On stability of linear time-varying second-order differential equations

Abstract

Abstract. We derive sufficient conditions for stability and asymptotic stability of second order, scalar differential equations with differentiable coefficients. Introduction.We study, for differentiable a 0 , a 1 : R ≥0 → R, stability properties of linear time-varying second-order differential equations of the form

Floquet analysis of linear dynamic RLC circuits

“…The study of periodic solutions and stability problems has attracted the attention of many authors in particular for that type and its mathematical variant extensions [15,21,37,42,41]. Ignatyev [18] discussed the asymptotic stability of zero solution of second-order differential equation and Duc et al [7] proved that the zero solution is also asymptotically stable if one of the hypotheses introduced in the study by Ignatyev [18] does not exist. Onitsuka [30] gave a natural generalization of results in [7,18] by proving that the zero solution is asymptotically stable if one of the hypotheses of [18] is fulfilled.…”

Non-uniform asymptotic stability for the damped linear oscillator

“…(1.2), since a(t) = 1/(1 + t) and b(t) = 1 for t ≥ 0, we see that conditions (1.4) and (1.5) are satisfied with Recently, Duc et al [5] have presented a sufficient condition for the equilibrium x = x = 0 of (E) to be asymptotically stable.…”

“…For example, those results can be found in [4][5][6][7][8][9][10] and the references cited therein. It is well known that if both a(t) and b(t) are periodic functions (or constants), then the equilibrium x = x = 0 of (E) is asymptotically stable if and only if it is uniformly asymptotically stable.…”

Many Types of Stability of Abstract First and Second Order Linear Dynamic Equations on Time Scales

“…In recent years, there has been an increasing interest in studying the asymptotic behavior of solutions of dynamic equations on time scales due to its applications in many fields especially in biology, economics. In [10] DUC, Ilchmann, Siegmund and Taraba derived sufficient conditions for stability and asymptotic stability of linear time varying second order scalar differential equations of the form: + 1 ( ) + 0 = 0. Drozdowicz and Popenda, in [9], investigated the asymptotic behavior of solutions of second order difference equations.…”