Dissipativity And Stability Of Several Classes Of Delay Differential Equations And Numerical Discretizations

Since the twentieth century, ordinary differential equations(ODEs) and partial differential equations(PDEs) with delays arise widely in scientific fields such as economics,biology, ecology, medicine, physics and fluid dynamics and so on. It is meaningful to investigate the qualitative theory and application of numerical methods for delay ODEs and PDEs. Considering the existence of different types of delays(constant delays, timevarying delays, finitely distributed delays and infinitely distributed delays), this paper investigates the dynamical behavior of neutral delay integro-differential equations, BAM neural networks with both time-varying and infinitely distributed delays, BAM neural networks with diffusion effects and mixed delays, advection-reaction-diffusion equations with both fixed and finitely distributed delays. Based on this, some numerical methods for neutral delay integro-differential equations and advection-reaction-diffusion equations with both fixed and finitely distributed delays are constructed. It is proved that the numerical methods discussed in the present paper have the ability to preserve dynamical behavior of the underlying systems. The main contents of this paper includes the following five aspects:Firstly, the dissipativity criteria for a class of neutral delay integro-differential equations are obtained by applying the generalization of the Halanay inequality. The one-legθ-methods together with repeated trapezoidal rule for the neutral delay integro-differential equations is introduced. It is shown that, for θ ∈(1/2, 1], the one-leg θ-methods can inherit the dissipativity of the underlying system. An adaptation of the linear θ-methods for the neutral delay integro-differential equations is introduced as well. By using the relationship between the one-leg methods and linear multistep methods, the dissipative of the linear θ-methods is obtained.Secondly, global dissipativity of a class of BAM neural networks with both timevarying and infinitely distributed delays is investigated. New generalized Halanay inequality and Lyapunov functional are employed to approach the su?cient conditions for the global dissipativity and the global exponential dissipativity of the BAM neural networks with both variable and infinitely distributed time delays. Moreover, the estimations of the positive invariant set, globally attractive set and globally exponential attractive set are carried out. Finally, by using Matlab linear matrix inequality(LMI) toolbox, some examples are given to demonstrate the effectiveness of the obtained results.Thirdly, diffusion effect is introduced to a class of BAM neural networks with both fixed delays and infinitely distributed delays. The existence and globally asymptotical stability of equilibrium of BAM neural networks with diffusion terms and mixed delays are studied. Under the assumption that the activation functions only satisfy global Lipschitz conditions, a novel LMI-based su?cient condition for the existence of equilibrium is obtained by using degree theory, LMI method and inequalities technique. After that, the globally asymptotical stability of the equilibrium is discussed by constructing a appropriate Lyapunov functional. In this paper, the assumptions for boundedness and monotonicity in existing papers on the activation functions are removed, and the validity of the obtained results is easily illustrated by using Matlab LMI toolbox.Fourthly, for a class of delay advection-reaction-diffusion equations with Dirichlet boundary conditions, some su?cient conditions for the dissipativity are given in the L2-norm. Linear θ-methods and one-leg θ-methods with compound quadrature formulae for the delayed advection-reaction-diffusion equations are introduced, respectively. It is shown that, for θ ∈ [1/2, 1], any linear θ-method and one-leg θ-method can inherit the dissipativity of the underlying systems.Fifthly, energy estimates, dissipativity, asymptotic stability, and contractivity of a class of non-Fickian delay advection-reaction-diffusion equations are considered. Some energy estimates for the L2-norm of the solution are proposed by constructing an energy function. Based on this, some su?cient conditions for the dissipativity, asymptotic stability, and contractivity of the underlying systems are obtained. Moreover, backward Euler method together with centered differences operator and the right rectangular rule has the ability to preserve asymptotic stability and contractivity of the underlying systems.