General mechanism for amplitude death in coupled systems

Generation of stochastic mixed-mode oscillations in a pair of VDP oscillators with direct-indirect coupling

Environmental noise can lead to complex stochastic dynamical behavior in nonlinear systems. In this paper, we studied the phenomenon of a pair of Van der Pol (VDP) oscillators with direct-indirect coupling affected by Gaussian white noise. That is to say, a noise-induced equilibrium transition oscillation was observed in three types of different parameter regions, where the deterministic system had two kinds of stable equilibrium points. Meanwhile, with the noise intensity increasing, we found that the stochastic system will constantly switch between two stable equilibrium points. To analyze the stochastic behavior, we used the stochastic sensitivity equation and confidence ellipse method. When the confidence ellipsoid crossed the boundary of the attraction basin of the equilibrium point, the system entered into the state of stochastic mixed-mode oscillations, which was consistent with the simulation results.

Interlayer antisynchronization in degree-biased duplex networks

With synchronization being one of nature's most ubiquitous collective behaviors, the field of network synchronization has experienced tremendous growth, leading to significant theoretical developments. However, most previous studies consider uniform connection weights and undirected networks with positive coupling. In the present article, we incorporate the asymmetry in a two-layer multiplex network by assigning the ratio of the adjacent nodes' degrees as the weights to the intralayer edges. Despite the presence of degree-biased weighting mechanism and attractive-repulsive coupling strengths, we are able to find the necessary conditions for intralayer synchronization and interlayer antisynchronization and test whether these two macroscopic states can withstand demultiplexing in a network. During the occurrence of these two states, we analytically calculate the oscillator's amplitude. In addition to deriving the local stability conditions for interlayer antisynchronization via the master stability function approach, we also construct a suitable Lyapunov function to determine a sufficient condition for global stability. We provide numerical evidence to show the necessity of negative interlayer coupling strength for the occurrence of antisynchronization, and such repulsive interlayer coupling coefficients cannot destroy intralayer synchronization.

Quenching of oscillations via attenuated coupling for dissimilar electrochemical systems

The coupled dynamics of two similar and disparate electrochemical cells oscillators are analyzed. For the similar case, the cells are intentionally operated at different system parameters such that they exhibit distinct oscillatory dynamics ranging from periodic to chaotic. It is observed that when such systems are subjected to an attenuated coupling, implemented bidirectionally, they undergo a mutual quenching of oscillations. The same holds true for the configuration wherein two entirely different electrochemical cells are coupled via bidirectional attenuated coupling. Therefore, the attenuated coupling protocol seems to be universally efficient in achieving oscillation suppression in coupled oscillators (similar or heterogeneous oscillators). The experimental observations were verified by numerical simulations using appropriate electrodissolution model systems. Our results indicate that quenching of oscillations via attenuated coupling is robust and therefore could be ubiquitous in coupled systems with a large spatial separation prone to transmission losses.

Transition from inhomogeneous limit cycles to oscillation death in nonlinear oscillators with similarity-dependent coupling

We consider a system of coupled nonlinear oscillators in which the interaction is modulated by a measure of the similarity between the oscillators. Such a coupling is common in treating spatially mobile dynamical systems where the interaction is distance dependent or in resonance-enhanced interactions, for instance. For a system of Stuart-Landau oscillators coupled in this manner, we observe a novel route to oscillation death via a Hopf bifurcation. The individual oscillators are confined to inhomogeneous limit cycles initially and are damped to different fixed points after the bifurcation. Analytical and numerical results are presented for this case, while numerical results are presented for coupled Rössler and Sprott oscillators.

Effect of parameter mismatch and dissipative coupling on amplitude death regime in a coupled nonlinear aeroelastic system

Amplitude death (AD) has been recently identified as a phenomenon that can be exploited to stop unwanted large amplitude oscillations arising from instabilities in engineering systems. These oscillations are a consequence of the occurrence of dynamic instability, for example, the flutter instability, which results in the manifestation of sustained limit cycle oscillations. Recent studies have demonstrated amplitude death in coupled aeroelastic systems with identical parameters using suitable reactive coupling. Deriving impetus from the same, the dynamical signatures of coupled non-identical aeroelastic systems under a variety of coupling characteristics are investigated in the present study. The coupling characteristics between the individual airfoils here are assumed to possess both reactive and dissipative terms and are represented via a linear torsional spring and a damper, respectively. Explicit parameter mismatch is introduced via the use of different structural parameters such as frequency ratio and air-mass ratio for the individual airfoils. We demonstrate that a nonlinear coupled aeroelastic system with parameter mismatch and combined coupling characteristics gives rise to broader regimes of AD in aeroelastic systems. Specifically, the possibility of encountering large amplitude oscillations, usually found with pure reactive coupling can be avoided by adding a dissipative coupling term. On introducing dissipative coupling, the regime of AD was found to increase substantially, for both identical and non-identical scenarios, which in turn aids in serving as an effective tool to be developed further toward the application of flutter instability suppression.

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.

