Sensitivity of Nonautonomous Dynamical Systems on Uniform Spaces

2021 ◽  
Vol 31 (01) ◽  
pp. 2150017
Author(s):  
Mohammad Salman ◽  
Xinxing Wu ◽  
Ruchi Das
Keyword(s):  
Dynamical Systems ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Uniform Spaces ◽  
Topological Transitivity ◽  
Nonautonomous Systems ◽  
Nonautonomous Dynamical Systems

We introduce the concepts of sensitivity, multisensitivity, cofinite sensitivity, and syndetic sensitivity for nonautonomous dynamical systems on uniform spaces and obtain some sufficient conditions under which topological transitivity and dense periodic points imply sensitivity for nonautonomous systems on Hausdorff uniform spaces. We also study sensitivity and other stronger versions of sensitivity for the systems induced on hyperspaces and for the product of nonautonomous dynamical systems on uniform spaces.

scholarly journals On some kinds of dense orbits in general nonautonomous dynamical systems

2020 ◽  
Vol 7 (1) ◽  
pp. 163-175
Author(s):  
Mehdi Pourbarat
Keyword(s):  
Dynamical Systems ◽  
Initial Conditions ◽  
Point Of View ◽  
Topological Conjugacy ◽  
Topological Transitivity ◽  
Nonautonomous Systems ◽  
Nonautonomous Dynamical Systems ◽  
Sensitive Dependence ◽  
Dense Orbits

AbstractWe study the theory of universality for the nonautonomous dynamical systems from topological point of view related to hypercyclicity. The conditions are provided in a way that Birkhoff transitivity theorem can be extended. In the context of generalized linear nonautonomous systems, we show that either one of the topological transitivity or hypercyclicity give sensitive dependence on initial conditions. Meanwhile, some examples are presented for topological transitivity, hypercyclicity and topological conjugacy.

scholarly journals Immediate conditional hyperbolicity in dynamical systems

1983 ◽  
Vol 3 (4) ◽  
pp. 627-647
Author(s):  
Joseph Rosenblatt ◽  
Richard Swanson
Keyword(s):  
Vector Bundle ◽  
Dynamical Systems ◽  
Compact Manifold ◽  
Hyperbolic Systems ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Almost Periodic ◽  
Uniquely Ergodic

AbstractFor many diffeomorphisms of a compact manifold X, eventual conditional hyperbolicity implies immediate conditional hyperbolicity in some (possibly new) Finsler structures. That is, if A and B are vector bundle isomorphisms over the mapping ƒ of the base X, such that uniformly on X, then there exist new norms for A and B such that uniformly on X, whenever the mapping ƒ satisfies the condition that there exist infinitely many N ≥ 1 such that any ƒ-invariant. For example, this condition on ƒ holds if any one of the following conditions holds: (1) ƒ is periodic; (2) ƒ is periodic on its non-wandering set; (3) ƒ has a finite non-wandering set (for example, ƒ is a Morse-Smale diffeomorphism); (4) ƒ is an almost periodic mapping of a connected base X; (5) ƒ is a mapping of the circle with no periodic points; or (6) ƒ and all its powers are uniquely ergodic. We consider various types of eventually conditionally hyperbolic systems and describe sufficient conditions on ƒ to have immediate conditional hyperbolicity of these systems in some new Finsler structures. Thus, for a sizable class of dynamical systems, we settle, in the affirmative, a question raised by Hirsch, Pugh, and Shub.

scholarly journals Dynamics of Weakly Mixing Nonautonomous Systems

2019 ◽  
Vol 29 (09) ◽  
pp. 1950123 ◽  
Author(s):  
Mohammad Salman ◽  
Ruchi Das
Keyword(s):  
Dynamical System ◽  
Sufficient Conditions ◽  
Unit Interval ◽  
Nonautonomous System ◽  
Probability Measures ◽  
Topological Transitivity ◽  
Weak Mixing ◽  
Nonautonomous Systems ◽  
Nonautonomous Dynamical System ◽  
Weakly Mixing

For a commutative nonautonomous dynamical system we show that topological transitivity of the nonautonomous system induced on probability measures (hyperspaces) is equivalent to the weak mixing of the induced systems. Several counter examples are given for the results which are true in autonomous but need not be true in nonautonomous systems. Wherever possible sufficient conditions are obtained for the results to hold true. For a commutative periodic nonautonomous system on intervals, it is proved that weak mixing implies Devaney chaos. Given a periodic nonautonomous system, it is shown that sensitivity is equivalent to some stronger forms of sensitivity on a closed unit interval.

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].

Sensitivity and Chaoticity on Nonautonomous Dynamical Systems

2020 ◽  
Vol 30 (10) ◽  
pp. 2050146
Author(s):  
Nan Li ◽  
Lidong Wang
Keyword(s):  
Dynamical Systems ◽  
Sufficient Conditions ◽  
Nonautonomous Dynamical Systems ◽  
First Time

This paper discusses the sensitivity and chaotic properties of nonautonomous dynamical systems. For the first time we obtain sufficient conditions of some stronger forms of sensitivity for nonautonomous dynamical systems. Then, some chaotic relations between two nonautonomous dynamical systems are proved.

scholarly journals Almost all $S$-integer dynamical systems have many periodic points

1998 ◽  
Vol 18 (2) ◽  
pp. 471-486 ◽  
Author(s):  
T. B. WARD
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Zeta Function ◽  
Periodic Points ◽  
Radius Of Convergence ◽  
Dynamical Zeta Function ◽  
Hyperbolic Dynamical Systems ◽  
Almost All

We show that for almost every ergodic $S$-integer dynamical system the radius of convergence of the dynamical zeta function is no larger than $\exp(-\frac{1}{2}h_{\rm top})<1$. In the arithmetic case almost every zeta function is irrational.We conjecture that for almost every ergodic $S$-integer dynamical system the radius of convergence of the zeta function is exactly $\exp(-h_{\rm top})<1$ and the zeta function is irrational.In an important geometric case (the $S$-integer systems corresponding to isometric extensions of the full $p$-shift or, more generally, linear algebraic cellular automata on the full $p$-shift) we show that the conjecture holds with the possible exception of at most two primes $p$.Finally, we explicitly describe the structure of $S$-integer dynamical systems as isometric extensions of (quasi-)hyperbolic dynamical systems.

scholarly journals Ruelle operator theorem for non-expansive systems

2009 ◽  
Vol 30 (2) ◽  
pp. 469-487 ◽  
Author(s):  
YUNPING JIANG ◽  
YUAN-LING YE
Keyword(s):  
Dynamical Systems ◽  
Sufficient Conditions ◽  
Iterated Function Systems ◽  
Ruelle Operator ◽  
Iterated Function

AbstractThe Ruelle operator theorem has been studied extensively both in dynamical systems and iterated function systems. In this paper we study the Ruelle operator theorem for non-expansive systems. Our theorems give some sufficient conditions for the Ruelle operator theorem to be held for a non-expansive system.

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