scholarly journals Global attractor for the lattice dynamical system of a nonlinear Boussinesq equation

2005 ◽  
Vol 2005 (6) ◽  
pp. 655-671 ◽  
Author(s):  
Ahmed Y. Abdallah
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Global Attractor ◽  
Fourth Order ◽  
Boussinesq Equation ◽  
Upper Semicontinuity ◽  
Time Variable ◽  
Finite Dimensional ◽  
Lattice Dynamical System ◽  
Second Order In Time

We will study the lattice dynamical system of a nonlinear Boussinesq equation. Our objective is to explore the existence of the global attractor for the solution semiflow of the introduced lattice system and to investigate its upper semicontinuity with respect to a sequence of finite-dimensional approximate systems. As far as we are aware, our result here is the first concerning the lattice dynamical system corresponding to a differential equation of second order in time variable and fourth order in spatial variable with nonlinearity involving the gradients.

ATTRACTORS FOR A LATTICE DYNAMICAL SYSTEM GENERATED BY NON-NEWTONIAN FLUIDS MODELING SUSPENSIONS

2010 ◽  
Vol 20 (09) ◽  
pp. 2681-2700 ◽  
Author(s):  
JOSÉ M. AMIGÓ ◽  
ÁNGEL GIMÉNEZ ◽  
FRANCISCO MORILLAS ◽  
JOSÉ VALERO
Keyword(s):  
Dynamical System ◽  
Upper Semicontinuity ◽  
Global Attractors ◽  
Euler Method ◽  
Equation Modeling ◽  
Suspension Flows ◽  
Upper Semicontinuous ◽  
Implicit Euler Method ◽  
Finite Dimensional ◽  
Lattice Dynamical System

In this paper we consider a lattice dynamical system generated by a parabolic equation modeling suspension flows. We prove the existence of a global compact connected attractor for this system and the upper semicontinuity of this attractor with respect to finite-dimensional approximations. Also, we obtain a sequence of approximating discrete dynamical systems by the implementation of the implicit Euler method, proving the existence and the upper semicontinuous convergence of their global attractors.

scholarly journals Upper semicontinuity of the attractor for lattice dynamical systems of partly dissipative reaction diffusion systems

2005 ◽  
Vol 2005 (3) ◽  
pp. 273-288 ◽  
Author(s):  
Ahmed Y. Abdallah
Keyword(s):  
Dynamical System ◽  
Hilbert Space ◽  
Upper Semicontinuity ◽  
Reaction Diffusion ◽  
Dimensional Lattice ◽  
Reaction Diffusion Systems ◽  
Infinite Dimensional ◽  
Diffusion Systems ◽  
Lattice Dynamical System ◽  
Lattice Dynamical Systems

We investigate the existence of a global attractor and its upper semicontinuity for the infinite-dimensional lattice dynamical system of a partly dissipative reaction diffusion system in the Hilbert spacel2×l2. Such a system is similar to the discretized FitzHugh-Nagumo system in neurobiology, which is an adequate justification for its study.

scholarly journals Existence and approximation of attractors for nonlinear coupled lattice wave equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Lianbing She ◽  
Mirelson M. Freitas ◽  
Mauricio S. Vinhote ◽  
Renhai Wang
Keyword(s):  
Asymptotic Behavior ◽  
Global Attractor ◽  
Wave Equations ◽  
Upper Semicontinuity ◽  
Asymptotic Behavior Of Solutions ◽  
Well Posedness ◽  
Asymptotic Compactness ◽  
Finite Dimensional ◽  
Lattice Wave ◽  
Infinite Lattices

<p style='text-indent:20px;'>This paper is concerned with the asymptotic behavior of solutions to a class of nonlinear coupled discrete wave equations defined on the whole integer set. We first establish the well-posedness of the systems in <inline-formula><tex-math id="M1">\begin{document}$ E: = \ell^2\times\ell^2\times\ell^2\times\ell^2 $\end{document}</tex-math></inline-formula>. We then prove that the solution semigroup has a unique global attractor in <inline-formula><tex-math id="M2">\begin{document}$ E $\end{document}</tex-math></inline-formula>. We finally prove that this attractor can be approximated in terms of upper semicontinuity of <inline-formula><tex-math id="M3">\begin{document}$ E $\end{document}</tex-math></inline-formula> by a finite-dimensional global attractor of a <inline-formula><tex-math id="M4">\begin{document}$ 2(2n+1) $\end{document}</tex-math></inline-formula>-dimensional truncation system as <inline-formula><tex-math id="M5">\begin{document}$ n $\end{document}</tex-math></inline-formula> goes to infinity. The idea of uniform tail-estimates developed by Wang (Phys. D, 128 (1999) 41-52) is employed to prove the asymptotic compactness of the solution semigroups in order to overcome the lack of compactness in infinite lattices.</p>

scholarly journals Exponential Attractor for Lattice System of Nonlinear Boussinesq Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Min Zhao ◽  
Shengfan Zhou
Keyword(s):  
Dynamical System ◽  
Continuous Semigroup ◽  
Boussinesq Equation ◽  
Lipschitz Continuity ◽  
Lattice System ◽  
Exponential Attractor ◽  
Lattice Dynamical System ◽  
The Difference

We study the lattice dynamical system of a nonlinear Boussinesq equation. We first verify the Lipschitz continuity of the continuous semigroup associated with the system. Then, we provide an estimation of the tail of the difference between two solutions of the system. Finally, we obtain the existence of an exponential attractor of the system.

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 The study of a fourth-order differential equation: existence, uniqueness and a dynamical system approach

2020 ◽  
Vol 34 ◽  
pp. 03005
Author(s):  
Cristian Paul Danet
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Fourth Order ◽  
Existence And Uniqueness ◽  
System Approach ◽  
Order Differential Equation ◽  
Stationary Points ◽  
Maximum Principles ◽  
Dynamical System Approach ◽  
Fourth Order Differential Equation

This paper is concerned with the problem of existence and uniqueness of solutions for the semilinear fourth-order differential equation uiv – ku′′ + a(x)u+c(x) f (u) = 0. Existence and uniqueness is proved using variational methods and maximum principles. We also give a dynamical system approach to the equation. We study the bifurcation of the system and show that the behaviour of the stationary points S (α, 0, 0, 0) depend on the relation between the parameter k and β = f ′(α).

scholarly journals A nonselfadjoint dynamical system

1974 ◽  
Vol 19 (1) ◽  
pp. 77-87 ◽  
Author(s):  
Kurt Kreith
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Fourth Order ◽  
Polar Coordinates ◽  
Image Position

This paper concerns criteria for assuring that every solution of a real fourth order nonselfadjoint differential equationis oscillatory at x = ∞. Our technique is a generalisation of that used by Whyburn (1) for the study of the selfadjoint equation,combined with the theory of H-oscillation of vector equations as introduced by Domšlak (2) and studied by Noussair and Swanson (3). Whyburn's technique consists of representing (1.2) as a dynamical system of the formand then studying (1.3) in terms of polar coordinates in the y, z-plane. In Section 2 below we show how to represent (1.1) as a dynamical system of the form

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