Jitsuro Sugie

Continuation results for differential equations by two Liapunov functions

SummaryIn this paper we give conditions under which all solutions of a system of differential equations are continuable in the future. We use two Liapunov functions which are not radially unbounded for fixed t. The results are more flexible than the work done by Conti and Strauss. Our theorems are applied to the Liénard equation without the assumption that xg(x)>0 if x ≠ 0.

CONVERGENCE OF SOLUTIONS OF TIME-VARYING LINEAR SYSTEMS WITH INTEGRABLE FORCING TERM

The following system is considered in this paper: x ′ =−e(t)x + f (t)y, y =−g(t)x − h(t)y + p(t). The primary goal is to establish conditions on time-varying coefficients e(t), f (t), g(t) and h(t) and a forcing term p(t) for all solutions to converge to the origin (0, 0) as t→∞. Here, the zero solution of the corresponding homogeneous linear system is assumed to be neither uniformly stable nor uniformly attractive. Sufficient conditions are given for asymptotic stability of the zero solution of the nonlinear perturbed system x ′ =−e(t)x + f (t)y, y =−g(t)x − h(t)y + q(t, x, y) under the assumption that q(t, 0, 0)= 0. 2000 Mathematics subject classification: 34D05, 34D10, 34D20.

Global asymptotic stability for predator-prey systems whose prey receives time-variation of the environment

A predator-prey model with prey receiving time-variation of the environment is considered. Such a system is shown to have a unique interior equilibrium that is globally asymptotically stable if the time-variation is bounded and weakly integrally positive. In particular, the result tells us that the equilibrium point can be stabilized even by nonnegative functions that make the limiting system structurally unstable. The method that is used to obtain the result is an analysis of asymptotic behavior of the solutions of an equivalent system to the predator-prey model.

Integral conditions on the uniform asymptotic stability for two-dimensional linear systems with time-varying coefficients

This paper is concerned with the uniform asymptotic stability of the zero solution of the linear system x′ = A(t)x with A(t) being a 2×2 matrix. Our result can be used without knowledge about a fundamental matrix of the system.

Uniform global asymptotic stability for half-linear differential systems with time-varying coefficients

x′ = − e(t)x+ f(t)φp∗(y), y′ = −(p− 1)g(t)φp(x)− (p− 1)h(t)y, where p and p∗ are positive numbers satisfying 1/p + 1/p∗ = 1, and φq(z) = |z|q−2z for q = p or q = p∗. This system is referred to as a half-linear system. We herein establish conditions on time-varying coefficients e(t), f(t), g(t) and h(t) for the zero solution to be uniformly globally asymptotically stable. If (e(t), f(t)) ≡ (h(t), g(t)), then the half-linear system is integrable. We consider two cases: the integrable case (e(t), f(t)) ≡ (h(t), g(t)) and the nonintegrable case (e(t), f(t)) ̸≡ (h(t), g(t)). Finally, some simple examples are presented to illustrate our results.

Liénard dynamics with an open limit orbit

Abstract. The purpose of this paper is to discuss the problem whether the Liénard system has a homoclinic loop as an

Smith-type criterion for the asymptotic stability of a pendulum with time-dependent damping

\omega

Uniqueness of Limit Cycles in a Rosenzweig–MacArthur Model with Prey Immigration

-limit set. This problem bears a near relation to global weak attractivity of the origin. Sufficient conditions for the origin to be global attractor are also presented.

Global asymptotic stability for predator–prey models with environmental time-variations

A necessary and sufficient condition is given for the asymptotic stability of origin of a pendulum with time-varying friction described by the equation x′′ + h(t)x′ + sinx = 0, where h(t) is continuous and nonnegative for t ≥ 0. This condition is expressed as a double integral on the friction h(t). The method that is used to obtain the result is Lyapunov’s stability theory and phase plane analysis of the positive orbits of an equivalent planar system to the above-mentioned equation.

Asymptotic stability for three-dimensional linear differential systems with time-varying coefficients

Many natural predator and prey populations persist while their densities show sustained oscillations. Hence these populations must be regulated in such a way that the densities are kept away from the values where extinction is likely to occur. On the other hand, nonspatial simple predator-prey models show vigorous oscillations that can bring the populations to the brink of extinction or beyond. Predator-prey systems that are kept in the laboratory also tend to show fluctuations in densities that are severe enough to drive them to extinction. Since the amount of space that laboratory populations live in is small compared to that of natural populations, one is readily led to the hypothesis that spatial interactions must contribute to the regulation of natural predator-prey systems. In this paper, we construct a simplest type of spatially interacting populations by taking into account constant immigration of prey for a predator-prey model with a Holling type II functional response and derive necessary and su...