Achieving criticality for reservoir computing using environment-induced explosive death

The network of oscillators coupled via a common environment has been widely studied due to its great abundance in nature. We exploit the occurrence of explosive oscillation quenching in a network of non-identical oscillators coupled to each other indirectly via an environment for efficient reservoir computing. At the very edge of explosive transition, the reservoir achieves criticality maximizing its information processing capacity. The efficiency of the reservoir at different configurations is determined by the computational accuracy for different tasks performed by it. We analyze the dependence of accuracy on the dynamical behavior of the reservoir in terms of an order parameter symbolizing the desynchronization of the system. We found that the reservoir achieves the criticality in the steady-state region right at the edge of the hysteresis area. By computing the entropy of the reservoir for different tasks, we confirm that maximum accuracy corresponds to the edge of chaos or the edge of stability for this reservoir.

Aging in global networks with competing attractive-Repulsive interaction

We study the dynamical inactivity of the global network of identical oscillators in the presence of mixed attractive and repulsive coupling. We consider that the oscillators are a priori in all to all attractive coupling and then upon increasing the number of oscillators interacting via repulsive interaction, the whole network attains a steady state at a critical fraction of repulsive nodes, p_c. The macroscopic inactivity of the network is found to follow a typical aging transition due to competition between attractive-repulsive interactions. The analytical expression connecting the coupling strength and p_c is deduced and corroborated with numerical outcomes. We also study the influence of asymmetry in the attractive-repulsive interaction, which leads to symmetry breaking. We detect chimera-like and mixed states for a certain ratio of coupling strengths. We have verified sequential and random modes to choose the repulsive nodes and found that the results are in agreement. The paradigmatic networks with diverse dynamics, viz., limit cycle (Stuart-Landau), chaos (Rössler), and bursting (Hindmarsh-Rose neuron), are analyzed.

Amplitude chimera and chimera death induced by external agents in two-layer networks

We report the emergence of stable amplitude chimeras and chimera death in a two-layer network where one layer has an ensemble of identical nonlinear oscillators interacting directly through local coupling and indirectly through dynamic agents that form the second layer. The nonlocality in the interaction among the dynamic agents in the second layer induces different types of chimera-related dynamical states in the first layer. The amplitude chimeras developed in them are found to be extremely stable, while chimera death states are prevalent for increased coupling strengths. The results presented are for a system of coupled Stuart-Landau oscillators and can, in general, represent systems with short-range interactions coupled to another set of systems with long-range interactions. In this case, by tuning the range of interactions among the oscillators or the coupling strength between two types of systems, we can control the nature of chimera states and the system can also be restored to homogeneous steady states. The dynamic agents interacting nonlocally with long-range interactions can be considered as a dynamic environment or a medium interacting with the system. We indicate how the second layer can act as a reinforcement mechanism on the first layer under various possible interactions for desirable effects.

Using critical curves to compute master stability islands for amplitude death in networks of delay-coupled oscillators

In this paper, we present a method to compute master stability islands (MSIs) for amplitude death in networks of delay-coupled oscillators using critical curves. We first demonstrate how critical curves can be used to compute boundaries and contours of MSIs in delay-coupling parameter space and then provide a general study on the effects of the oscillator dynamics and network topology on the number, size, and contour types of all MSIs. We find that the oscillator dynamics can be used to determine the number and size of MSIs and that there are six possible contour types that depend on the choice of oscillator dynamics and the network topology. We introduce contour sequences and use these sequences to study the contours of all MSIs. Finally, we provide example MSIs for several classical nonlinear systems including the van der Pol system, the Rucklidge system, and the Rössler system.

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.

Explosive death in nonlinear oscillators coupled by quorum sensing

Many biological and chemical systems exhibit collective behavior in response to the change in their population density. These elements or cells communicate with each other via dynamical agents or signaling molecules. In this work, we explore the dynamics of nonlinear oscillators, specifically Stuart-Landau oscillators and Rayleigh oscillators, interacting globally through dynamical agents in the surrounding environment modeled as a quorum sensing interaction. The system exhibits the typical continuous second-order transition from oscillatory state to death state, when the oscillation amplitude is small. However, interestingly, when the amplitude of oscillations is large we find that the system shows an abrupt transition from oscillatory to death state, a transition termed "explosive death." So the quorum-sensing form of interaction can induce the usual second-order transition, as well as sudden first-order transitions. Further, in the case of the explosive death transitions, the oscillatory state and the death state coexist over a range of coupling strengths near the transition point. This emergent regime of hysteresis widens with increasing strength of the mean-field feedback, and is relevant to hysteresis that is widely observed in biological, chemical, and physical processes.

Traveling amplitude death in coupled pendula

We investigate the phenomenon of amplitude death [in two scenarios-traveling (TAD) and stationary] in coupled pendula with escapement mechanisms. The possible dynamics of the network is examined in coupling parameters' plane, and the corresponding examples of attractors are discussed. We analyze the properties of the observed patterns, studying the period of one full cycle of TAD under the influence of system's parameters, as well as the mechanism of its existence. It is shown, using the energy balance method, that the strict energy transfer between the pendula determines the direction in which the amplitude death travels from one unit to another. The occurrence of TAD is investigated as a result of a simple perturbation procedure, which shows that the transient dynamics on the road from complete synchronization to amplitude death is not straightforward. The pendula behavior during the transient processes is studied, and the influence of parameters and perturbation magnitude on the possible network's response is described. Finally, we analyze the energy transfer during the transient motion, indicating the potential triggers leading to the desired state. The obtained results suggest that the occurrence of traveling amplitude death is related to the chaotic dynamics and the phenomenon appears as a result of completely random process.

