Effect of oscillator and initial condition differences in the dynamics of a ring of dissipative coupled van der Pol oscillators

Abstract

This paper investigates the dynamical behavior of coupled van der Pol oscillators in a ring to understand vibrations that may occur in systems such as turbine blades mounted on a single shaft. The objective is to investigate the effect of spatial differences in oscillator parameters and initial conditions that occur in realistic systems. The coupling between the neighboring oscillators is modeled as a linear dissipative element, and the mathematical model is analyzed asymptotically and numerically. Synchronization of self excited oscillators in mechanical systems has been predominantly investigated in recent literature by focusing on its parameter dependence. This work investigates the dependence of dynamics of such systems on initial conditions. The analysis is conducted for identical oscillators as well as oscillators with a frequency mismatch, along with three different sets of initial conditions. The dynamics of the system is discussed based on time plots, frequency plots, instantaneous dynamics of each oscillator by Hilbert transform and the phase equation obtained by asymptotic expansion. The study reveals interesting phenomena like amplitude death, oscillation suppression, oscillation resurrection, frequency locking and beat frequency in the model when subjected to the different set of initial conditions.