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.

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.

Random attractors for second-order stochastic lattice dynamical systems

2010 ◽  
Vol 72 (1) ◽  
pp. 483-494 ◽  
Author(s):  
Xiaohu Wang ◽  
Shuyong Li ◽  
Daoyi Xu
Keyword(s):  
Dynamical Systems ◽  
Second Order ◽  
Random Attractors ◽  
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).

scholarly journals Existence of Random Attractors for a Class of Second-Order Lattice Dynamical Systems with Brownian Motions

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Yamin Wang ◽  
Ziqiao Huang ◽  
Fuad E. Alsaadi ◽  
Stanislao Lauria ◽  
Yurong Liu
Keyword(s):  
Dynamical Systems ◽  
Second Order ◽  
Random Attractor ◽  
Random Dynamical Systems ◽  
Brownian Motions ◽  
Asymptotic Compactness ◽  
Random Attractors ◽  
Lattice Dynamical Systems ◽  
Uniqueness And Existence ◽  
Special Case

This paper is concerned with the random attractors for a class of second-order stochastic lattice dynamical systems. We first prove the uniqueness and existence of the solutions of second-order stochastic lattice dynamical systems in the spaceF=lλ2×l2. Then, by proving the asymptotic compactness of the random dynamical systems, we establish the existence of the global random attractor. The system under consideration is quite general, and many existing results can be regarded as the special case of our results.

SYNCHRONIZATION OF A CLASS OF SECOND-ORDER NONLINEAR SYSTEMS

2008 ◽  
Vol 18 (11) ◽  
pp. 3461-3471 ◽  
Author(s):  
A. P. MIJOLARO ◽  
L. F. C. ABERTO ◽  
N. G. BRETAS
Keyword(s):  
Asymptotic Behavior ◽  
Vector Field ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Second Order ◽  
Coupled Systems ◽  
Nonlinear Dynamical ◽  
Unbounded Solutions ◽  
Synchronization Region ◽  
Coupling Parameters

The asymptotic behavior of a class of coupled second-order nonlinear dynamical systems is studied in this paper. Using very mild assumptions on the vector-field, conditions on the coupling parameters that guarantee synchronization are provided. The proposed result does not require solutions to be ultimately bounded in order to prove synchronization, therefore it can be used to study coupled systems that do not globally synchronize, including synchronization of unbounded solutions. In this case, estimates of the synchronization region are obtained. Synchronization of two-coupled nonlinear pendulums and two-coupled Duffing systems are studied to illustrate the application of the proposed theory.

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