Control of coherence resonance by self-induced stochastic resonance in a multiplex neural network

We consider a two-layer multiplex network of diffusively coupled FitzHugh-Nagumo (FHN) neurons in the excitable regime. We show that the phenomenon of coherence resonance (CR) in one layer can not only be controlled by the network topology, the intra- and interlayer time-delayed couplings, but also by another phenomenon, namely, self-induced stochastic resonance (SISR) in the other layer. Numerical computations show that when the layers are isolated, each of these noise-induced phenomena is weakened (strengthened) by a sparser (denser) ring network topology, stronger (weaker) intralayer coupling forces, and longer (shorter) intralayer time delays. However, CR shows a much higher sensitivity than SISR to changes in these control parameters. It is also shown, in contrast to SISR in a single isolated FHN neuron, that the maximum noise amplitude at which SISR occurs in the network of coupled FHN neurons is controllable, especially in the regime of strong coupling forces and long time delays. In order to use SISR in the first layer of the multiplex network to control CR in the second layer, we first choose the control parameters of the second layer in isolation such that in one case CR is poor and in another case, nonexistent. It is then shown that a pronounced SISR can not only significantly improve a poor CR, but can also induce a pronounced CR, which was nonexistent in the isolated second layer. In contrast to strong intralayer coupling forces, strong interlayer coupling forces are found to enhance CR, while long interlayer time delays, just as long intralayer time delays, deteriorate CR. Most importantly, we find that in a strong interlayer coupling regime, SISR in the first layer performs better than CR in enhancing CR in the second layer. But in a weak interlayer coupling regime, CR in the first layer performs better than SISR in enhancing CR in the second layer. Our results could find novel applications in noisy neural network dynamics and engineering.

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.

Explosive death in complex network

We report the emergence of an explosive death transition in a network of identical oscillators interacting to other oscillators through nonlocal coupling in the presence of a common environment. This transition has an abrupt and irreversible characteristic in parameter space which has been a common signature of first order phase transition. For the similar coupling scheme, both ensemble of chaotic and periodic oscillators showed qualitatively similar kind of transition, hence making it a universal transition. The details of which along with dependence of environmental and nonlocal coupling on this first-order like phase transition is also discussed.

Role of time scales and topology on the dynamics of complex networks

The interplay between time scales and structural properties of complex networks of nonlinear oscillators can generate many interesting phenomena, like amplitude death, cluster synchronization, frequency synchronization, etc. We study the emergence of such phenomena and their transitions by considering a complex network of dynamical systems in which a fraction of systems evolves on a slower time scale on the network. We report the transition to amplitude death for the whole network and the scaling near the transitions as the connectivity pattern changes. We also discuss the suppression and recovery of oscillations and the crossover behavior as the number of slow systems increases. By considering a scale free network of systems with multiple time scales, we study the role of heterogeneity in link structure on dynamical properties and the consequent critical behaviors. In this case with hubs made slow, our main results are the escape time statistics for loss of complete synchrony as the slowness spreads on the network and the self-organization of the whole network to a new frequency synchronized state. Our results have potential applications in biological, physical, and engineering networks consisting of heterogeneous 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.

Emergent dynamics in delayed attractive-repulsively coupled networks

We investigate different emergent dynamics, namely, oscillation quenching and revival of oscillation, in a global network of identical oscillators coupled with diffusive (positive) delay coupling as it is perturbed by symmetry breaking localized repulsive delayed interaction. Starting from the oscillatory state (OS), we systematically identify three types of transition phenomena in the parameter space: (1) The system may reach inhomogeneous steady states from the homogeneous steady state sometimes called as the transition from amplitude death (AD) to oscillation death (OD) state, i.e., OS-AD-OD scenario, (2) Revival of oscillation (OS) from the AD state (OS-AD-OS), and (3) Emergence of the OD state from the oscillatory state (OS) without passing through AD, i.e., OS-OD. The dynamics of each node in the network is assumed to be governed either by the identical limit cycle Stuart-Landau system or by the chaotic Rössler system. Based on clustering behavior observed in the oscillatory network, we derive a reduced low-dimensional model of the large network. Using the reduced model, we investigate the effect of time delay on these transitions and demarcate OS, AD, and OD regimes in the parameter space. We also explore and characterize the bifurcation transitions present in both systems. The generic behavior of the low dimensional model and full network is found to match satisfactorily.

