On Generic Properties of Nonautonomous Dynamical Systems

2018 ◽  
Vol 28 (08) ◽  
pp. 1850102 ◽  
Author(s):  
Francisco Balibrea ◽  
Jaroslav Smítal ◽  
Marta Štefánková
Keyword(s):  
Dynamical Systems ◽  
Topological Entropy ◽  
Compact Space ◽  
Metric Space ◽  
Generic Properties ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Nonautonomous Dynamical Systems ◽  
Uniformly Convergent ◽  
Generic System

We consider nonautonomous dynamical systems consisting of sequences of continuous surjective maps of a compact metric space [Formula: see text]. Let [Formula: see text], [Formula: see text] and [Formula: see text] denote the space of systems [Formula: see text], which are uniformly convergent, or equicontinuous, or pointwise convergent (to a continuous map), respectively. We show that for [Formula: see text], the generic system in any of the spaces has infinite topological entropy, while, if [Formula: see text] is the Cantor set, the generic system in any of the spaces has zero topological entropy. This shows, among others, that the general results of the above type for arbitrary compact space [Formula: see text] are difficult if not impossible.

scholarly journals A remark on positive topological entropy ofN-buffer switched flow networks

2005 ◽  
Vol 2005 (2) ◽  
pp. 93-99 ◽  
Author(s):  
Xiao-Song Yang
Keyword(s):  
Topological Entropy ◽  
Metric Space ◽  
Elementary Proof ◽  
Flow Networks ◽  
Compact Metric Space ◽  
Positive Topological Entropy ◽  
Continuous Map ◽  
Simple Theorem

We present a simpler elementary proof on positive topological entropy of theN-buffer switched flow networks based on a new simple theorem on positive topological entropy of continuous map on compact metric space.

CHAOTIC MAP WITH THE PROPERTY OF RECURRENCE

2007 ◽  
Vol 21 (15) ◽  
pp. 2711-2721 ◽  
Author(s):  
LIDONG WANG ◽  
ZHENYAN CHU ◽  
XIAODONG DUAN
Keyword(s):  
Topological Entropy ◽  
Metric Space ◽  
Chaotic Map ◽  
Compact Metric Space ◽  
Continuous Map

In this paper, we consider a continuous map f: X→X, where X is a compact metric space, and discuss the existence of a chaotic set of f specially (as X=[0,1]). We prove that f has a positively topological entropy if and only if it has an uncountably chaotic set in which each point is recurrent and is not weakly periodic.

scholarly journals The chain recurrent set, attractors, and explosions

1985 ◽  
Vol 5 (3) ◽  
pp. 321-327 ◽  
Author(s):  
Louis Block ◽  
John E. Franke
Keyword(s):  
Compact Space ◽  
Metric Space ◽  
Periodic Points ◽  
Limit Set ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Special Case ◽  
Equivalent Conditions ◽  
Chain Recurrent Set ◽  
Recurrent Set

AbstractCharles Conley has shown that for a flow on a compact metric space, a point x is chain recurrent if and only if any attractor which contains the & ω-limit set of x also contains x. In this paper we show that the same statement holds for a continuous map of a compact metric space to itself, and additional equivalent conditions can be given. A stronger result is obtained if the space is locally connected.It follows, as a special case, that if a map of the circle to itself has no periodic points then every point is chain recurrent. Also, for any homeomorphism of the circle to itself, the chain recurrent set is either the set of periodic points or the entire circle. Finally, we use the equivalent conditions mentioned above to show that for any continuous map f of a compact space to itself, if the non-wandering set equals the chain recurrent set then f does not permit Ω-explosions. The converse holds on manifolds.

