New methods for proving the existence and stability of periodic solutions in singularly perturbed delay systems

We carry out a detailed analysis of the existence, asymptotics, and stability problems for periodic solutions that bifurcate from the zero equilibrium state in systems with large delay. The account is based on a specific meaningful example given by a certain scalar nonlinear second-order differential-difference equation that is a mathematical model of a single-circuit RCL oscillator with delay in a feedback loop.     

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.     

Exponential stability of singularly perturbed switched systems with time delay

This paper investigates exponential stability of singularly perturbed switched systems with time delay. The multiple Lyapunov functions technique and dwell time approach are used to establish stability criteria for a switched system consisting of both stable and unstable subsystems. Examples are presented to illustrate the criteria.     

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 existence,uniqueness and stability of periodic solutions of large scale systems with unbounded delay

In this paper, we consider the periodic solution problems for the systems with unbounded delay, and the existence, uniqueness and stability of the periodic solution are dealt with unitedly. First we establish the suitable delay-differential inequality, then study seperately the problems of periodic solution for the systems with bounded delay, with unbounded delay and the Volterra integral-differential systems with infinite delay by using the character of matrix measure and the asymptotic fixed point theorem of poincaré's periodic operator in the dfferent phase spaces. A series of simple criteria for the existence, uniqueness and stability of these systems are obtained.     

Canard solutions of two-dimensional singularly perturbed systems

In this paper, some new lemmas on asymptotic analysis are established. We apply an asymptotic method to study generalized two-dimensional singularly perturbed systems with one parameter, whose critical manifold has an $\frac{m-2}{2}$ th-order degenerate extreme point. Certain sufficient conditions are obtained for the existence of canard solutions, which are the extension and correction of some existing results. Finally, one numerical example is given.     

On some periodic solutions of singularly perturbed parabolic equations

We present results from the theory of singular perturbations and, in particular, from a new branch of this theory (contrast alternating-type structures). __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 3, pp. 359–369, March, 2007.     

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.     

On the existence of almost periodic solutions of a singularly perturbed differential equation with piecewise constant argument

Based on the investigation of almost periodic solutions to difference equations, the existence of almost periodic solution for a nonautonomous, singularly perturbed differential equation with piecewise constant argument is considered. In addition, the stability properties of these solutions are characterized by the construction of manifolds of initial data.     

Bifurcation of periodic solutions of a codimension one set of singularly perturbed parabolic systems

Using the single-frequency method we construct periodic in time solutions of parabolic systems of reaction-diffusion type with a small matrix diffusion.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 8, pp. 1037–1042, August, 1990.     

The existence and uniqueness of positive periodic solutions of Nicholson-type delay systems

This paper is concerned with a class of Nicholson blowfly systems with multiple time-varying delays. By applying the method of the Lyapunov functional, some criteria are established for the existence and uniqueness of positive periodic solutions of the system. Moreover, an example is given to illustrate the main results.     

Globally exponential stability of periodic solutions for nonlinear impulsive delay systems

In this paper, the existence and globally exponential stability of periodic solutions for nonlinear impulsive delay systems are studied. Our results improve and generalize some of the previous results found in the literature. Two examples are discussed to illustrate our results.     

Homoclinic,heteroclinic and periodic orbits of singularly perturbed systems

The main aims of this paper are to study the persistence of homoclinic and heteroclinic orbits of the reduced systems on normally hyperbolic critical manifolds, and also the limit cycle bifurcations either from the homoclinic loop of the reduced systems or from a family of periodic orbits of the layer systems. For the persistence of homoclinic and heteroclinic orbits, and the limit cycles bifurcating from a homolinic loop of the reduced systems, we provide a new and readily detectable method to characterize them compared with the usual Melnikov method when the reduced system forms a generalized rotated vector field. To determine the limit cycles bifurcating from the families of periodic orbits of the layer systems, we apply the averaging methods.We also provide two four-dimensional singularly perturbed differential systems, which have either heteroclinic or homoclinic orbits located on the slow manifolds and also three limit cycles bifurcating from the periodic orbits of the layer system.     

Asymptotic expansions of solutions to singularly perturbed systems

Under the condition that a degenerate system has an exponentially stable integral manifold, an asymptotic expansion of the Cauchy problem that generalizes the well known Vasil'eva expansion is constructed for a perturbed system.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 4, pp. 552–560, April, 1993.     

Diagonalization and stability of multi-time-scale singularly perturbed linear systems

In this paper, an alternate approach to the method of asymptotic expansions for the study of a singularly perturbed, linear system with multiparameters and multi-time scales, is developed. The method consists of developing a linear, nonsingular transformation that enables to decouple the original system completely. This process of diagonalization thus provides a very simple and suitable tool to investigate (i) the stability analysis and (ii) approximations of solutions of original system in terms of the overall reduced system and the corresponding boundary layer systems.     

New existence of periodic solutions for second order non-autonomous Hamiltonian systems

By using mountain pass theorem and local link theorem, some existence theorems are obtained for periodic solutions of second order non-autonomous Hamiltonian systems under local superquadratic condition and other suitable conditions.     

Asymptotic stability via the Krein-Rutman theorem for singularly perturbed parabolic periodic dirichlet problems

We consider singularly perturbed semilinear parabolic periodic problems and assume the existence of a family of solutions. We present an approach to establish the exponential asymptotic stability of these solutions by means of a special class of lower and upper solutions. The proof is based on a corollary of the Krein-Rutman theorem.