Synchronization in networks with heterogeneous coupling delays

Synchronization in networks of identical oscillators with heterogeneous coupling delays is studied. A decomposition of the network dynamics is obtained by block diagonalizing a newly introduced adjacency lag operator which contains the topology of the network as well as the corresponding coupling delays. This generalizes the master stability function approach, which was developed for homogenous delays. As a result the network dynamics can be analyzed by delay differential equations with distributed delay, where different delay distributions emerge for different network modes. Frequency domain methods are used for the stability analysis of synchronized equilibria and synchronized periodic orbits. As an example, the synchronization behavior in a system of delay-coupled Hodgkin-Huxley neurons is investigated. It is shown that the parameter regions where synchronized periodic spiking is unstable expand when increasing the delay heterogeneity.

Mathematical Modeling for Description of Oscillation Suppression Induced by Deep Brain Stimulation

A mathematical modeling for description of oscillation suppression by deep brain stimulation (DBS) is explored in this paper. High-frequency DBS introduced to the basal ganglia network can suppress pathological neural oscillations that occur in the Parkinsonian state. However, selecting appropriate stimulation parameters remains a challenging issue due to the limited understanding of the underlying mechanisms of the Parkinsonian state and its control. In this paper, we use a describing function analysis to provide an intuitive way to select the optimal stimulation parameters based on a biologically plausible computational model of the Parkinsonian neural network. By the stability analysis using the describing function method, effective DBS parameter regions for inhibiting the pathological oscillations can be predicted. Additionally, it is also found that a novel sinusoidal-shaped DBS may become an alternative stimulation pattern and expends less energy, but with a different mechanism. This paper provides new insight into the possible mechanisms underlying DBS and a prediction of optimal DBS parameter settings, and even suggests how to select novel DBS wave patterns for the treatment of movement disorders, such as Parkinson's disease.

Stabilization of a spatially uniform steady state in two systems exhibiting Turing patterns

This paper deals with the stabilization of a spatially uniform steady state in two coupled one-dimensional reaction-diffusion systems with Turing instability. This stabilization corresponds to amplitude death that occurs in a coupled system with Turing instability. Stability analysis of the steady state shows that stabilization does not occur if the two reaction-diffusion systems are identical. We derive a sufficient condition for the steady state to be stable for any length of system and any boundary conditions. Our analytical results are supported with numerical examples.

Distinct collective states due to trade-off between attractive and repulsive couplings

We investigate the effect of repulsive coupling together with an attractive coupling in a network of nonlocally coupled oscillators. To understand the complex interaction between these two couplings we introduce a control parameter in the repulsive coupling which plays a crucial role in inducing distinct complex collective patterns. In particular, we show the emergence of various cluster chimera death states through a dynamically distinct transition route, namely the oscillatory cluster state and coherent oscillation death state as a function of the repulsive coupling in the presence of the attractive coupling. In the oscillatory cluster state, the oscillators in the network are grouped into two distinct dynamical states of homogeneous and inhomogeneous oscillatory states. Further, the network of coupled oscillators follow the same transition route in the entire coupling range. Depending upon distinct coupling ranges, the system displays different number of clusters in the death state and oscillatory state. We also observe that the number of coherent domains in the oscillatory cluster state exponentially decreases with increase in coupling range and obeys a power-law decay. Additionally, we show analytical stability for observed solitary state, synchronized state, and incoherent oscillation death state.

Quenching oscillating behaviors in fractional coupled Stuart-Landau oscillators

Oscillation quenching has been widely studied during the past several decades in fields ranging from natural sciences to engineering, but investigations have so far been restricted to oscillators with an integer-order derivative. Here, we report the first study of amplitude death (AD) in fractional coupled Stuart-Landau oscillators with partial and/or complete conjugate couplings to explore oscillation quenching patterns and dynamics. It has been found that the fractional-order derivative impacts the AD state crucially. The area of the AD state increases along with the decrease of the fractional-order derivative. Furthermore, by introducing and adjusting a limiting feedback factor in coupling links, the AD state can be well tamed in fractional coupled oscillators. Hence, it provides one an effective approach to analyze and control the oscillating behaviors in fractional coupled oscillators.

The impact of propagation and processing delays on amplitude and oscillation deaths in the presence of symmetry-breaking coupling

We numerically investigate the impacts of both propagation and processing delays on the emergences of amplitude death (AD) and oscillation death (OD) in one system of two Stuart-Landau oscillators with symmetry-breaking coupling. In either the absence of or the presence of propagation delay, the processing delay destabilizes both AD and OD by revoking the stability of the stable homogenous and inhomogenous steady states. In the AD to OD transition, the processing delay destabilizes first OD from large values of coupling strength until its stable regime completely disappears and then AD from both the upper and lower bounds of the stable coupling interval. Our numerical study sheds new insight lights on the understanding of nontrivial effects of time delays on dynamic activity of coupled nonlinear systems.

Amplitude death in a pair of one-dimensional complex Ginzburg-Landau systems coupled by diffusive connections

This paper shows that, in a pair of one-dimensional complex Ginzburg-Landau (CGL) systems, diffusive connections can induce amplitude death. Stability analysis of a spatially uniform steady state in coupled CGL systems reveals that amplitude death never occurs in a pair of identical CGL systems coupled by no-delay connection, but can occur in the case of delay connection. Moreover, amplitude death never occurs in coupled identical CGL systems with zero nominal frequency. Based on these analytical results, we propose a procedure for designing the connection delay time and the coupling strength to induce spatial-robust stabilization, that is, a stabilization of the steady state for any system size and any boundary condition. Numerical simulations are performed to confirm the analytical results.

