Exponential stability of singularly perturbed impulsive delay differential equations

In this paper, the exponential stability of singularly perturbed impulsive delay differential equations (SPIDDEs) is concerned. We first establish a delay differential inequality, which is useful to deal with the stability of SPIDDEs, and then by the obtained inequality, a sufficient condition is provided to ensure that any solution of SPIDDEs is exponentially stable for sufficiently small ε>0. A numerical example and the simulation result show the effectiveness of our theoretical result.     

Uniform asymptotic stability for perturbed neutral delay differential equations

One-dimensional perturbed neutral delay differential equations of the form (x(t)−P(t,x(t−τ)))′=f(t,x_t)+g(t,x_t) are considered assuming that f satisfies −v(t)M(φ)?f(t,φ)?v(t)M(−φ), where M(φ)=max{0,max_s∈[−r,0]φ(s)}. A typical result is the following: if ‖g(t,φ)‖?w(t)‖φ‖ and , then the zero solution is uniformly asymptotically stable providing that the zero solution of the corresponding equation without perturbation (x(t)−P(t,x(t−τ)))′=f(t,x_t) is uniformly asymptotically stable. Some known results associated with this equation are extended and improved.     

Dissipativity of delay functional differential equations with bounded lag

Delay functional differential equations are essentially different from ordinary differential equations because their phase space is infinite dimensional. We first establish a sufficient condition for delay functional differential equations with bounded lag to be dissipative. Then we construct a one-leg θ-method to solve such dissipative equations and prove that it is dissipative if θ=1. One numerical example is given to confirm our theoretical result.     

Stability conditions for singularly perturbed delay differential equations

Singular perturbation problems containing a small positive parameter ε occur in many areas, including biochemical kinetics, genetics, plasma physics, and mechanical and electrical systems. A uniformly valid, reliable interpretable approximation of such problems is required. This paper provides sufficient conditions to ensure the exponential stability of the analytical and numerical solutions of the singularly perturbed delay differential equations with a bounded time-lag for suf.ciently small ε > 0. The Halanay inequality is used to prove the main results of the paper. A numerical example is provided to illustrate the methodology and clarify the need for a stiff solver for numerical solutions of these problems.     

Dynamics of a singularly perturbed system of two differential equations with delay

Theoretical and Mathematical Physics - We study a two-dimensional singularly perturbed system with delay, which is a simplification of models used in laser physics. We analyze several cases of a...     

A parameter-uniform scheme for singularly perturbed partial differential equations with a time lag

A numerical scheme for a class of singularly perturbed delay parabolic partial differential equations which has wide applications in the various branches of science and engineering is suggested. The solution of these problems exhibits a parabolic boundary layer on the lateral side of the rectangular domain which continuously depends on the perturbation parameter. For the small perturbation parameter, the standard numerical schemes for the solution of these problems fail to resolve the boundary layer(s) and the oscillations occur near the boundary layer. Thus, in this paper to resolve the boundary layer the extended cubic B-spline basis functions consisting of a free parameter λ are used on a fitted-mesh. The extended B-splines are the extension of classical B-splines. To find the best value of λ the optimization technique is adopted. The extended cubic B-splines are an advantage over the classical B-splines as for some optimized value of λ the solution obtained by the extended B-splines is better than the solution obtained by classical B-splines. The method is shown to be first-order accurate in t and almost the second-order accurate in x. It is also shown that this method is better than some existing methods. Several test problems are encountered to validate the theoretical results.     

Asymptotics of solutions of a singularly perturbed system of differential equations with delay

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 5, pp. 652–658, May, 1989.     

Uniform difference method for parameterized singularly perturbed delay differential equations

This paper deals with the singularly perturbed initial value problem for quasilinear first-order delay differential equation depending on a parameter. A numerical method is constructed for this problem which involves an appropriate piecewise-uniform meshes on each time subinterval. The difference scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter. Some numerical experiments illustrate in practice the result of convergence proved theoretically.     

Fitted finite difference method for singularly perturbed delay differential equations

This paper deals with singularly perturbed initial value problem for linear second-order delay differential equation. An exponentially fitted difference scheme is constructed in an equidistant mesh, which gives first order uniform convergence in the discrete maximum norm. The difference scheme is shown to be uniformly convergent to the continuous solution with respect to the perturbation parameter. A numerical example is solved using the presented method and compared the computed result with exact solution of the problem.     

Periodic solutions and stability for a singularly perturbed linear delay differential equation

We describe the bifurcation hypersurfaces for periodic solutions of a singularly perturbed linear differential difference equation in the space of the coefficients of the equation. For low dimension we show that the locus of stability of that equation approaches the locus of stability of a limit difference equation.     

Asymptotic expansion for the solution of singularly perturbed delay differential equations

Singular perturbation problems occur in many areas, including biochemical kinetics, genetics, plasma physics, and mechanical and electrical systems. For practical problems, one seeks a uniformly valid, readily interpretable approximation to a solution that does not behave uniformly. In this paper we extend singular perturbation theory in ordinary differential equations to delay differential equations with a fixed lag. We aim to give an explicit sufficient condition so that the solution of a class of singularly perturbed delay differential equations can be asymptotically expanded. O'Malley-Hoppensteadt technique is adopted in the construction of approximate solutions for such problems. Some particular phenomena different from singularly perturbed ordinary differential equations are discovered.     

Absolute exponential stability of nonlinear systems of differential equations with lag

Constructive sufficient conditions of absolute exponential stability for a class of nonlinear systems of differential equations with lag are found.     

Periodic solutions of singularly perturbed delay equations

We consider a general class of singularly perturbed delay differential systems depending on a singular parameter and another parameter . For =0, the equation defines a mapT which undergoes a generic period doubling at =0. If the bifurcation is supercritical (subcritical), these period two points define a stable period two square wave (unstable period two pulse wave). We give conditions on the vector field such that there is a sectorS in the , plane such that there is a unique periodic orbit if the parameters are inS, the orbit is stable (unstable) if the period doubling bifurcation is supercritical (subcritical) and approaches the square (pulse) wave as 0.Partially supported by NSF and DARPA.     

On the internal layer for a singularly perturbed system of second-order delay differential equations

We consider a nonlinear singularly perturbed boundary value problem with delay. By using the method of boundary functions and the theory of contrast structures, we prove the existence of a smooth solution with an internal transition layer and construct its uniform asymptotic expansion in a small parameter.     

Robust exponential attractors for singularly perturbed Hodgkin-Huxley equations

Here we consider a singular perturbation of the Hodgkin-Huxley system which is derived from the Lieberstein's model. We study the associated dynamical system on a suitable bounded phase space, when the perturbation parameter ε (i.e., the axon specific inductance) is sufficiently small. We prove the existence of bounded absorbing sets as well as of smooth attracting sets. We deduce the existence of a smooth global attractor Aε. Finally we prove the main result, that is, the existence of a family of exponential attractors {Eε} which is Hölder continuous with respect to ε.     

On discontinuous Galerkin finite element method for singularly perturbed delay differential equations

A nonsymmetric discontinuous Galerkin FEM with interior penalties has been applied to one-dimensional singularly perturbed problem with a constant negative shift. Using higher order polynomials on Shishkin-type layer-adapted meshes, a robust convergence has been proved in the corresponding energy norm. Numerical experiments support theoretical findings.