li–Yorke sense if and only if it is distributively chaotic in a sequence. 相似文献
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Michal Málek 《Journal of Difference Equations and Applications》2016,22(3):468-473
Omega-limit sets play an important role in one-dimensional dynamics. During last fifty year at least three definitions of basic set has appeared. Authors often use results with different definition. Here we fill in the gap of missing proof of equivalency of these definitions. Using results on basic sets we generalize results in paper [P. Oprocha, Invariant scrambled sets and distributional chaos, Dyn. Syst. 24 (2009), no. 1, 31–43.] to the case continuous maps of finite graphs. The Li-Yorke chaos is weaker than positive topological entropy. The equivalency arises when we add condition of invariance to Li-Yorke scrambled set. In this note we show that for a continuous graph map properties positive topological entropy; horseshoe; invariant Li-Yorke scrambled set; uniform invariant distributional chaotic scrambled set and distributionaly chaotic pair are mutually equivalent. 相似文献
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Lidong Wang Gongfu Liao ZhenYan Chu XiaoDong Duan 《Journal of Applied Mathematics and Computing》2004,14(1-2):277-288
In this paper, we discuss a continuous self-map of an interval and the existence of an uncountable strongly chaotic set. It is proved that if a continuous self-map of an interval has positive topological entropy, then it has an uncountable strongly chaotic set in which each point is recurrent, but is not almost periodic. 相似文献
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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
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Jonq Juang 《Journal of Mathematical Analysis and Applications》2008,341(2):1055-1067
A concept related to total variation termed H1 condition was recently proposed to characterize the chaotic behavior of an interval map f by Chen, Huang and Huang [G. Chen, T. Huang, Y. Huang, Chaotic behavior of interval maps and total variations of iterates, Internat. J. Bifur. Chaos 14 (2004) 2161-2186]. In this paper, we establish connections between H1 condition, sensitivity and topological entropy for interval maps. First, we introduce a notion of restrictiveness of a piecewise-monotone continuous interval map. We then prove that H1 condition of a piecewise-monotone continuous map implies the non-restrictiveness of the map. In addition, we also show that either H1 condition or sensitivity then gives the positivity of the topological entropy of f. 相似文献
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Ll. Alsedà D. Juher P. Mumbrú 《Proceedings of the American Mathematical Society》2001,129(10):2941-2946
This paper deals with the relationship between the periodic orbits of continuous maps on graphs and the topological entropy of the map. We show that the topological entropy of a graph map can be approximated by the entropy of its periodic orbits.
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A convenient measure of a map or flow’s chaotic action is the topological entropy. In many cases, the entropy has a homological origin: it is forced by the topology of the space. For example, in simple toral maps, the topological entropy is exactly equal to the growth induced by the map on the fundamental group of the torus. However, in many situations the numerically computed topological entropy is greater than the bound implied by this action. We associate this gap between the bound and the true entropy with ‘secondary folding’: material lines undergo folding which is not homologically forced. We examine this phenomenon both for physical rod stirring devices and toral linked twist maps, and show rigorously that for the latter secondary folds occur. 相似文献
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José M. Amigó Peter E. Kloeden Ángel Giménez 《Journal of Difference Equations and Applications》2013,19(11):1872-1888
Switching systems are non-autonomous dynamical systems obtained by switching between two or more autonomous dynamical systems as time goes on. They can be mainly found in control theory, physics, economy, biomathematics, chaotic cryptography and of course in the theory of dynamical systems, in both discrete and continuous time. Much of the recent interest in these systems is related to the emergence of new properties by the mechanism of switching, a phenomenon known in the literature as Parrondo's paradox. In this paper we consider a discrete-time switching system composed of two affine transformations and show that the switched dynamics has the same topological entropy as the switching sequence. The complexity of the switching sequence, as measured by the topological entropy, is fully transferred, for example, to the switched dynamics in this particular case. 相似文献
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《Topology and its Applications》2005,146(5-6):735-746
The aim of this paper is to introduce a definition of topological entropy for continuous maps such that, at least for continuous real maps, it keeps the following general philosophy: positive topological entropy implies that the map has a complicated dynamical behaviour. Besides, we pursue that our definition keeps some properties which are hold by the classic definition of topological entropy introduced for compact sets. 相似文献
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按序列分布混沌与拓扑混合 总被引:2,自引:0,他引:2
本文讨论了按序列分布混沌与拓扑混合的关系,并证明了:若X为至少两点的可分局部紧致度量空间,连续映射f:X→X是拓扑混合的,则对于任一正整数递增序列{mi},存在X的c-稠密Fσ子集D是f按{mi}的某子序列的分布混沌集. 相似文献
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By a topological dynamical system, we mean a pair (X,f), where X is a compactum and f is a continuous self-map on X. A system is said to be null if its topological sequence entropies are zero along all strictly increasing sequences of natural numbers. We show that there exists a null system which is distributionally chaotic. This system admits open distributionally scrambled sets, and its collection of all maximal distributionally scrambled sets has the same cardinality as the collection of all subsets of the phase space. Finally such system can even exist on continua. 相似文献
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Periodica Mathematica Hungarica - For a given integer $$k\ge 3$$ , a sequence A of nonnegative integers is called an $$AP_k$$ -covering sequence if there exists an integer $$n_0$$ such... 相似文献
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We study the topological entropy for dynamical systems with discrete or continuous multiple time. Due to the generalization of a well-known one time-dimensional result we show that the definition of topological entropy, using the approach for subshifts, leads to the zero entropy for many systems different from subshift. We define a new type of relative topological entropy to avoid this phenomenon. The generalization of Bowen’s power rule allows us to define topological and relative topological entropies for systems with continuous multiple time. As an application, we find a relation between the relative topological entropy and controllability of linear systems with continuous multiple time. 相似文献
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A class of new simple Hopfield neural networks is revisited. To confirm the chaotic behavior in these Hopfield neural networks demonstrated in numerical studies, we resort to Poincaré section and Poincaré map technique and present a rigorous verification of existence of horseshoe chaos by virtue of topological horseshoes theory and estimates of topological entropy in the derived Poincaré maps. 相似文献