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

When in an ensemble of oscillatory units the interaction occurs through a diffusion-like manner, the intrinsic oscillations can be quenched through two structurally different scenarios: amplitude death (AD) and oscillation death (OD). Unveiling the underlying principles of stable rhythmic activity against AD and OD is a challenging issue of substantial practical significance. Here, by developing a low-pass filter (LPF) to track the output signals of the local system in the coupling, we show that it can revoke both AD and OD, and even the AD to OD transition, thereby giving rise to oscillations in coupled nonlinear oscillators under diverse death scenarios. The effectiveness of the local LPF is proven to be valid in an arbitrary network of coupled oscillators with distributed propagation delays. The constructive role of the local LPF in revoking deaths provides a potential dynamic mechanism of sustaining a reliable rhythmicity in real-world systems.

The Synergistic Role of Light-Feeding Phase Relations on Entraining Robust Circadian Rhythms in the Periphery

The feeding and fasting cycles are strong behavioral signals that entrain biological rhythms of the periphery. The feeding rhythms synchronize the activities of the metabolic organs, such as liver, synergistically with the light/dark cycle primarily entraining the suprachiasmatic nucleus. The likely phase misalignment between the feeding rhythms and the light/dark cycles appears to induce circadian disruptions leading to multiple physiological abnormalities motivating the need to investigate the mechanisms behind joint light-feeding circadian entrainment of peripheral tissues. To address this question, we propose a semimechanistic mathematical model describing the circadian dynamics of peripheral clock genes in human hepatocyte under the control of metabolic and light rhythmic signals. The model takes the synergistically acting light/dark cycles and feeding rhythms as inputs and incorporates the activity of sirtuin 1, a cellular energy sensor and a metabolic enzyme activated by nicotinamide adenine dinucleotide. The clock gene dynamics was simulated under various light-feeding phase relations and intensities, to explore the feeding entrainment mechanism as well as the convolution of light and feeding signals in the periphery. Our model predicts that the peripheral clock genes in hepatocyte can be completely entrained to the feeding rhythms, independent of the light/dark cycle. Furthermore, it predicts that light-feeding phase relationship is a critical factor in robust circadian oscillations.

Environmental coupling in ecosystems: From oscillation quenching to rhythmogenesis

How landscape fragmentation affects ecosystems diversity and stability is an important and complex question in ecology with no simple answer, as spatially separated habitats where species live are highly dynamic rather than just static. Taking into account the species dispersal among nearby connected habitats (or patches) through a common dynamic environment, we model the consumer-resource interactions with a ring type coupled network. By characterizing the dynamics of consumer-resource interactions in a coupled ecological system with three fundamental mechanisms such as the interaction within the patch, the interaction between the patches, and the interaction through a common dynamic environment, we report the occurrence of various collective behaviors. We show that the interplay between the dynamic environment and the dispersal among connected patches exhibits the mechanism of generation of oscillations, i.e., rhythmogenesis, as well as suppression of oscillations, i.e., amplitude death and oscillation death. Also, the transition from homogeneous steady state to inhomogeneous steady state occurs through a codimension-2 bifurcation. Emphasizing a network of a spatially extended system, the coupled model exposes the collective behavior of a synchrony-stability relationship with various synchronization occurrences such as in-phase and out-of-phase.

A common lag scenario in quenching of oscillation in coupled oscillators

A large parameter mismatch can induce amplitude death in two instantaneously coupled oscillators. Alternatively, a time delay in the coupling can induce amplitude death in two identical oscillators. We unify the mechanism of quenching of oscillation in coupled oscillators, either by a large parameter mismatch or a delay coupling, by a common lag scenario that is, surprisingly, different from the conventional lag synchronization. We present numerical as well as experimental evidence of this unknown kind of lag scenario when the lag increases with coupling and at a critically large value at a critical coupling strength, amplitude death emerges in two largely mismatched oscillators. This is analogous to amplitude death in identical systems with increasingly large coupling delay. In support, we use examples of the Chua oscillator and the Bonhoeffer-van der Pol system. Furthermore, we confirm this lag scenario during the onset of amplitude death in identical Stuart-Landau system under various instantaneous coupling forms, repulsive, conjugate, and a type of nonlinear coupling.

Master stability islands for amplitude death in networks of delay-coupled oscillators

This paper presents a master stability function (MSF) approach for analyzing the stability of amplitude death (AD) in networks of delay-coupled oscillators. Unlike the familiar MSFs for instantaneously coupled networks, which typically have a single input encoding for the effects of the eigenvalues of the network Laplacian matrix, for delay-coupled networks we show that such MSFs generally require two additional inputs: the time delay and the coupling strength. To utilize the MSF for determining the stability of AD of general networks for a chosen nonlinear system (node dynamics) and coupling function, we introduce the concept of master stability islands (MSIs), which are two-dimensional stability islands of the delay-coupling parameter space together with a third dimension ("altitude") encoding for eigenvalues that result in stable AD. We numerically compute the MSFs and visualize the corresponding MSIs for several common chaotic systems including the Rössler, the Lorenz, and Chen's system and find that it is generally possible to achieve AD and that a nonzero time delay is necessary for the stabilization of the AD states.

