Periodic orbits in the Ott-Antonsen manifold

Abstract

In their seminal paper [Chaos 18, 037113 (2008)], E. Ott and T. M. Antonsen showed that large groups of phase oscillators driven by a certain type of common force display low dimensional long-term dynamics, which is described by a small number of ordinary differential equations. This fact was later used as a simplifying reduction technique in many studies of synchronization phenomena occurring in networks of coupled oscillators and in neural networks. Most of these studies focused mainly on partially synchronized states corresponding to the equilibrium-type dynamics in the so called Ott-Antonsen manifold. Going beyond this paradigm, here we propose a universal approach for the efficient analysis of partially synchronized states with non-equilibrium periodic collective dynamics. Our method is based on the observation that the Poincaré map of the complex Riccati equation, which describes the dynamics in the Ott-Antonsen manifold, coincides with the well-known Möbius transformation. To illustrate the possibilities of our method, we use it to calculate a complete bifurcation diagram of travelling chimera states in a ring network of phase oscillators with asymmetric nonlocal coupling.

References in corpus (8)

• Synchronization in complex networks
• Low Dimensional Behavior of Large Systems of Globally Coupled Oscillators
• Paths to Synchronization on Complex Networks
• Macroscopic description for networks of spiking neurons
• Exact Results for the Kuramoto Model with a Bimodal Frequency Distribution
• Thermodynamic limit of the first-order phase transition in the Kuramoto model
• Chimera states in heterogeneous networks
• Mathematical framework for breathing chimera states

Cited by in corpus (2)

• Hopf Bifurcations of Twisted States in Phase Oscillators Rings with Nonpairwise Higher-Order Interactions
• Two-phase quadratic integrate-and-fire neurons: Exact low-dimensional description for ensembles of finite-voltage neurons