Bistability in two simple symmetrically coupled oscillators with symmetry-broken amplitude- and phase-locking

Abstract

In the model system of two instantaneously and symmetrically coupled identical Stuart-Landau oscillators, we demonstrate that there exist stable solutions with symmetry-broken amplitude- and phase-locking. These states are characterized by a non-trivial fixed phase or amplitude relationship between both oscillators, while simultaneously maintaining perfectly harmonic oscillations of the same frequency. While some of the surrounding bifurcations have been previously described, we present the first detailed analytical and numerical description of these states and present analytically and numerically how they are embedded in the bifurcation structure of the system, arising both from the in-phase and the anti-phase solutions, as well as through a saddle-node bifurcation. The dependence of both the amplitude and the phase on parameters can be expressed explicitly with analytic formulas. As opposed to the previous reports, we find that these symmetry-broken states are stable, which can even be shown analytically. As an example of symmetry-breaking solutions in a simple and symmetric system, these states have potential applications as bistable states for switches in a wide array of coupled oscillatory systems.

Adaptive frequency-based modeling of whole-brain oscillations: Predicting regional vulnerability and hazardousness rates

“…Accordingly, we model the local dynamics of each brain area with a modified Stuart-Landau equation. The Stuart-Landau equation describes the behavior of a nonlinear oscillating system near the Hopf bifurcation, and it can be thought of as the principal model for nonlinear oscillators since it is the simplest possible model to describe amplitude dynamics (Röhm, Lüdge, & Schneider, 2018). When coupled together, the collective behavior of interacting oscillator systems has been shown to reproduce features of brain dynamics (Deco, Kringelbach, Jirsa, & Ritter, 2017; Freyer et al, 2011).…”

Multiplexed networks: reservoir computing with virtual and real nodes

“…Networks of such oscillators can exhibit a wide range of different dynamics [37][38][39][40][41][42][43][44]. Here, we study the connection between reservoir computing capabilities and the dynamics of the underlying network [45]. In figure 3 we have numerically integrated the system given by equation (1) but with a real-part coupling to highlight dynamical complexity.…”

Cluster singularity: The unfolding of clustering behavior in globally coupled Stuart-Landau oscillators

“…The asymmetric solutions u 1,2 (see Appendix A for their derivation) have the property that the amplitudes of the two oscillators differ and the oscillators have a phase difference between 0 and π, see also Ref. 25 . Considering the symmerty of the cluster solutions, we note that for any 2-cluster state u cl , the isotropy subgroup Σ u cl has the form S N1 × S N2 ⊆ S N .…”