# Emerging long orbits and self-similar temporal sequences in classical oscillators

## TL;DR

This paper investigates how repulsively coupled Kuramoto oscillators with frequency disorder exhibit long, self-similar orbits and complex transient behaviors, revealing how distribution width influences emergent time scales.

## Contribution

It uncovers the emergence of long, self-similar orbits in repulsively coupled oscillators and links their time scales to the distribution of natural frequencies.

## Key findings

- Long-period orbits can be orders of magnitude longer than individual oscillator periods.
- Self-similar sequences of phase-locked motion are observed across different time scales.
- The ratio of time scales correlates with the ratio of frequency distribution widths.

## Abstract

We analyze repulsively coupled Kuramoto oscillators, which are exposed to a distribution of natural frequencies. This source of disorder leads to closed orbits with a variety of different periods, which can be orders of magnitude longer than periods of individual oscillators. By construction the attractor space is quite rich. This may cause long transients until the deterministic trajectories find their stationary orbits. The smaller the width of the distribution about the common natural frequency is, the longer are the emerging time scales on average. Among the long-period orbits we find self-similar sequences of temporary phase-locked motion on different time scales. The ratio of time scales is determined by the ratio of widths of the distributions.