On the asymptotic stability for a two-dimensional linear nonautonomous differential system

Onitsuka

2010

“…For example, sin 2 t is an integrally positive function (see [5,8]). Throughout this paper, we assume that f (t)g(t) > 0 and g(t)/f (t) is differentiable for t ≥ 0.…”

Section: The Main Resultsmentioning

confidence: 99%

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

Onitsuka

2010

“…For example, if h(t) = sin 2 t, then the zero solution of (17) is uniformly asymptotically stable. On the other hand, if h(t) = 1/(1 + t) or h(t) = sin 2 t/(1 + t), then the zero solution of (17) is asymptotically stable, but it is not uniformly asymptotically stable (for details, see [2,13,28]). If the zero solution of a linear system is uniformly asymptotically stable, then the zero solution of the corresponding quasi-linear system is also uniformly asymptotically stable.…”

Section: Discussionmentioning

confidence: 99%

“…To state our main result, we define a family of functions. We say that a nonnegative function φ is weakly integrally positive if [12,13,27,28,29]). It is easy to see that the family of weakly integrally positive functions includes nonnegative functions which converge to 0 as t → ∞.…”

Section: Introductionmentioning

confidence: 99%

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

Saito

Fan

2011

Abstract.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.

“…A nonnegative function φ(t) is said to be weakly integrally positive if [τ n , σ n ] such that τ n + δ < σ n < τ n+1 < σ n + ∆ for some δ > 0 and ∆ > 0. For example, 1/(1 + t) and sin 2 t/(1 + t) are weakly integrally positive functions (see [6,7,13,14,15]). …”

Section: R(s)ds and ψ(T) = 2(q(t) − R(t))mentioning

confidence: 99%

Attractivity for two-dimensional linear systems whose anti-diagonal coefficients are periodic

Endo

2009

Abstract. This paper deals with the linear system x = A(t)x with A(t) being a 2 × 2 matrix. The anti-diagonal components of A(t) are assumed to be periodic, but the diagonal components are not necessarily periodic. Our concern is to establish sufficient conditions for the zero solution to be attractive. Floquet theory is of no use in solving our problem, because not all components are periodic. Another approach is adopted. Some simple examples are included to illustrate the main result.