Revival and death of oscillation under mean-field coupling: Interplay of intrinsic and extrinsic filtering

Effect of fractional derivatives on amplitude chimeras and symmetry-breaking death states in networks of limit-cycle oscillators

We study networks of coupled oscillators whose local dynamics are governed by the fractional-order versions of the paradigmatic van der Pol and Rayleigh oscillators. We show that the networks exhibit diverse amplitude chimeras and oscillation death patterns. The occurrence of amplitude chimeras in a network of van der Pol oscillators is observed for the first time. A form of amplitude chimera, namely, "damped amplitude chimera" is observed and characterized, where the size of the incoherent region(s) increases continuously in the course of time, and the oscillations of drifting units are damped continuously until they are quenched to steady state. It is found that as the order of the fractional derivative decreases, the lifetime of classical amplitude chimeras increases, and there is a critical point at which there is a transition to damped amplitude chimeras. Overall, a decrease in the order of fractional derivatives reduces the propensity to synchronization and promotes oscillation death phenomena including solitary oscillation death and chimera death patterns that were unobserved in networks of integer-order oscillators. This effect of the fractional derivatives is verified by the stability analysis based on the properties of the master stability function of some collective dynamical states calculated from the block-diagonalized variational equations of the coupled systems. The present study generalizes the results of our recently studied network of fractional-order Stuart-Landau oscillators.

Collective behavior of identical Stuart-Landau oscillators in a star network with coupling asymmetry effects

In this study, we investigated the impact of the asymmetry of a coupling scheme on oscillator dynamics in a star network. We obtained stability conditions for the collective behavior of the systems, ranging from an equilibrium point over complete synchronization (CS) and quenched hub incoherence to remote synchronization states using both numerical and analytical methods. The coupling asymmetry factor α significantly influences and determines the stable parameter region of each state. For α ≠ 1, the equilibrium point can emerge when the Hopf bifurcation parameter a is positive, which is impossible for diffusive coupling. However, CS can occur even if a is negative under α < 1. Unlike diffusive coupling, we observe more behavior when α ≠ 1, including additional in-phase remote synchronization. These results are supported by theoretical analysis and validated through numerical simulations and independent of network size. The findings may offer practical methods for controlling, restoring, or obstructing specific collective behavior.

Quenching of oscillation by the limiting factor of diffusively coupled oscillators

A simple limiting factor in the intrinsic variable of the normal diffusive coupling is known to facilitate the phenomenon of reviving of oscillation [Zou et al., Nat. Commun. 6, 7709 (2015)2041-172310.1038/ncomms8709], where the limiting factor destabilizes the stable steady states, thereby resulting in the manifestation of the stable oscillatory states. In contrast, in this work we show that the same limiting factor can indeed facilitate the manifestation of the stable steady states by destabilizing the stable oscillatory state. In particular, the limiting factor in the intrinsic variable facilitates the genesis of a nontrivial amplitude death via a saddle-node infinite-period limit (SNIPER) bifurcation and symmetry-breaking oscillation death via a saddle-node bifurcation among the coupled identical oscillators. The limiting factor facilities the onset of symmetric oscillation death among the coupled nonidentical oscillators. It is known that the nontrivial amplitude death state manifests via a subcritical pitchfork bifurcation in general. Nevertheless, here we observe the transition to the nontrivial amplitude death via a SNIPER bifurcation. The in-phase oscillatory state loses its stability via the SNIPER bifurcation, resulting in the manifestation of the nontrivial amplitude death state, whereas the out-of-phase oscillatory state loses its stability via a homoclinic bifurcation, resulting in an unstable oscillatory state. Multistabilities among the various dynamical states are also observed. We have also deduced the evolution equation for the perturbation governing the stability of the observed dynamical states and stability conditions for SNIPER and pitchfork bifurcations. The generic nature of the effect of the limiting factor is also reinforced using two distinct nonlinear oscillators.

Effects of propagation delay in coupled oscillators under direct-indirect coupling: Theory and experiment

Propagation delay arises in a coupling channel due to the finite propagation speed of signals and the dispersive nature of the channel. In this paper, we study the effects of propagation delay that appears in the indirect coupling path of direct (diffusive)-indirect (environmental) coupled oscillators. In sharp contrast to the direct coupled oscillators where propagation delay induces amplitude death, we show that in the case of direct-indirect coupling, even a small propagation delay is conducive to an oscillatory behavior. It is well known that simultaneous application of direct and indirect coupling is the general mechanism for amplitude death. However, here we show that the presence of propagation delay hinders the death state and helps the revival of oscillation. We demonstrate our results by considering chaotic time-delayed oscillators and FitzHugh-Nagumo oscillators. We use linear stability analysis to derive the explicit conditions for the onset of oscillation from the death state. We also verify the robustness of our results in an electronic hardware level experiment. Our study reveals that the effect of time delay on the dynamics of coupled oscillators is coupling function dependent and, therefore, highly non-trivial.

Tuning coupling rate to control oscillation quenching in fractional-order coupled oscillators

Introducing the fractional-order derivative into the coupled dynamical systems intrigues gradually the researchers from diverse fields. In this work, taking Stuart-Landau and Van der Pol oscillators as examples, we compare the difference between fractional-order and integer-order derivatives and further analyze their influences on oscillation quenching behaviors. Through tuning the coupling rate, as an asymmetric parameter to achieve the change from scalar coupling to non-scalar coupling, we observe that the onset of fractional-order not only enlarges the range of oscillation death, but attributes to the transition from fake amplitude death to oscillation death for coupled Stuart-Landau oscillators. We go on to show that for a coupled Van der Pol system only in the presence of a fractional-order derivative, oscillation quenching behaviors will occur. The results pave a way for revealing the control mechanism of oscillation quenching, which is critical for further understanding the function of fractional-order in a coupled nonlinear model.

Symmetry breaking dynamics induced by mean-field density and low-pass filter

The phenomenon of spontaneous symmetry breaking facilitates the onset of a plethora of nontrivial dynamical states/patterns in a wide variety of dynamical systems. Spontaneous symmetry breaking results in amplitude and phase variations in a coupled identical oscillator due to the breaking of the prevailing permutational/translational symmetry of the coupled system. Nevertheless, the role and the competing interaction of the low-pass filter and the mean-field density parameter on the symmetry breaking dynamical states are unclear and yet to be explored explicitly. The effect of low pass filtering along with the mean-field parameter is explored in conjugately coupled Stuart-Landau oscillators. The dynamical transitions are examined via bifurcation analysis. We show the emergence of a spontaneous symmetry breaking (asymmetric) oscillatory state, which coexists with a nontrivial amplitude death state. Through the basin of attraction, the multi-stable nature of the spontaneous symmetry breaking state is examined, which reveals that the asymmetric distribution of the initial state favors the spontaneous symmetry breaking dynamics, while the symmetric distribution of initial states gives rise to the nontrivial amplitude death state. In addition, the trade-off between the cut-off frequency of the low-pass filter along with the mean-field density induces and enhances the symmetry breaking dynamical states. Global dynamical transitions are discussed as a function of various system parameters. Analytical stability curves corresponding to the nontrivial amplitude death and oscillation death states are deduced.