Different kinds of chimera death states in nonlocally coupled oscillators

We investigate the significance of nonisochronicity parameter in a network of nonlocally coupled Stuart-Landau oscillators with symmetry breaking form. We observe that the presence of nonisochronicity parameter leads to structural changes in the chimera death region while varying the strength of the interaction. This gives rise to the existence of different types of chimera death states such as multichimera death state, type I periodic chimera death (PCD) state, and type II periodic chimera death state. We also find that the number of periodic domains in both types of PCD states decreases exponentially with an increase of coupling range and obeys a power law under nonlocal coupling. Additionally, we also analyze the structural changes of chimera death states by reducing the system of dynamical equations to a phase model through the phase reduction. We also briefly study the role of nonisochronicity parameter on chimera states, where the existence of a multichimera state with respect to the coupling range is pointed out. Moreover, we also analyze the robustness of the chimera death state to perturbations in the natural frequencies of the oscillators.

Mixed-mode oscillation suppression states in coupled oscillators

We report a collective dynamical state, namely the mixed-mode oscillation suppression state where the steady states of the state variables of a system of coupled oscillators show heterogeneous behaviors. We identify two variants of it: The first one is a mixed-mode death (MMD) state, which is an interesting oscillation death state, where a set of variables show dissimilar values, while the rest arrive at a common value. In the second mixed death state, bistable and monostable nontrivial homogeneous steady states appear simultaneously to a different set of variables (we refer to it as the MNAD state). We find these states in the paradigmatic chaotic Lorenz system and Lorenz-like system under generic coupling schemes. We identify that while the reflection symmetry breaking is responsible for the MNAD state, the breaking of both the reflection and translational symmetries result in the MMD state. Using a rigorous bifurcation analysis we establish the occurrence of the MMD and MNAD states, and map their transition routes in parameter space. Moreover, we report experimental observation of the MMD and MNAD states that supports our theoretical results. We believe that this study will broaden our understanding of oscillation suppression states; subsequently, it may have applications in many real physical systems, such as laser and geomagnetic systems, whose mathematical models mimic the Lorenz system.

Oscillation suppression in indirectly coupled limit cycle oscillators

We study the phenomena of oscillation quenching in a system of limit cycle oscillators which are coupled indirectly via a dynamic environment. The dynamics of the environment is assumed to decay exponentially with some decay parameter. We show that for appropriate coupling strength, the decay parameter of the environment plays a crucial role in the emergent dynamics such as amplitude death (AD) and oscillation death (OD). The critical curves for the regions of oscillation quenching as a function of coupling strength and decay parameter of the environment are obtained analytically using linear stability analysis and are found to be consistent with the numerics.

Feedback as a mechanism for the resurrection of oscillations from death states

The quenching of oscillations in interacting systems leads to several unwanted situations, which necessitate a suitable remedy to overcome the quenching. In this connection, this work addresses a mechanism that can resurrect oscillations in a typical situation. Through both numerical and analytical studies, we show that the candidate which is capable of resurrecting oscillations is nothing but the feedback, the one which is profoundly used in dynamical control and in biotherapies. Even in the case of a rather general system, we demonstrate analytically the applicability of the technique over one of the oscillation quenched states called amplitude death states. We also discuss some of the features of this mechanism such as adaptability of the technique with the feedback of only a few of the oscillators.

Transitions among the diverse oscillation quenching states induced by the interplay of direct and indirect coupling

We report the transitions among different oscillation quenching states induced by the interplay of diffusive (direct) coupling and environmental (indirect) coupling in coupled identical oscillators. This coupling scheme was introduced by Resmi et al. [Phys. Rev. E 84, 046212 (2011)] as a general scheme to induce amplitude death (AD) in nonlinear oscillators. Using a detailed bifurcation analysis we show that, in addition to AD, which actually occurs only in a small region of parameter space, this coupling scheme can induce other oscillation quenching states, namely oscillation death (OD) and a novel nontrvial AD (NAD) state, which is a nonzero bistable homogeneous steady state; more importantly, this coupling scheme mediates a transition from the AD state to the OD state and a new transition from the AD state to the NAD state. We identify diverse routes to the NAD state and map all the transition scenarios in the parameter space for periodic oscillators. Finally, we present the first experimental evidence of oscillation quenching states and their transitions induced by the interplay of direct and indirect coupling.

Mobility and density induced amplitude death in metapopulation networks of coupled oscillators

We investigate the effects of mobility and density on the amplitude death of coupled Landau-Stuart oscillators and Brusselators in metapopulation networks, wherein each node represents a subpopulation occupied any number of mobile individuals. By numerical simulations in scale-free topology, we find that the systems undergo phase transitions from incoherent state to amplitude death, and then to frequency synchronization with increasing the mobility rate or density of oscillators. Especially, there exists an extent of intermediate mobility rate and density that can lead to global oscillator death. Furthermore, we show that such nontrivial phenomena are robust to diverse network topologies. Our findings may invoke further efforts and attentions to explore the underlying mechanism of collective behaviors in coupled metapopulation systems.

