Stability and Oscillatory Behavior of the Solutions on a Class of Coupled Van der Pol-Duffing Equations with Delays

Chunhua Feng

doi:10.11648/j.mcs.20190406.12

Abstract

In the present paper, a class of coupled van der Pol-Duffing oscillators with a nonlinear friction of higher polynomial order model which involves time delays is investigated. The coefficients of the highest order of the polynomial determine the boundedness of the solutions. With special attention to the boundedness of the solutions and the instability of the unique equilibrium point of linearized system, some sufficient conditions to guarantee the existence of oscillatory solutions for the model are obtained based on the generalized Chafee's criterion. Convergence of the trivial solution is determined by the negative real part of eigenvalues of the linearized system. Examples are provided to demonstrate the reduced conservativeness for the parameters of the proposed results. The results obtained shown that the passive decay rate in the model affects the oscillatory frequency and amplitude. When a permanent oscillation occurred, time delays affect mainly oscillatory frequency and amplitude slightly.

Highlights

It is known that the van der Pol (VDP) oscillator could model the typical self-excited or self-sustained oscillation
Barron has considered the stability of a ring of coupled van der Pol oscillators with non-uniform distribution of the coupling parameter as follows: (2)
Motivated by the above models, in this paper we consider the following a ring of time delays Duffing-van der Pol-like oscillator with a nonlinear friction of higher polynomial order system: 6 7 + ̃ 9:; 7+ − < 7* + 7 − = >7 + %̃ 7 + < 7

Summary

Introduction

It is known that the van der Pol (VDP) oscillator could model the typical self-excited or self-sustained oscillation. Barron has considered the stability of a ring of coupled van der Pol oscillators with non-uniform distribution of the coupling parameter as follows:. A driven van der Pol-like oscillator with a nonlinear friction of higher polynomial order model as follows:. The time delay coupled van der Pol equations have been extensively studied by many researchers [13,14,15,16,17,18,19,20,21,22,23,24]. Motivated by the above models, in this paper we consider the following a ring of time delays Duffing-van der Pol-like oscillator with a nonlinear friction of higher polynomial order system:. Our aim is to investigate the dynamical behavior of n coupled oscillators by means of the generalized Chafee's criterion [25, 26]

Preliminaries

Main Result

Discussion

Conclusion