Abstract
It is known that the solutions of a second order linear differential equation with periodic coefficients are almost always analytically impossible to obtain and in order to study its properties we often require a computational approach. In this paper we compare graphically, using the Arnold Tongues, some sufficient criteria for the stability of periodic differential equations. We also present a brief explanation on how the authors, of each criterion, obtained them. And a comparison between four sufficient stability criteria and the stability zones found by perturbation methods is presented.
Highlights
The second order differential equations are often encountered in engineering and physical problems, and they have been studied for more than a hundred years
4 This work is structured as follows: Section 2 is dedicated to introduce some basic concepts on periodic differential equations and its solutions; in section 3 we give the discriminant approximation made by Lyapunov and the first two criteria are presented; section 4 is devoted to the study of canonical (Hamiltonian) systems, some properties of such systems solutions are described and four criteria are presented; In section 5 two criteria based on Sturm-Liouville equation properties are exhibit, both criteria follows from solutions proposed by Hochstadt in [13]; in section 6 we briefly present a new approximation of the discriminant of the Hill equation obtained by Shi and a criterion do to Xu
For more stability criteria see for example: [4] where Starzhinskii collect sufficient stability criteria for Equation (1) but he consider second order periodic differential equations with dissipation, n-th order systems and some particular cases of the vector equation y + μP (t ) y = 0 ; In [[28], Chapter 2 § 4.3] some conditions for stability of solutions of (1) that belongs to the first stability zone are presented and some general stability criteria are described; And in [[14], Chapters 7 and 8] Yakubovich and Starshinskii, in their outstanding monograph on Linear differential equations with periodic coefficients, they presented and proved some stability criteria starting with the Lyapunov approach, and doing a deep analysis on canonical
Summary
The second order differential equations are often encountered in engineering and physical problems, and they have been studied for more than a hundred years. Π2 the solutions of the periodic system are stable Lyapunov in his celebrated work “The general problem of the stability of motion” [5] developed an approximation of the discriminant of the periodic system (see section 2) and obtained a variety of sufficient stability conditions, being the most known the one here presented. Authors such as Borg [4] Yakubovich [6] [7] and.
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