Transition from amplitude to oscillation death in a network of oscillators

We report a transition from a homogeneous steady state (HSS) to inhomogeneous steady states (IHSSs) in a network of globally coupled identical oscillators. We perturb a synchronized population of oscillators in the network with a few local negative or repulsive mean field links. The whole population splits into two clusters for a certain number of repulsive mean field links and a range of coupling strength. For further increase of the strength of interaction, these clusters collapse into a HSS followed by a transition to IHSSs where all the oscillators populate either of the two stable steady states. We analytically determine the origin of HSS and its transition to IHSS in relation to the number of repulsive mean-field links and the strength of interaction using a reductionism approach to the model network. We verify the results with numerical examples of the paradigmatic Landau-Stuart limit cycle system and the chaotic Rössler oscillator as dynamical nodes. During the transition from HSS to IHSSs, the network follows the Turing type symmetry breaking pitchfork or transcritical bifurcation depending upon the system dynamics.

Synchronization and amplitude death in hypernetworks

We study dynamical systems on a hypernetwork, namely by coupling them through several variables. For the case when the coupling(s) are all linear, a comprehensive analysis of the master stability function (MSF) for synchronized dynamics is presented and, through application to a number of paradigmatic examples, the typical forms of the MSF are discussed. The MSF formalism for hypernetworks also provides a framework to study synchronization in systems that are diffusively coupled through dissimilar variables-the so-called conjugate coupling that can lead to amplitude or oscillation death.

Amplitude death of coupled hair bundles with stochastic channel noise

Hair cells conduct auditory transduction in vertebrates. In lower vertebrates such as frogs and turtles, due to the active mechanism in hair cells, hair bundles (stereocilia) can be spontaneously oscillating or quiescent. Recently an amplitude death phenomenon has been proposed [K.-H. Ahn, J. R. Soc. Interface, 10, 20130525 (2013)] as a mechanism for auditory transduction in frog hair-cell bundles, where sudden cessation of the oscillations arises due to the coupling between nonidentical hair bundles. The gating of the ion channel is intrinsically stochastic due to the stochastic nature of the configuration change of the channel. The strength of the noise due to the channel gating can be comparable to the thermal Brownian noise of hair bundles. Thus, we perform stochastic simulations of the elastically coupled hair bundles. In spite of stray noisy fluctuations due to its stochastic dynamics, our simulation shows the transition from collective oscillation to amplitude death as interbundle coupling strength increases. In its stochastic dynamics, the formation of the amplitude death state of coupled hair bundles can be seen as a sudden suppression of the displacement fluctuation of the hair bundles as the coupling strength increases. The enhancement of the signal-to-noise ratio through the amplitude death phenomenon is clearly seen in the stochastic dynamics. Our numerical results demonstrate that the multiple number of transduction channels per hair bundle is an important factor to the amplitude death phenomenon, because the phenomenon may disappear for a small number of transduction channels due to strong gating noise.

Analytical criterion for amplitude death in nonautonomous systems with piecewise nonlinear coupling

We investigate the amplitude death phenomenon in a nonautonomous chained network with complicated piecewise nonlinear coupling functions. An analytical criterion for the boundary of the amplitude death region is derived by using the average method. The mechanism of the amplitude death in the nonautonomous networks is very different from that of autonomous systems and rapid dynamic transitions could halt the amplitude death. Numerical verifications are carried out to check jump transitions among different solution branches and further confirm the correctness of the theoretical results.

Diverse routes of transition from amplitude to oscillation death in coupled oscillators under additional repulsive links

We report the existence of diverse routes of transition from amplitude death to oscillation death in three different diffusively coupled systems, which are perturbed by a symmetry breaking repulsive coupling link. For limit-cycle systems the transition is through a pitchfork bifurcation, as has been noted before, but in chaotic systems it can be through a saddle-node or a transcritical bifurcation depending on the nature of the underlying dynamics of the individual systems. The diversity of the routes and their dependence on the complex dynamics of the coupled systems not only broadens our understanding of this important phenomenon but can lead to potentially new practical applications.

Amplitude death and synchronized states in nonlinear time-delay systems coupled through mean-field diffusion

We explore and experimentally demonstrate the phenomena of amplitude death (AD) and the corresponding transitions through synchronized states that lead to AD in coupled intrinsic time-delayed hyperchaotic oscillators interacting through mean-field diffusion. We identify a novel synchronization transition scenario leading to AD, namely transitions among AD, generalized anticipatory synchronization (GAS), complete synchronization (CS), and generalized lag synchronization (GLS). This transition is mediated by variation of the difference of intrinsic time-delays associated with the individual systems and has no analogue in non-delayed systems or coupled oscillators with coupling time-delay. We further show that, for equal intrinsic time-delays, increasing coupling strength results in a transition from the unsynchronized state to AD state via in-phase (complete) synchronized states. Using Krasovskii-Lyapunov theory, we derive the stability conditions that predict the parametric region of occurrence of GAS, GLS, and CS; also, using a linear stability analysis, we derive the condition of occurrence of AD. We use the error function of proper synchronization manifold and a modified form of the similarity function to provide the quantitative support to GLS and GAS. We demonstrate all the scenarios in an electronic circuit experiment; the experimental time-series, phase-plane plots, and generalized autocorrelation function computed from the experimental time series data are used to confirm the occurrence of all the phenomena in the coupled oscillators.