Weakly Almost Periodic Points and Some Chaotic Properties of Dynamical Systems

2015 ◽  
Vol 25 (09) ◽  
pp. 1550115 ◽  
Author(s):  
Jiandong Yin ◽  
Zuoling Zhou
Keyword(s):  
Dynamical Systems ◽  
Metric Space ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Full Measure ◽  
Almost Periodic ◽  
Countable Base ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Weakly Almost Periodic

Let X be a compact metric space and f : X → X be a continuous map. In this paper, ergodic chaos and strongly ergodic chaos are introduced, and it is proven that f is strongly ergodically chaotic if f is transitive but not minimal and has a full measure center. In addition, some sufficient conditions for f to be Ruelle–Takens chaotic are presented. For instance, we prove that f is Ruelle–Takens chaotic if f is transitive and there exists a countable base [Formula: see text] of X such that for each i > 0, the meeting time set N(Ui, Ui) for Ui with respect to itself has lower density larger than [Formula: see text].

scholarly journals Sensitivity for topologically double ergodic dynamical systems

2021 ◽  
Vol 6 (10) ◽  
pp. 10495-10505
Author(s):  
Risong Li ◽  
◽  
Xiaofang Yang ◽  
Yongxi Jiang ◽  
Tianxiu Lu ◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Metric Space ◽  
Inverse Limit ◽  
Compact Metric Space ◽  
Limit Space ◽  
Factor Map ◽  
Continuous Surjection ◽  
Inverse Limit Space

<abstract><p>As a stronger form of multi-sensitivity, the notion of ergodic multi-sensitivity (resp. strongly ergodically multi-sensitivity) is introduced. In particularly, it is proved that every topologically double ergodic continuous selfmap (resp. topologically double strongly ergodic selfmap) on a compact metric space is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive). And for any given integer $ m\geq 2 $, $ f $ is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive) if and only if so is $ f^{m} $. Also, it is shown that if $ f $ is a continuous surjection, then $ f $ is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive) if and only if so is $ \sigma_{f} $, where $ \sigma_{f} $ is the shift selfmap on the inverse limit space $ \lim\limits_{\leftarrow}(X, f) $. Moreover, it is proved that if $ f:X\rightarrow X $ (resp. $ g:Y\rightarrow Y $) is a map on a nontrivial metric space $ (X, d) $ (resp. $ (Y, d') $), and $ \pi $ is a semiopen factor map between $ (X, f) $ and $ (Y, g) $, then the ergodic multi-sensitivity (resp. the strongly ergodic multi-sensitivity) of $ g $ implies the same property of $ f $.</p></abstract>

scholarly journals Li-Yorke Sensitivity of Set-Valued Discrete Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Heng Liu ◽  
Fengchun Lei ◽  
Lidong Wang
Keyword(s):  
Metric Space ◽  
Short Proof ◽  
Hausdorff Metric ◽  
Discrete Systems ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Nonempty Compact ◽  
Compact Subsets

Consider the surjective, continuous mapf:X→Xand the continuous mapf¯of𝒦(X)induced byf, whereXis a compact metric space and𝒦(X)is the space of all nonempty compact subsets ofXendowed with the Hausdorff metric. In this paper, we give a short proof that iff¯is Li-Yoke sensitive, thenfis Li-Yorke sensitive. Furthermore, we give an example showing that Li-Yorke sensitivity offdoes not imply Li-Yorke sensitivity off¯.

EQUI-ATTRACTION AND THE CONTINUOUS DEPENDENCE OF PULLBACK ATTRACTORS ON PARAMETERS

2004 ◽  
Vol 04 (03) ◽  
pp. 373-384 ◽  
Author(s):  
DESHENG LI ◽  
P. E. KLOEDEN
Keyword(s):  
Metric Space ◽  
Continuous Dependence ◽  
Hausdorff Metric ◽  
Regularity Conditions ◽  
Pullback Attractors ◽  
Compact Metric Space ◽  
Nonautonomous Dynamical Systems ◽  
The Family ◽  
Nonempty Compact ◽  
Autonomous Dynamical System

The equi-attraction properties of uniform pullback attractors [Formula: see text] of nonautonomous dynamical systems (θ,ϕλ) with a parameter λ∈Λ, where Λ is a compact metric space, are investigated; here θ is an autonomous dynamical system on a compact metric space P which drives the cocycle ϕλon a complete metric state space X. In particular, under appropriate regularity conditions, it is shown that the equi-attraction of the family [Formula: see text] uniformly in p∈P is equivalent to the continuity of the setvalued mappings [Formula: see text] in λ with respect to the Hausdorff metric on the nonempty compact subsets of X.

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