scholarly journals PULLBACK EXPONENTIAL ATTRACTORS FOR NONAUTONOMOUS DYNAMICAL SYSTEM IN SPACE OF HIGHER REGULARITY

2016 ◽  
Vol 6 (1) ◽  
pp. 242-253 ◽  
Author(s):  
Yongjun Li ◽  
◽  
Tinggang Zhao ◽  
Hongqing Wu ◽  
Jinying Wei
Keyword(s):  
Dynamical System ◽  
Exponential Attractors ◽  
Nonautonomous Dynamical System ◽  
Higher Regularity

scholarly journals The Existence of WeakD-Pullback Exponential Attractor for Nonautonomous Dynamical System

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Yongjun Li ◽  
Xiaona Wei ◽  
Yanhong Zhang
Keyword(s):  
Dynamical System ◽  
Diffusion Equation ◽  
External Force ◽  
Exponential Growth ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Exponential Attractor ◽  
Exponential Attractors ◽  
Nonautonomous Dynamical System

First, for a processU(t,τ)∣t≥τ, we introduce a new concept, called the weakD-pullback exponential attractor, which is a family of setsM(t)∣t≤T, for anyT∈R, satisfying the following: (i)M(t)is compact, (ii)M(t)is positively invariant, that is,U(t,τ)M(τ)⊂M(t), and (iii) there existk,l>0such thatdist(U(t,τ)B(τ),M(t))≤ke-(t-τ); that is,M(t)pullback exponential attractsB(τ). Then we give a method to obtain the existence of weakD-pullback exponential attractors for a process. As an application, we obtain the existence of weakD-pullback exponential attractor for reaction diffusion equation inH01with exponential growth of the external force.

scholarly journals Metric Entropy of Nonautonomous Dynamical Systems

2014 ◽  
Vol 1 (1) ◽  
Author(s):  
Christoph Kawan
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Metric Entropy ◽  
Nonautonomous Dynamical Systems ◽  
Nonautonomous Dynamical System

AbstractWe introduce the notion of metric entropy for a nonautonomous dynamical system given by a sequence (X

scholarly journals SHRINKING TARGETS FOR NONAUTONOMOUS DYNAMICAL SYSTEMS CORRESPONDING TO CANTOR SERIES EXPANSIONS

2015 ◽  
Vol 92 (2) ◽  
pp. 205-213 ◽  
Author(s):  
LIOR FISHMAN ◽  
BILL MANCE ◽  
DAVID SIMMONS ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Hausdorff Dimension ◽  
Diophantine Approximation ◽  
Wide Class ◽  
Series Expansions ◽  
Closed Formula ◽  
Nonautonomous Dynamical Systems ◽  
Nonautonomous Dynamical System ◽  
Cantor Series

We provide a closed formula of Bowen type for the Hausdorff dimension of a very general shrinking target scheme generated by the nonautonomous dynamical system on the interval$[0,1)$, viewed as$\mathbb{R}/\mathbb{Z}$, corresponding to a given method of Cantor series expansion. We also examine a wide class of examples utilising our theorem. In particular, we give a Diophantine approximation interpretation of our scheme.

Pullback and Uniform Exponential Attractors for Nonautonomous Boussinesq Lattice System

2015 ◽  
Vol 25 (08) ◽  
pp. 1550100 ◽  
Author(s):  
Min Zhao ◽  
Shengfan Zhou
Keyword(s):  
Dynamical System ◽  
Boussinesq Equations ◽  
Lattice System ◽  
Time Dependent ◽  
Pullback Attractor ◽  
Exponential Attractor ◽  
Uniform Attractor ◽  
Exponential Attractors ◽  
Lattice Dynamical System ◽  
External Forces

We first prove the existence of a pullback attractor and a pullback exponential attractor for a nonautonomous lattice dynamical system of nonlinear Boussinesq equations affected by time-dependent coupled coefficients and forces. Then, we prove the existence of a uniform attractor and a uniform exponential attractor for the system driven by quasi-periodic external forces.

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.

Remarks on Topological Entropy of Nonautonomous Dynamical Systems

2015 ◽  
Vol 25 (12) ◽  
pp. 1550158 ◽  
Author(s):  
Zhiming Li
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Topological Entropy ◽  
Nonautonomous Dynamical Systems ◽  
Nonautonomous Dynamical System

In this paper, we give several classical definitions of topological entropy (on a noncompact and noninvariant subset) for nonautonomous dynamical system. Furthermore, their relationships are established.

Directional entropy of ℤ+k-actions

2015 ◽  
Vol 16 (01) ◽  
pp. 1650004 ◽  
Author(s):  
Jinlian Zhang ◽  
Wenda Zhang
Keyword(s):  
Dynamical System ◽  
Variational Principle ◽  
Lyapunov Exponents ◽  
Riemannian Manifolds ◽  
Metric Space ◽  
Compact Metric Space ◽  
Finite Graphs ◽  
Nonautonomous Dynamical System

In this paper, topological and measure-theoretic directional entropies are investigated for [Formula: see text]-actions. Let [Formula: see text] be a [Formula: see text]-action on a compact metric space. For each ray [Formula: see text] in [Formula: see text] we introduce a notion of positive expansivity for [Formula: see text] along [Formula: see text]. We apply the technique of “coding” which was given by Boyle and Lind in [1] to show that these directional entropies are both continuous at positively expansive directions. We relate the directional entropies of a [Formula: see text]-action at a ray [Formula: see text] to the entropies of a nonautonomous dynamical system which induced by the compositions of a sequence of maps along [Formula: see text]. And hence the variational principle relating topological and measure-theoretic directional entropies is given at positively expansive directions. Applying some known results relating entropies and other invariants (such as preimage entropies, degrees and Lyapunov exponents), we obtain the formulas of directional entropies for some classic examples, such as the [Formula: see text]-subshift actions on [Formula: see text], [Formula: see text]-actions on finite graphs and certain smooth [Formula: see text]-actions on Riemannian manifolds.

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