Embedding the dynamics of a single delay system into a feed-forward ring

TL;DR

This paper explores how the dynamics of a single delay-coupled oscillator can be embedded into a ring of such oscillators, revealing relationships between their periodic solutions, stability, and bifurcation scenarios, supported by experimental electronic circuit demonstrations.

Contribution

It establishes a connection between delayed oscillator dynamics and ring networks, including stability and bifurcation transfer, with experimental validation in electronic circuits.

Findings

Periodic solutions of delayed oscillators lead to rotating waves in rings.

Stability of single oscillator solutions relates to ring wave stability.

Multi-jittering bifurcations can be transferred from single to ring systems.

Abstract

We investigate the relation between the dynamics of a single oscillator with delayed self-feedback and a feed-forward ring of such oscillators, where each unit is coupled to its next neighbor in the same way as in the self-feedback case. We show that periodic solutions of the delayed oscillator give rise to families of rotating waves with different wave numbers in the corresponding ring. In particular, if for the single oscillator the periodic solution is resonant to the delay, it can be embedded into a ring with instantaneous couplings. We discover several cases where stability of periodic solution for the single unit can be related to the stability of the corresponding rotating wave in the ring. As a specific example we demonstrate how the complex bifurcation scenario of simultaneously emerging multi-jittering solutions can be transferred from a single oscillator with delayed pulse feedback to multi-jittering rotating waves in a sufficiently large ring of oscillators with instantaneous pulse coupling. Finally, we present an experimental realization of this dynamical phenomenon in a system of coupled electronic circuits of FitzHugh-Nagumo type.