Phase-flip bifurcation induced by time delay

Abstract

We present a general bifurcation in the synchronized dynamics of time-delay-coupled nonlinear oscillators. The relative phase between the oscillators jumps from zero to pi as a function of the coupling; this phase-flip bifurcation is accompanied by a discontinuous change in the frequency of the synchronized oscillators. This phenomenon is of broad relevance, being observed in regimes of oscillator death as well as in periodic, quasiperiodic, and chaotic dynamics. Time-delay coupling is necessary for the phase-flip bifurcation. We illustrate the phenomenon, and present analytical results for paradigmatic nonlinear systems. Possible applications are discussed.

Amplitude death: The emergence of stationarity in coupled nonlinear systems

“…[26,27,28,70,67] in a variety of systems, as well as in different dynamical regimes. Furthermore, when there are more than two coupled oscillators, the phase-difference is not necessarily π, but can depends upon the specifics of the system as well as on the coupling topology of the network [105].…”

“…Furthermore, since the different subsystems are coupled, a study of the synchronization properties [25] is also of interest. In a number of examples it has been noted that the individual systems, while being synchronized, can nevertheless be either in-phase or out of phase, with a transition between these states at a specific value of the coupling [26,27,28]. We review this phase-flip transition and its associated behaviour in Sec.…”

Dynamics of globally delay-coupled neurons displaying subthreshold oscillations

“…At the bifurcation point, there is a transition from in-phase to out-of-phase (or vice versa) synchronized oscillations, which is accompanied by an abrupt change in the common oscillation frequency. This phase-flip bifurcation was recently investigated in two mutually delay-coupled oscillators (Prasad et al 2006(Prasad et al , 2008 and can play a role in the mechanisms by which the neurons switch among different activity patterns. The study of the influence of network heterogeneities and noise is the subject of future work.…”

“…At the bifurcation point, the synchronized dynamics changes from in-phase to out-of-phase oscillations (or vice versa), and this change is accompanied by an abrupt variation in the common oscillation frequency. The phase-flip bifurcation was originally reported by Schuster & Wagner (1989) in a simple model of two delay-coupled phase oscillators and was recently investigated in detail by Prasad et al (2006Prasad et al ( , 2008 in various types of delay-coupled oscillators, in which not only the phase but also the amplitude of the oscillators are affected by the coupling. Prasad et al considered models of excitable lasers, Fitzhugh-Nagumo neurons, predator-prey models and chaotic Chua electronic circuits, thus demonstrating the broad relevance of this bifurcation.…”

“…(4.7) Prasad et al (2006Prasad et al ( , 2008 reported a phase-flip bifurcation induced by variation of the time delay: at a critical value of the delay time, the oscillators switch, from being in-phase to out-of-phase. The phase-flip bifurcation, which was shown to occur in different dynamical systems, was always accompanied by an abrupt jump in the frequency of the synchronized oscillators.…”