On center sets of ODEs determined by moments of their coefficients

The classical H. Poincaré Center-Focus problem asks about the characterization of planar polynomial vector fields such that all their integral trajectories are closed curves whose interiors contain a fixed point, a center. This problem can be reduced to a center problem for some ordinary differential equation whose coefficients are trigonometric polynomials depending polynomially on the coefficients of the field. In this paper we show that the set of centers in the Center-Focus problem can be determined as the set of zeros of some continuous functions from the moments of coefficients of this equation.     

POWERS OF REAL SYMMETRIC DIFFERENTIAL EXPRESSIONS WITHOUT SMOOTHNESS ASSUMPTIONS

Abstract

In recent years many authors have studied properties of powers of symmetric (formally self-adjoint) ordinary linear differential expressions. So that these powers can be formed in the classical way these authors have placed heavy smoothness assumptions on the coefficients. Here we show that no differentiability conditions whatsoever are needed on the coefficients in order to form powers of a given expression-provided these powers are formed in the quasi-differential sense rather than the classical one.     

A model of immune system with time-dependent immune reactivity

In this paper we deal with the Marchuk model of an immune system. Among the main parameters of the model are the coefficients which describe the state of infected organism and the rate of production of antibodies. In the classical model these coefficients are constants. We consider the case when these coefficients are time-dependent. In particular, we are interested in the case of periodic coefficients which can describe periodic changes of the immune reactivity due to periodic changes of the environment. We examine the asymptotic behaviour of solutions. Under some assumptions we prove that the solutions tend to periodic functions. We also present the results of numerical simulations to illustrate the behaviour of solutions.     

Estimates on the growth of meromorphic solutions of linear differential equations

We give a pointwise estimate of meromorphic solutions of linear differential equations with coefficients meromorphic in a finite disk or in the open plane. Our results improve some earlier estimates of Bank and Laine. In particular we show that the growth of meromorphic solutions with ()>0 can be estimated in terms of initial conditions of the solution at or near the origin and the characteristic functions of the coefficients. Examples show that the estimates are sharp in a certain sense. Our results give an affirmative answer to a question of Milne Anderson. Our method consists of two steps. In Theorem 2.1 we construct a path (₀, , t) consisting of the ray followed by the circle on which the coefficients are all bounded in terms of the sum of their characteristic functions on a larger circle. In Theorem 2.2 we show how such an estimate for the coefficients leads to a corresponding bound for the solution on z = t. Putting these two steps together we obtain our main result, Theorem 2.3.     

Reflectionless Sturm–Liouville equations

We consider compactly supported perturbations of periodic Sturm–Liouville equations. In this context, one can use the Floquet solutions of the periodic background to define scattering coefficients. We prove that if the reflection coefficient is identically zero, then the operators corresponding to the periodic and perturbed equations, respectively, are unitarily equivalent. In some appendices, we also provide the proofs of several basic estimates, e.g., bounds and asymptotics for the relevant m$m$-functions.     

Non-real eigenvalues of indefinite Sturm–Liouville problems

The present paper deals with non-real eigenvalues of regular indefinite Sturm–Liouville problems. A priori bounds and sufficient conditions of the existence for non-real eigenvalues are obtained under mild integrable conditions of coefficients.     

On nonuniform exponential dichotomy of evolution operators in Banach spaces

The problem of nonuniform exponential dichotomy of evolution operators in Banach spaces is considered. Connections between this concept and admissibility of the pair (C ₀,C ₀) are established. Generalizations to the nonuniform case of some results of Van Min, Räbiger and Schnaubelt ([MRS]) are obtained. It is shown that an implication from the uniform case is not true for nonuniform exponential dichotomy. The results are applicable for general time-varying linear equations with unbounded coefficients in Banach spaces.     

PRODUCTS OF DIFFERENTIAL EXPRESSIONS WITHOUT SMOOTHNESS ASSUMPTIONS

Abstract

To form products of differential expressions in the classical way it is necessary to place heavy differentiability assumptions on the coefficients. Here we consider symmetric (formally self-adjoint) expressions defined, not in the classical way, but in terms of quasi-derivatives. With this very general notion of symmetry we show that products such as M1M2MI of symmetric expressions M1, Hp can be formed vithout any smoothness assumptions on the coefficients and such products are symmetric expressions.     

Weak and strong composition conditions for the Abel differential equation

We establish an equivalence between two forms of the composition condition for the Abel differential equation with trigonometric coefficients.     

AN OSCILLATION THEOREM FOR A TWO-PARAMETER SYSTEM OF DIFFERENTIAL EQUATIONS

Abstract

The oscillation theorem for two simultaneous Sturm-Liouville systems in two parameters is well known when the coefficients of the differential equations are subjected to the usual definiteness condition. However, in practical applications the usual definiteness condition may fail to hold, and hence in this paper we consider the oscillation theorem under another important definiteness condition.     

