Travelling Waves in the Ring of Coupled Oscillators with Delayed Feedback

Abstract

We studied travelling waves in N nonlinear differential equations with a delay and large parameter. This system is important because it can be regarded as a phenomenological model of N-coupled neuron-like oscillators with delay. The problem of the existence of travelling-wave-type solutions was reduced to the study of the dynamics of an auxiliary equation with two delays. Using a special asymptotic method for the large parameter we proved that this equation has a relaxation cycle, studied its properties (amplitude, period and asymptotics) and found the sufficient stability conditions. Based on this periodic solution the travelling waves of the initial model were constructed.

Keywords

• multiple delays
• relaxation oscillations
• travelling wave
• rotating wave
• asymptotics
• large parameter