Oscillation death in diffusively coupled oscillators by local repulsive link

A death of oscillation is reported in a network of coupled synchronized oscillators in the presence of additional repulsive coupling. The repulsive link evolves as an averaging effect of mutual interaction between two neighboring oscillators due to a local fault and the number of repulsive links grows in time when the death scenario emerges. Analytical condition for oscillation death is derived for two coupled Landau-Stuart systems. Numerical results also confirm oscillation death in chaotic systems such as a Sprott system and the Rössler oscillator. We explore the effect in large networks of globally coupled oscillators and find that the number of repulsive links is always fewer than the size of the network.

Frustration induced oscillator death on networks

An array of identical maps with Ising symmetry, with both positive and negative couplings, is studied. We divide the maps into two groups, with positive intra-group couplings and negative inter-group couplings. This leads to antisynchronization between the two groups which have the same stability properties as the synchronized state. Introducing a certain degree of randomness in signs of these couplings destabilizes the anti-synchronized state. Further increasing the randomness in signs of these couplings leads to oscillator death. This is essentially a frustration induced phenomenon. We explain the observed results using the theory of random matrices with nonzero mean. We briefly discuss applications to coupled differential equations.

Amplitude death in nonlinear oscillators with mixed time-delayed coupling

Amplitude death (AD) is an emergent phenomenon whereby two or more autonomously oscillating systems completely lose their oscillations due to coupling. In this work, we study AD in nonlinear oscillators with mixed time-delayed coupling, which is a combination of instantaneous and time-delayed couplings. We find that the mixed time-delayed coupling favors the onset of AD for a larger set of parameters than in the limiting cases of purely instantaneous or completely time-delayed coupling. Coupled identical oscillators experience AD under instantaneous coupling mixed with a small proportion of time-delayed coupling. Our work gives a deeper understanding of delay-induced AD in coupled nonlinear oscillators.

Amplitude death phenomena in delay-coupled Hamiltonian systems

Hamiltonian systems, when coupled via time-delayed interactions, do not remain conservative. In the uncoupled system, the motion can typically be periodic, quasiperiodic, or chaotic. This changes drastically when delay coupling is introduced since now attractors can be created in the phase space. In particular, for sufficiently strong coupling there can be amplitude death (AD), namely, the stabilization of point attractors and the cessation of oscillatory motion. The approach to the state of AD or oscillation death is also accompanied by a phase flip in the transient dynamics. A discussion and analysis of the phenomenology is made through an application to the specific cases of harmonic as well as anharmonic coupled oscillators, in particular the Hénon-Heiles system.

Collective dynamics of a network of ratchets coupled via a stochastic dynamical environment

We investigate the collective dynamics of a network of inertia particles diffusing in a ratchet potential and interacting indirectly through their stochastic dynamical environment. We obtain analytically the condition for the existence of a stable collective state, and we show that the number N of particles in the network, and the strength k of their interaction with the environment, play key roles in synchronization and transport processes. Synchronization is preceded by symmetry-breaking associated with double-resonance oscillations and is shown to be strongly dependent on the network size: convergence to the synchronization manifold occurs much faster with a large network. For small networks, increasing the noise level enhances synchronization in the weakly coupled regime, while particles in a large network are weakly synchronized. Similarly, in the strongly coupled regime, particles in a small network are weakly synchronized; whereas the synchronization is strong and robust against noise when the network-size is large. Small and moderate networks maximize and stabilize efficient transport. Although the dynamics for larger networks is highly correlated, the transport current is erratic.

Phase-flip transition in nonlinear oscillators coupled by dynamic environment

We study the dynamics of nonlinear oscillators indirectly coupled through a dynamical environment or a common medium. We observed that this form of indirect coupling leads to synchronization and phase-flip transition in periodic as well as chaotic regime of oscillators. The phase-flip transition from in- to anti-phase synchronization or vise-versa is analyzed in the parameter plane with examples of Landau-Stuart and Rössler oscillators. The dynamical transitions are characterized using various indices such as average phase difference, frequency, and Lyapunov exponents. Experimental evidence of the phase-flip transition is shown using an electronic version of the van der Pol oscillators.

Amplitude death with mean-field diffusion

We study the dynamics of nonlinear oscillators under mean-field diffusive coupling. We observe that this form of coupling leads to amplitude death via a synchronization transition in the parameter space of the coupling strength and mean-field control parameter. A general criterion for amplitude death for any given dynamical system with mean-field diffusion is obtained, and these dynamical transitions are characterized using various indices such as average phase difference, Lyapunov exponents, and average amplitude. This behavior is analyzed in the parameter plane by numerical studies of specific cases of the Landau-Stuart limit-cycle oscillator, and Rössler, Lorenz, FitzHugh-Nagumo excitable, and Chua systems.