Attractors For Retarded Lattice Dynamical Systems

Recently, the dynamics of lattice systems has drawn the attention of many researchers, and they mainly focus on coupled lattices and coupled ordinary differentialequations (see [1, 2, 3, 4]). Lattice dynamical systems arise in a wide variety of applications (see[5, 6. 7, 8, 9]), ranging from biology to chemical reactiontheory, pattern recognition, electrical engineering and laser systems. Lattice dynamical systems (LDSs) are spatiotemporal systems with discretization in some variables including coupled ordinary differential equations. In some cases. LDSs occur as spatial discretizations of partial differential equations on unbounded domains.In many cases, the long time behavior of LDSs which is extended by evolutionequations is described by the attractors of semigroup. However, the mathematicalstudies of PDEs happened to be very complicated, and basically all the successes in that area for a very long time were restricted to the finding of some simple solutions and to the studies of their stability. LDSs can be regarded as an "approximation" to the corresponding continuous dissipative PDEs if they arise as spatial discretizations of dissipative PDEs, therefore, it is significant to study the dynamics of the lattice systems corresponding to the initial value problem of dissipative PDEs on unbounded domains. In[4], using the idea of "tail end", B.X. Wang study the following LDSs:(?)_i=v(u_i-1-2u_i+u_i+1+ f(u_i)+g_i, u_i(0)=u_0i, i∈Z,and obtain the global attractor of above systems. Where f satisfying f(0) = 0, f(u)u≤-αu² +β, f′(u)≤γ.In[17], CD. Zhao also use the idea of "tail end" for studying the following retarded lattice dynamical systems:(?)_i + 2u_i - v(u_i-1 - u_i+1) +λ_iu_i+f_i(u_it) = g_i, u_i(0) =u_i(s), s∈[-v,0] i∈Z.Base on above papers, we develop the concept of asymptotically null, under more suitable dissipative conditions than [17],(1). f is a C′function:(2). Let Y is a subset of Banach space, then for any u,v∈Y, there exist a positive constant C such that||f(u)-f(v)||≤C||u-v||;(3) There exist positive constantsα,β,η, (?),ρ, such thatf(0) = 0, |f′|≤ρ, f(u)v≥αu² + v(βuv +ηv²) - (?), for all u, v∈R.and wherewe study the following retarded lattice systems:(?)_i + 2u_i - u_i-1- u_i+1+λu_i+ f(u_it) = g_i, u_i(0) =u_i(s), and obtain the existence result of the global attractor:定理3. Let g = (g_i)_i∈Z∈l_σ² and (1)-(3) hold. Then, the semigroup {S_v(t)}_t≥0 associated with equations (2.1) possesses a global attractor A_v (?) X_v, for each fixed v > 0.By comparing with the global attractor of the following lattice systems:(?)+ Au(t)+λu(t)+f(u(t))=g, t>0, u(0)=u₀, we obtain the upper semi-continuity:定理4. Let g = (g_i)_i∈Z∈l_σ² and (1)-(3) hold. Then,Our results show that a small delay has little effect on the asymptotic dynamicsof first order lattice dynamics systems.