Quenching and revival of oscillations induced by coupling through adaptive variables

Regulating dynamics through intermittent interactions

In this article we experimentally demonstrate an efficient scheme to regulate the behavior of coupled nonlinear oscillators through dynamic control of their interaction. It is observed that introducing intermittency in the interaction term as a function of time or the system state predictably alters the dynamics of the constituent oscillators. Choosing the nature of the interaction, attractive or repulsive, allows for either suppression of oscillations or stimulation of activity. Two parameters Δ and τ, that reign the extent of interaction among subsystems, are introduced. They serve as a harness to access the entire range of possible behaviors from fixed points to chaos. For fixed values of system parameters and coupling strength, changing Δ and τ offers fine control over the dynamics of coupled subsystems. We show this experimentally using coupled Chua's circuits and elucidate their behavior for a range of coupling parameters through detailed numerical simulations.

Oscillation behavior driven by processing delay in diffusively coupled inactive systems: Cluster synchronization and multistability

Couplings involving time delay play a relevant role in the dynamical behavior of complex systems. In this work, we address the effect of processing delay, which is a specific kind of coupling delay, on the steady state of general nonlinear systems and prove that it may drive the system to Hopf bifurcation and, in turn, to a rich oscillatory behavior. Additionally, one may observe multistable states and size-dependent cluster synchronization. We derive the analytic conditions to obtain an oscillatory regime and confirm the result by numerically simulated experiments on different oscillator networks. Our results demonstrate the importance of processing delay for complex systems and pave the way for a better understanding of dynamical control and synchronization in oscillatory networks.

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.

Aging transition in the absence of inactive oscillators

The role of counter-rotating oscillators in an ensemble of coexisting co- and counter-rotating oscillators is examined by increasing the proportion of the latter. The phenomenon of aging transition was identified at a critical value of the ratio of the counter-rotating oscillators, which was otherwise realized only by increasing the number of inactive oscillators to a large extent. The effect of the mean-field feedback strength in the symmetry preserving coupling is also explored. The parameter space of aging transition was increased abruptly even for a feeble decrease in the feedback strength, and, subsequently, aging transition was observed at a critical value of the feedback strength surprisingly without any counter-rotating oscillators. Further, the study was extended to symmetry breaking coupling using conjugate variables, and it was observed that the symmetry breaking coupling can facilitate the onset of aging transition even in the absence of counter-rotating oscillators and for the unit value of the feedback strength. In general, the parameter space of aging transition was found to increase by increasing the frequency of oscillators and by increasing the proportion of the counter-rotating oscillators in both symmetry preserving and symmetry breaking couplings. Further, the transition from oscillatory to aging occurs via a Hopf bifurcation, while the transition from aging to oscillation death state emerges via the pitchfork bifurcation. Analytical expressions for the critical ratio of the counter-rotating oscillators are deduced to find the stable boundaries of the aging transition.

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

Mean-field diffusive coupling was known to induce the phenomenon of quenching of oscillations even in identical systems, where the standard diffusive coupling (without mean-field) fails to do so [Phys. Rev. E 89, 052912 (2014)PLEEE81539-375510.1103/PhysRevE.89.052912]. In particular, the mean-field diffusive coupling facilitates the transition from amplitude to oscillation death states and the onset of a nontrivial amplitude death state via a subcritical pitchfork bifurcation. In this paper, we show that an adaptive coupling using a low-pass filter in both the intrinsic and extrinsic variables in the coupling is capable of inducing the counterintuitive phenomenon of reviving of oscillations from the death states induced by the mean-field coupling. In particular, even a weak filtering of the extrinsic (intrinsic) variable in the mean-field coupling facilitates the onset of revival (quenching) of oscillations, whereas a strong filtering of the extrinsic (intrinsic) variable results in quenching (revival) of oscillations. Our results reveal that the degree of filtering plays a predominant role in determining the effect of filtering in the extrinsic or intrinsic variables, thereby engineering the dynamics as desired. We also extend the analysis to networks of mean-field coupled limit-cycle and chaotic oscillators along with the low-pass filters to illustrate the generic nature of our results. Finally, we demonstrate the observed dynamical transition experimentally to elucidate the robustness of our results despite the presence of inherent parameter fluctuations and noise.

Coexistence of oscillation and quenching states: Effect of low-pass active filtering in coupled oscillators

Effects of a low-pass active filter (LPAF) on the transition processes from oscillation quenching to asymmetrical oscillation are explored for diffusively coupled oscillators. The low-pass filter part and the active part of LPAF exhibit different effects on the dynamics of these coupled oscillators. With the amplifying active part only, LPAF keeps the coupled oscillators staying in a nontrivial amplitude death (NTAD) and oscillation state. However, the additional filter is beneficial to induce a transition from a symmetrical oscillation death to an asymmetrical oscillation death and then to an asymmetrical oscillation state which is oscillating with different amplitudes for two oscillators. Asymmetrical oscillation state is coexisting with a synchronous oscillation state for properly presented parameters. With the attenuating active part only, LPAF keeps the coupled oscillators in rich oscillation quenching states such as amplitude death (AD), symmetrical oscillation death (OD), and NTAD. The additional filter tends to enlarge the AD domains but to shrink the symmetrical OD domains by increasing the areas of the coexistence of the oscillation state and the symmetrical OD state. The stronger filter effects enlarge the basin of the symmetrical OD state which is coexisting with the synchronous oscillation state. Moreover, the effects of the filter are general in globally coupled oscillators. Our results are important for understanding and controlling the multistability of coupled systems.

Effect of filtered feedback on birhythmicity: Suppression of birhythmic oscillation

The birhythmic oscillation, generally known as birhythmicity, arises in a plethora of physical, chemical, and biological systems. In this paper we investigate the effect of filtered feedback on birhythmicity as both are relevant in many living and engineering systems. We show that the presence of a low-pass filter in the feedback path of a birhythmic system suppresses birhythmicity and supports monorhythmic oscillations depending on the filtering parameter. Using harmonic decomposition and energy balance methods we determine the conditions for which birhythmicity is removed. We carry out a detailed bifurcation analysis to unveil the mechanism behind the quenching of birhythmic oscillations. Finally, we demonstrate our theoretical findings in analog simulation with electronic circuit. This study may have practical applications in quenching birhythmicity in several biochemical and physical systems.