# Hysteretic behavior of spatially coupled phase-oscillators

## TL;DR

This paper studies how spatially coupled phase oscillators with inertia exhibit hysteretic synchronization behavior, influenced by local interactions and kernel radii, with implications for biological systems like firefly swarms.

## Contribution

It introduces a generalized Kuramoto model with inertia and spatially dependent coupling, analyzing how local interactions affect hysteresis in synchronization.

## Key findings

- Locally locked clusters form at low coupling strengths.
- Hysteresis disappears with small kernel radii.
- Kernel radius relative to maximal oscillator distance determines network behavior.

## Abstract

Motivated by phenomena related to biological systems such as the synchronously flashing swarms of fireflies, we investigate a network of phase oscillators evolving under the generalized Kuramoto model with inertia. A distance-dependent, spatial coupling between the oscillators is considered. Zeroth and first order kernel functions with finite kernel radii were chosen to investigate the effect of local interactions. The hysteretic dynamics of the synchronization depending on the coupling parameter was analyzed for different kernel radii. Numerical investigations demonstrate that (1) locally locked clusters develop for small coupling strength values, (2) the hysteretic behavior vanishes for small kernel radii, (3) the ratio of the kernel radius and the maximal distance between the oscillators characterizes the behavior of the network.