Exponential convergence behavior of shunting inhibitory cellular neural networks with time-varying coefficients

In this paper the convergence behavior of delayed shunting inhibitory cellular neural networks with time-varying coefficients are considered. Some sufficient conditions are established to ensure that all solutions of the networks converge exponentially to the zero point, which are new and complement previously known results.     

Stability and instability criteria for Kaplan–Yorke solutions

We derive sufficient conditions for the stability and instability of periodic solutions of Kaplan–Yorke type to the equation where f is even in the first and odd in the second argument. The criteria are based on the monotonicity of the coefficient in a transformed version of the variational equation. For the special case of cubic f, we show that this monotonicity property is satisfied if and only if the set is contained in a region E defined by a quadratic form (bounded by an an ellipse or a hyperbola). The coefficients of this quadratic form are expressible in terms of the Taylor coefficients of f. Further, the parameter α in the equation and the amplitude z of the periodic solution are related by an elliptic integral. Using the relation between this integral and the arithmeticgeometric mean, we obtain upper and lower estimates on this relation, and on the inverse function. Combining these estimates with the inequality that defines the region E, we obtain stability criteria explicit in terms of the Taylor coefficients of f. These criteria go well beyond local stability analysis, as examples show. This research was supported by the Alexander von Humboldt Foundation (Germany) Received: March 14, 2005; revised: August 16, 2005     

New convergence behavior of high-order hopfield neural networks with time-varying coefficients

In this paper the convergence behavior of the delayed high-order Hopfield neural networks (HHNNs) with time-varying coefficients are considered. Some sufficient conditions are established to ensure that all solutions of the networks converge to zero point, which are new and complement of previously known results.     

AN OSCILLATION THEOREM FOR A TWO-PARAMETER SYSTEM OF DIFFERENTIAL EQUATIONS WITH PERIODIC BOUNDARY CONDITIONS

Abstract

The oscillation theory for two simultaneous systems of second order linear differential equations in two parameters with periodic boundary conditions is well known when the coefficients of the differential equations are subjected to the usual definiteness condition. However, in practical applications the usual definiteness condition may fail to hold, and hence in this paper we consider the oscillation theory under another important definiteness condition.     

Limit cycle bifurcation by perturbing a cuspidal loop of order 2 in a Hamiltonian system

This paper deals with the analytical property of the first Melnikov function for general Hamiltonian systems possessing a cuspidal loop of order 2 and its expansion at the Hamiltonian value corresponding to the loop. The explicit formulas for the first coefficients of the expansion have been given. We prove that at least 13 limit cycles can bifurcate from the cuspidal loop of order 2 under certain conditions. Then we consider the cyclicity of a cuspidal loop in some Liénard and Hamiltonian systems, and determine the number of limit cycles that can bifurcate from the perturbed system.     

Many-point boundary-value problem for some differential-operational equations

The existence and uniqueness of strong solution of many-point boundary-value problems for some differential-operational equations and its continuous dependence on the coefficients are investigated. The tool employed in our analysis is the method of an energy inequality.     

Wave propagation for a two-component lattice dynamical system arising in strong competition models

We study a Lotka-Volterra type competition system with bistable nonlinearity in which the habitat is divided into discrete niches. We show that there exist non-monotone stationary solutions when the migration coefficients are sufficiently small. Also, we prove that the propagation failure phenomenon occurs. Finally, we focus on the traveling wave with nonzero wave speed. By investigating the asymptotic behavior of tails of wave profiles, we show that nonzero speed wave profiles are monotone. Moreover, the nonzero wave speed is unique in the sense that the wave cannot propagate with two different nonzero wave speeds.     

On the stability of solutions of linear boundary value problems for a system of ordinary differential equations

Linear boundary value problems for a system of ordinary differential equations are considered. The stability of the solution with respect to small perturbations of coefficients and boundary values is investigated.     

Asymptotic existence theorems for formal power series whose coefficients satisfy certain partial differential recursions

We study power series whose coefficients are holomorphic functions of another complex variable and a nonnegative real parameter s, and are given by a differential recursion equation. For positive integer s, series of this form naturally occur as formal solutions of some partial differential equations with constant coefficients, while for s=0 they satisfy certain perturbed linear ordinary differential equations. For arbitrary s?0, these series solve a differential-integral equation. Such power series, in general, are not multisummable. However, we shall prove existence of solutions of the same differential-integral equation that in sectors of, in general, maximal opening have the formal series as their asymptotic expansion. Furthermore, we shall indicate that the solutions so obtained can be related to one another in a fairly explicit manner, thus exhibiting a Stokes phenomenon.     

Periodic solutions of competition systems with delays by an average approach

A delayed competition system of Lotka-Volterra equations, with periodic coefficients, is considered. Such a differential system admits at least a periodic positive solution if and only if the corresponding autonomous, averaged system has a positive stationary solution.