Kolmogorov entropy of global attractor for dissipative lattice dynamical systems

2003 ◽  
Vol 44 (12) ◽  
pp. 5804 ◽  
Author(s):  
Qiuli Jia ◽  
Shengfan Zhou ◽  
Fuqi Yin
Keyword(s):  
Dynamical Systems ◽  
Global Attractor ◽  
Kolmogorov Entropy ◽  
Lattice Dynamical Systems

scholarly journals Lattice dynamical systems: dissipative mechanism and fractal dimension of global and exponential attractors

2019 ◽  
Vol 20 (2) ◽  
pp. 485-515
Author(s):  
Jan W. Cholewa ◽  
Radosław Czaja
Keyword(s):  
Fractal Dimension ◽  
Dynamical Systems ◽  
Global Attractor ◽  
American Mathematical Society ◽  
Reaction Diffusion Equations ◽  
Dissipative Systems ◽  
Exponential Attractor ◽  
Lattice Systems ◽  
Exponential Attractors ◽  
Lattice Dynamical Systems

Abstract In this work, we examine first-order lattice dynamical systems, which are discretized versions of reaction–diffusion equations on the real line. We prove the existence of a global attractor in $$\ell ^2$$ℓ2, and using the method by Chueshov and Lasiecka (Dynamics of quasi-stable dissipative systems, Springer, Berlin, 2015; Memoirs of the American Mathematical Society, vol 195(912), AMS, 2008), we estimate its fractal dimension. We also show that the global attractor is contained in a finite-dimensional exponential attractor. The approach relies on the interplay between the discretized diffusion and reaction, which has not been exploited as yet for the lattice systems. Of separate interest is a characterization of positive definiteness of the discretized Schrödinger operator, which refers to the well-known Arendt and Batty’s result (Differ Int Equ 6:1009–1024, 1993).

ATTRACTORS FOR LATTICE DYNAMICAL SYSTEMS

2001 ◽  
Vol 11 (01) ◽  
pp. 143-153 ◽  
Author(s):  
PETER W. BATES ◽  
KENING LU ◽  
BIXIANG WANG
Keyword(s):  
Asymptotic Behavior ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Global Attractor ◽  
Upper Semicontinuity ◽  
Global Attractors ◽  
Asymptotic Behavior Of Solutions ◽  
Lattice Differential Equations ◽  
Finite Dimensional ◽  
Lattice Dynamical Systems

We study the asymptotic behavior of solutions for lattice dynamical systems. We first prove asymptotic compactness and then establish the existence of global attractors. The upper semicontinuity of the global attractor is also obtained when the lattice differential equations are approached by finite-dimensional systems.

scholarly journals Pullback attractors via quasi-stability for non-autonomous lattice dynamical systems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Radosław Czaja
Keyword(s):  
Dynamical Systems ◽  
Global Attractor ◽  
Exponential Attractor ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Pullback Attractors ◽  
Diffusion Operator ◽  
First Order ◽  
Lattice Dynamical Systems ◽  
Long Time

<p style='text-indent:20px;'>In this paper we study long-time behavior of first-order non-autono-mous lattice dynamical systems in square summable space of double-sided sequences using the cooperation between the discretized diffusion operator and the discretized reaction term. We obtain existence of a pullback global attractor and construct pullback exponential attractor applying the introduced notion of quasi-stability of the corresponding evolution process.</p>

scholarly journals Asymptotic behavior for second order lattice dynamical systems

2001 ◽  
Vol 6 (2) ◽  
pp. 137-143
Author(s):  
Shengfan Zhou
Keyword(s):  
Asymptotic Behavior ◽  
Dynamical Systems ◽  
Global Attractor ◽  
Second Order ◽  
Lattice Dynamical Systems

We consider the existence of the global attractor for a second order lattice dynamical systems.

Development of mixing and isotropy in inviscid homogeneous turbulence

1982 ◽  
Vol 120 ◽  
pp. 155-183 ◽  
Author(s):  
Jon Lee
Keyword(s):  
Dynamical Systems ◽  
Lower Order ◽  
Stokes System ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Fourier Space ◽  
Kolmogorov Entropy ◽  
Navier Stokes Equations ◽  
Constant Energy

We have investigated a sequence of dynamical systems corresponding to spherical truncations of the incompressible three-dimensional Navier-Stokes equations in Fourier space. For lower-order truncated systems up to the spherical truncation of wavenumber radius 4, it is concluded that the inviscid Navier-Stokes system will develop mixing (and a fortiori ergodicity) on the constant energy-helicity surface, and also isotropy of the covariance spectral tensor. This conclusion is, however, drawn not directly from the mixing definition but from the observation that one cannot evolve the trajectory numerically much beyond several characteristic corre- lation times of the smallest eddy owing to the accumulation of round-off errors. The limited evolution time is a manifestation of trajectory instability (exponential orbit separation) which underlies not only mixing, but also the stronger dynamical charac- terization of positive Kolmogorov entropy (K-system).

Attractors for multi-valued lattice dynamical systems with nonlinear diffusion terms

2021 ◽  
Author(s):  
Panpan Zhang ◽  
Anhui Gu
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Diffusion ◽  
Upper Semicontinuity ◽  
Growth Conditions ◽  
Pullback Attractor ◽  
Random Lattice ◽  
Lipschitz Continuous ◽  
Lattice Dynamical Systems ◽  
Long Term Behavior

This paper is devoted to the long-term behavior of nonautonomous random lattice dynamical systems with nonlinear diffusion terms. The nonlinear drift and diffusion terms are not expected to be Lipschitz continuous but satisfy the continuity and growth conditions. We first prove the existence of solutions, and establish the existence of a multi-valued nonautonomous cocycle. We then show the existence and uniqueness of pullback attractors parameterized by sample parameters. Finally, we establish the measurability of this pullback attractor by the method based on the weak upper semicontinuity of the solutions.

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