Nature of the phase-flip transition in the synchronized approach to amplitude death

Abstract

We study the dynamics of time-delay coupled limit-cycle oscillators in the amplitude death regime. Through a detailed analysis of the Jacobian at the fixed point, we show that the phase-flip transition, namely, the abrupt change from in-phase synchronized dynamics to antiphase synchronized dynamics, is associated with an interchange of the imaginary parts of complex pairs of eigenvalues at an "avoided crossing" of Lyapunov exponents as a parameter is varied. An order parameter for the transition is constructed through the eigenvectors of the Jacobian.

Amplitude and phase dynamics in oscillators with distributed-delay coupling

“…One possible extension of this work would be the analysis of a phase-flip transition [49,85], where in-phase and anti-phase phase-locked branches exchange stability. This phenomenon has been observed in systems with discrete time delays, and it has even been observed in the transient dynamics preceding amplitude death.…”

“…This phenomenon, named amplitude death [37], oscillator death, or 'death by delay' [39], has been demonstrated experimentally in nonlinear electronic circuits [40], in the dynamics of the slime mould Physarum polycephalum [41] and in thermo-optical oscillators linearly coupled by heat transfer [42]. Amplitude death has been subsequently studied for a number of different systems and couplings [31,[43][44][45][46][47][48][49][50][51][52].…”

Amplitude death: The emergence of stationarity in coupled nonlinear systems

“…In addition, there are other new features such as the suppression of chaos, and riddling [11,12,70] due to conjugate coupling. In addition, this is particularly suited to experiments and both in electrochemical systems [71] as well as in electronic circuits [72], this coupling can be realized.…”

“…With such coupling, the necessity for systems being either mismatched or having time-delayed interactions can be dropped, and AD occurs in identical systems with instantaneous coupling. This was first studied in a model system by Karnatak et al [11] who considered two Landau-Stuart oscillators with the evolution equationṡ…”

“…Different situations where amplitude death occurs have been described, and it is now known that the phenomenon can be seen when the interacting systems are identical [7,11,12,13], mismatched [1,6], when the coupling is dynamical [14], or nonlinear (nondiffusive) [15]. We review these various situations and also discuss the general mechanism for AD in Sec.…”