Travelling Waves in the Ring of Coupled Oscillators with Delayed Feedback

Alexandra Kashchenko (), Ilia Kashchenko and Sergey Kondratiev
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Alexandra Kashchenko: Regional Scientific and Educational Mathematical Center «Centre of Integrable Systems», P. G. Demidov Yaroslavl State University, 150003 Yaroslavl, Russia
Ilia Kashchenko: Regional Scientific and Educational Mathematical Center «Centre of Integrable Systems», P. G. Demidov Yaroslavl State University, 150003 Yaroslavl, Russia
Sergey Kondratiev: Regional Scientific and Educational Mathematical Center «Centre of Integrable Systems», P. G. Demidov Yaroslavl State University, 150003 Yaroslavl, Russia

Mathematics, 2023, vol. 11, issue 13, 1-17

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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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