Revoking amplitude and oscillation deaths by low-pass filter in coupled oscillators

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.

Oscillation quenching in diffusively coupled dynamical networks with inertial effects

Self-sustained oscillations are ubiquitous and of fundamental importance for a variety of physical and biological systems including neural networks, cardiac dynamics, and circadian rhythms. In this work, oscillation quenching in diffusively coupled dynamical networks including "inertial" effects is analyzed. By adding inertia to diffusively coupled first-order oscillatory systems, we uncover that even small inertia is capable of eradicating the onset of oscillation quenching. We consolidate the generality of inertia in eradicating oscillation quenching by extensively examining diverse quenching scenarios, where macroscopic oscillations are extremely deteriorated and even completely lost in the corresponding models without inertia. The presence of inertia serves as an additional scheme to eradicate the onset of oscillation quenching, which does not need to tailor the coupling functions. Our findings imply that inertia of a system is an enabler against oscillation quenching in coupled dynamical networks, which, in turn, is helpful for understanding the emergence of rhythmic behaviors in complex coupled systems with amplitude degree of freedom.

Revival of oscillation and symmetry breaking in coupled quantum oscillators

Restoration of oscillations from an oscillation suppressed state in coupled oscillators is an important topic of research and has been studied widely in recent years. However, the same in the quantum regime has not been explored yet. Recent works established that under certain coupling conditions, coupled quantum oscillators are susceptible to suppression of oscillations, such as amplitude death and oscillation death. In this paper, for the first time, we demonstrate that quantum oscillation suppression states can be revoked and rhythmogenesis can be established in coupled quantum oscillators by controlling a feedback parameter in the coupling path. However, in sharp contrast to the classical system, we show that in the deep quantum regime, the feedback parameter fails to revive oscillations, and rather results in a transition from a quantum amplitude death state to the recently discovered quantum oscillation death state. We use the formalism of an open quantum system and a phase space representation of quantum mechanics to establish our results. Therefore, our study establishes that the revival scheme proposed for classical systems does not always result in restoration of oscillations in quantum systems, but in the deep quantum regime, it may give counterintuitive behaviors that are of a pure quantum mechanical origin.

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.

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.

Low pass filtering mechanism enhancing dynamical robustness in coupled oscillatory networks

A network that consists of a set of active and inactive nodes is called a damaged network and this type of network shows an aging effect (degradation of dynamical activity). This dynamical deterioration affects the normal functioning of the network and also its performance. Therefore, it is necessary to design a proper mechanism to avoid undesired dynamical activity like degradation. In this work, an efficient mechanism, called the low pass filtering technique, is proposed to enhance the dynamical activity of damaged networks of coupled oscillators. Using this mechanism, the dynamic behavior of the damaged network of coupled active and inactive dynamical units is improved and the network survivability is ensured. Because a minor deviation of the controlling parameter is sufficient to restore the oscillatory behavior when the entire network undergoes an aging transition. Even when the whole network degrades due to the deterioration of each node, the larger values of the interaction strength and the controlling parameter play a key role in favor of the revival of dynamic activity in the entire network. Our proposed mechanism is very simple and effective to recover the dynamic features of a damaged network. The effectiveness of this technique has been testified in globally coupled and Erdős Rényi random networks of Stuart-Landau oscillators.

Quenching and revival of oscillations induced by coupling through adaptive variables

An adaptive coupling based on a low-pass filter (LPF) is proposed to manipulate dynamic activity of diffusively coupled dynamical systems. A theoretical analysis shows that tracking either the external or internal signal in the coupling via a LPF gives rise to distinctly different ways of regulating the rhythmicity of the coupled systems. When the external signals of the coupling are attenuated by a LPF, the macroscopic oscillations of the coupled system are quenched due to the emergence of amplitude or oscillation death. If the internal signals of the coupling are further filtered by a LPF, amplitude and oscillation deaths are effectively revoked to restore dynamic behaviors. The applicability of this approach is demonstrated in laboratory experiments of coupled oscillatory electrochemical reactions by inducing coupling through LPFs. Our study provides additional insight into (ar)rhythmogenesis in diffusively coupled systems.

Spatiotemporal Structure and Dynamics of Spontaneous Oscillatory Synchrony in the Vagal Complex

Fundamental structure and dynamics of spontaneous neuronal activities without apparent peripheral inputs were analyzed in the vagal complex (VC), whose activities had been generally thought to be produced almost passively to peripheral cues. The analysis included the caudal nucleus of the tractus solitarius—a main gateway for viscerosensory peripheral afferents and involved dynamically and critically in cardiorespiratory brainstem networks. In the present study, a possibility of self-organized brain activity was addressed in the VC. While VC neurons exhibited sparse firing in anesthetized rats and in in vitro preparations, we identified peculiar features of the emergent electrical population activity: (1) Spontaneous neuronal activity, in most cases, comprised both respiration and cardiac cycle components. (2) Population potentials of polyphasic high amplitudes reaching several millivolts emerged in synchrony with the inspiratory phase of respiratory cycles and exhibited several other characteristic temporal dynamics. (3) The spatiotemporal dynamics of local field potentials (LFPs), recorded simultaneously over multiple sites, were characterized by a stochastic emergence of high-amplitude synchrony. By adjusting amplitude and frequency (phase) over both space and time, the traveling synchrony exhibited varied degrees of coherence and power with a fluctuating balance between mutual oscillators of respiratory and cardiac frequency ranges. Full-fledged large-scale oscillatory synchrony over a wide region of the VC emerged after achieving a maximal stable balance between the two oscillators. Distinct somatic (respiratory; ~1 Hz) and visceral (autonomic; ~5 Hz) oscillators seemed to exist and communicate co-operatively in the brainstem network. Fluctuating oscillatory coupling may reflect varied degrees of synchrony influenced by the varied amplitude and frequency of neuronal activity in the VC. Intranuclear micro-, intrabulbar meso-, and wide-ranging macro-circuits involving the VC are likely to form nested networks and strategically interact to maintain a malleable whole-body homeostasis. These two brainstem oscillators could orchestrate neuronal activities of the VC, and other neuronal groups, through a phase-phase coupling mechanism to perform specific physiological functions.

Transition from homogeneous to inhomogeneous limit cycles: Effect of local filtering in coupled oscillators

We report an interesting symmetry-breaking transition in coupled identical oscillators, namely, the continuous transition from homogeneous to inhomogeneous limit cycle oscillations. The observed transition is the oscillatory analog of the Turing-type symmetry-breaking transition from amplitude death (i.e., stable homogeneous steady state) to oscillation death (i.e., stable inhomogeneous steady state). This novel transition occurs in the parametric zone of occurrence of rhythmogenesis and oscillation death as a consequence of the presence of local filtering in the coupling path. We consider paradigmatic oscillators, such as Stuart-Landau and van der Pol oscillators, under mean-field coupling with low-pass or all-pass filtered self-feedback and through a rigorous bifurcation analysis we explore the genesis of this transition. Further, we experimentally demonstrate the observed transition, which establishes its robustness in the presence of parameter fluctuations and noise.