Periodic Solutions and Nontrivial Periodic Solutions for a Class of Rayleigh-Type Equation with Two Deviating Arguments

Meiqiang Feng

doi:10.1155/2013/414901

Abstract

The Rayleigh equation with two deviating argumentsx′′(t)+f(x'(t))+g1(t,x(t-τ1(t)))+g2(t,x(t-τ2(t)))=e(t)is studied. By using Leray-Schauder index theorem and Leray-Schauder fixed point theorem, we obtain some new results on the existence of periodic solutions, especially for the existence of nontrivial periodic solutions to this equation. The results are illustrated with two examples, which cannot be handled using the existing results.

Highlights

Consider the Rayleigh equation with two deviating arguments in the form of x󸀠󸀠 (t) + f (x󸀠 (t)) + g1 (t, x (t − τ1 (t))) + g2 (t, x (t − τ2 (t))) = e (t), (1)where f ∈ C (R, R), gi ∈ C (R × R, R), i = 1, 2, e, τi ∈ C (R, R), i = 1, 2, (2)gi (t + T, x) = gi (t, x), τi (t + T) = τi (t), i = 1, 2, e (t + T) = e (t) .The dynamic behaviors of Rayleigh equation have been widely investigated due to their applications in many fields such as physics, mechanics, and the engineering technique fields
An excess voltage of ferroresonance, known as some kind of nonlinear resonance having long duration that arises from the magnetic saturation of inductance in an oscillating circuit of a power system, and a boosted excess voltage can give rise to some problems in relay protection
A mathematical model was proposed in [1,2,3], which is a special case of the Rayleigh equation with two delays

Summary

Introduction

Consider the Rayleigh equation with two deviating arguments in the form of x󸀠󸀠 (t) + f (x󸀠 (t)) + g1 (t, x (t − τ1 (t))) + g2 (t, x (t − τ2 (t))). An excess voltage of ferroresonance, known as some kind of nonlinear resonance having long duration that arises from the magnetic saturation of inductance in an oscillating circuit of a power system, and a boosted excess voltage can give rise to some problems in relay protection To probe this mechanism, a mathematical model was proposed in [1,2,3], which is a special case of the Rayleigh equation with two delays. If the periodic solution is unique, it must be trivial It is worth discussing the existence of the nontrivial periodic solutions of Rayleigh equations with two deviating arguments in this case. If g1(t, 0) + g2(t, 0) ≢ e(t), the periodic solution obtained by Theorem 1 must be nontrivial.

Preliminaries

Proof of Theorem 1

Proof of Theorem 3

Two Examples

Conclusion