# Isochronous Dynamics in Pulse Coupled Oscillator Networks with Delay

## TL;DR

This paper investigates how delay-induced discontinuities in pulse-coupled oscillator networks lead to the formation of isochronous regions filled with periodic orbits, providing analytical and numerical insights into their properties.

## Contribution

It introduces the concept of isochronous regions in pulse-coupled oscillator networks with delay and offers a detailed analytical and numerical analysis of their existence and characteristics.

## Key findings

- Identification of isochronous regions with same-period orbits
- Analytical proof of isochronous region existence
- Numerical characterization of multiple isochronous regions

## Abstract

We consider a network of identical pulse-coupled oscillators with delay and all-to-all coupling. We demonstrate that the discontinuous nature of the dynamics induces the appearance of isochronous regions---subsets of the phase space filled with periodic orbits having the same period. For fixed values of the network parameters each such isochronous region corresponds to a subset of initial states on an appropriate surface of section with non-zero dimension such that all periodic orbits in this set have qualitatively similar dynamical behaviour. We analytically and numerically study in detail such an isochronous region, give a proof of its existence, and describe its properties. We further describe other isochronous regions that appear in the system.