Breaking of time-translation invariance in Kuramoto dynamics with multiple time scales

¹ Department of Physics, Virginia Tech - Blacksburg, VA 24061-0435, USA
² Center for Soft Matter and Biological Physics, Virginia Tech - Blacksburg, VA 24061-0435, USA
³ Department of Physics and Earth Sciences, Jacobs University - Bremen, Germany
⁴ Academy of Integrated Science, Virginia Tech - Blacksburg, VA 24061-0405, USA

Received: 26 May 2017
Accepted: 4 July 2017

Abstract

We identify the breaking of time-translation invariance in a deterministic system of repulsively coupled Kuramoto oscillators, which are exposed to a distribution of natural frequencies. We analyze random and regular implementations of frequency distributions and consider grid sizes with different characteristics of the attractor space, which is by construction quite rich. This may cause long transients until the deterministic trajectories find their stationary orbits. The stationary orbits are limit cycles with periods that extend over orders of magnitude. It is the long transient times that cause the breaking of time-translation invariance in autocorrelation functions of oscillator phases. This feature disappears close to the transition to the monostable phase, where the phase trajectories are just irregular and no stationary behavior can be identified.