Uniform Exponential Attractor for Second Order Lattice System with Quasi-Periodic External Forces in Weighted Space

2014 ◽  
Vol 24 (01) ◽  
pp. 1450006 ◽  
Author(s):  
Shengfan Zhou ◽  
Min Zhao
Keyword(s):  
Weighted Space ◽  
Second Order ◽  
Lattice System ◽  
Exponential Attractor ◽  
Uniform Attractor ◽  
External Forces ◽  
Infinite Sequences

In this paper, we study the existence of a uniform exponential attractor for second order lattice system with quasi-periodic external forces in weighted space of infinite sequences. We first prove that the system possesses a uniform attractor. Then we obtain the existence of a uniform exponential attractor for the system.

Random exponential attractor for second-order nonautonomous stochastic lattice systems with multiplicative white noise

2019 ◽  
Vol 19 (06) ◽  
pp. 1950044
Author(s):  
Haijuan Su ◽  
Shengfan Zhou ◽  
Luyao Wu
Keyword(s):  
White Noise ◽  
Weighted Space ◽  
Sufficient Conditions ◽  
Second Order ◽  
Lattice System ◽  
Exponential Attractor ◽  
Lattice Systems ◽  
Random System ◽  
Multiplicative White Noise ◽  
Infinite Sequences

We studied the existence of a random exponential attractor in the weighted space of infinite sequences for second-order nonautonomous stochastic lattice system with linear multiplicative white noise. Firstly, we present some sufficient conditions for the existence of a random exponential attractor for a continuous cocycle defined on a weighted space of infinite sequences. Secondly, we transferred the second-order stochastic lattice system with multiplicative white noise into a random lattice system without noise through the Ornstein–Uhlenbeck process, whose solutions generate a continuous cocycle on a weighted space of infinite sequences. Thirdly, we estimated the bound and tail of solutions for the random system. Fourthly, we verified the Lipschitz continuity of the continuous cocycle and decomposed the difference between two solutions into a sum of two parts, and carefully estimated the bound of the norm of each part and the expectations of some random variables. Finally, we obtained the existence of a random exponential attractor for the considered system.

scholarly journals Pullback Exponential Attractor for Second Order Nonautonomous Lattice System

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Shengfan Zhou ◽  
Hong Chen ◽  
Zhaojuan Wang
Keyword(s):  
Fractal Dimension ◽  
Product Space ◽  
Weighted Space ◽  
Continuous Process ◽  
Sufficient Conditions ◽  
Second Order ◽  
Lattice System ◽  
Time Dependent ◽  
Exponential Attractor ◽  
Infinite Sequences

We first present some sufficient conditions for the existence of a pullback exponential attractor for continuous process on the product space of the weighted spaces of infinite sequences. Then we prove the existence and continuity of a pullback exponential attractor for second order lattice system with time-dependent coupled coefficients in the weighted space of infinite sequences. Moreover, we obtain the upper bound of fractal dimension and attracting rate for the attractor.

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.

Asymptotic Dynamics of Second Order Nonautonomous Systems on Infinite Lattices

2016 ◽  
Vol 26 (01) ◽  
pp. 1650003
Author(s):  
Ahmed Y. Abdallah
Keyword(s):  
Sufficient Conditions ◽  
Second Order ◽  
Exponential Attractor ◽  
Nonlinear Part ◽  
Nonautonomous Systems ◽  
Lattice Dynamical System ◽  
Lattice Dynamical Systems ◽  
Asymptotic Dynamics ◽  
Infinite Lattices ◽  
Infinite Sequences

We have introduced abstract sufficient conditions for the existence of a uniform exponential attractor for a special family of second order nonautonomous lattice dynamical systems with quasiperiodic symbols in a standard space of infinite sequences. Compared with the lattice dynamical system in [Zhou & Zhao, 2014], here a generalized nonlinear part and weaker assumptions have been presented, kindly see Remark Remark 2.1 for more details.

On the spectra of non-self-adjoint realisations of second-order elliptic operators

1981 ◽  
Vol 90 (1-2) ◽  
pp. 71-105 ◽  
Author(s):  
W. D. Evans
Keyword(s):  
Spherical Shell ◽  
Essential Spectrum ◽  
Weighted Space ◽  
Elliptic Operators ◽  
Second Order ◽  
Sectorial Operator ◽  
Image Position ◽  
Second Order Elliptic Operators ◽  
Complex Valued

SynopsisLet τ denote the second-order elliptic expressionwhere the coefficients bj and q are complex-valued, and let Ω be a spherical shell Ω = {x:x ∈ ℝn, l <|x|<m} with l≧0, m≦∞. Under the conditions assumed on the coefficients of τ and with either Dirichlet or Neumann conditions on the boundary of Ω, τ generates a quasi-m-sectorial operator T in the weighted space L2(Ω;w). The main objective is to locate the spectrum and essential spectrum of T. Best possible results are obtained.

scholarly journals Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems

2005 ◽  
Vol 135 (4) ◽  
pp. 703-730 ◽  
Author(s):  
M. Efendiev ◽  
S. Zelik ◽  
A. Miranville
Keyword(s):  
Dynamical Systems ◽  
Reaction Diffusion ◽  
Exponential Attractor ◽  
Hölder Continuous ◽  
Exponential Attractors ◽  
Finite Dimensional Reduction ◽  
Finite Dimensional ◽  
Equations Of Mathematical Physics ◽  
External Forces ◽  
Autonomous Dynamical Systems

We suggest in this paper a new explicit algorithm allowing us to construct exponential attractors which are uniformly Hölder continuous with respect to the variation of the dynamical system in some natural large class. Moreover, we extend this construction to non-autonomous dynamical systems (dynamical processes) treating in that case the exponential attractor as a uniformly exponentially attracting, finite-dimensional and time-dependent set in the phase space. In particular, this result shows that, for a wide class of non-autonomous equations of mathematical physics, the limit dynamics remains finite dimensional no matter how complicated the dependence of the external forces on time is. We illustrate the main results of this paper on the model example of a non-autonomous reaction–diffusion system in a bounded domain.

A DETAILED MONTE CARLO INVESTIGATION OF THE TRICRITICAL REGION OF A BIAXIAL LIQUID CRYSTAL SYSTEM

1999 ◽  
Vol 10 (02n03) ◽  
pp. 469-476 ◽  
Author(s):  
CESARE CHICCOLI ◽  
PAOLO PASINI ◽  
FRANCO SEMERIA ◽  
CLAUDIO ZANNONI
Keyword(s):  
Monte Carlo ◽  
Liquid Crystal ◽  
Second Order ◽  
Lattice System ◽  
Crystal System ◽  
Monte Carlo Calculations ◽  
Liquid Crystal System ◽  
Monte Carlo Investigation ◽  
Limiting Case

We study a lattice system of biaxial particles interacting with a second-rank anisotropic potential. We have performed detailed Monte Carlo calculations in the vicinity of the prolate–oblate dual value of molecular biaxiality. Our results confirm the second-order character of the transition in this limiting case.

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