Amplitude death with mean-field diffusion

Realization of logic gates in bi-directionally coupled nonlinear oscillators

Implementation of logic gates has been investigated in nonlinear dynamical systems from various perspectives over the years. Specifically, logic gates have been implemented in both single nonlinear systems and coupled nonlinear oscillators. The majority of the works in the literature have been done on the evolution of single oscillators into OR/AND or NOR/NAND logic gates. In the present study, we demonstrate the design of logic gates in bi-directionally coupled double-well Duffing oscillators by applying two logic inputs to the drive system alone along with a fixed bias. The nonlinear system, comprising both bi-directional components, exhibits varied logic behaviors within an optimal range of coupling strength. Both attractive and repulsive couplings yield similar and complementary logic behaviors in the first and second oscillators. These couplings play a major role in exhibiting fundamental and universal logic gates in simple nonlinear systems. Under a positive bias, both the first and second oscillators demonstrate OR logic gate for the attractive coupling, while exhibiting OR and NOR logic gates, respectively, for the repulsive coupling. Conversely, under a negative bias, both the first and second oscillators display AND logic gate for the attractive coupling, and AND and NAND logical outputs for the repulsive coupling. Furthermore, we confirm the robustness of the bi-directional oscillators against moderate noise in maintaining the desired logical outputs.

Learning unidirectional coupling using an echo-state network

Reservoir Computing has found many potential applications in the field of complex dynamics. In this article, we explore the exceptional capability of the echo-state network (ESN) model to make it learn a unidirectional coupling scheme from only a few time series data of the system. We show that, once trained with a few example dynamics of a drive-response system, the machine is able to predict the response system's dynamics for any driver signal with the same coupling. Only a few time series data of an A-B type drive-response system in training is sufficient for the ESN to learn the coupling scheme. After training, even if we replace drive system A with a different system C, the ESN can reproduce the dynamics of response system B using the dynamics of new drive system C only.

Enhanced signal response in globally coupled networks of bistable oscillators: Effects of mean field density and signal shape

This paper studies a set of globally coupled bistable oscillators, all subjected to the same weak periodic signal and identical coupling. The effect of mean field density (MFD) on global dynamics is analyzed. The oscillators switch from intra- to interwell motion as MFD increases, clearly demonstrating MFD-enhanced signal amplification. A maximum amplification also occurs at a moderate level of MFD, indicating that the response exhibits a nonmonotonic sensitivity to MFD. The MFD-enhanced response depends mainly on the signal intensity but not on the signal frequency or the network topology. The analytical investigation provides a simplified model to study the mechanism underlying this resonancelike behavior. It is shown that by modifying the bistability nature of the potential energy, the mean field density can promote well-to-well oscillations and larger amplitude motions. Finally, the robustness of this phenomenon to various signal waveforms is examined. It can therefore be used alternatively to efficiently amplify weak signals in practical situations with large network sizes.

Using coupling imperfection to control amplitude death

Previous studies of nonlinear oscillator networks have shown that amplitude death (AD) occurs after tuning oscillator parameters and coupling properties. Here, we identify regimes where the opposite occurs and show that a local defect (or impurity) in network connectivity leads to AD suppression in situations where identically coupled oscillators cannot. The critical impurity strength value leading to oscillation restoration is an explicit function of network size and system parameters. In contrast to homogeneous coupling, network size plays a crucial role in reducing this critical value. This behavior can be traced back to the steady-state destabilization through a Hopf's bifurcation, which occurs for impurity strengths below this threshold. This effect is illustrated across different mean-field coupled networks and is supported by simulations and theoretical analysis. Since local inhomogeneities are ubiquitous and often unavoidable, such imperfections can be an unexpected source of oscillation control.

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.

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.

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.

Occasional coupling enhances amplitude death in delay-coupled oscillators

This paper aims to study amplitude death in time delay coupled oscillators using the occasional coupling scheme that implies intermittent interaction among the oscillators. An enhancement of amplitude death regions (i.e., an increment of the width of the amplitude death regions along the control parameter axis) can be possible using the occasional coupling in a pair of delay-coupled oscillators. Our study starts with coupled limit cycle oscillators (Stuart-Landau) and coupled chaotic oscillators (Rössler). We further examine coupled horizontal Rijke tubes, a prototypical model of thermoacoustic systems. Oscillatory states are highly detrimental to thermoacoustic systems such as combustors. Consequently, a state of amplitude death is always preferred. We employ the on-off coupling (i.e., a square wave function), as an occasional coupling scheme, to these coupled oscillators. On monotonically varying the coupling strength (as a control parameter), we observe an enhancement of amplitude death regions using the occasional coupling scheme compared to the continuous coupling scheme. In order to study the contribution of the occasional coupling scheme, we perform a detailed linear stability analysis and analytically explain this enhancement of the amplitude death region for coupled limit cycle oscillators. We also adopt the frequency ratio of the oscillators and the time delay between the oscillators as the control parameters. Intriguingly, we obtain a similar enhancement of the amplitude death regions using the frequency ratio and time delay as the control parameters in the presence of the occasional coupling. Finally, we use a half-wave rectified sinusoidal wave function (motivated by practical reality) to introduce the occasional coupling in time delay coupled oscillators and get similar results.

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 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.

Emergent rhythms in coupled nonlinear oscillators due to dynamic interactions

The role of a new form of dynamic interaction is explored in a network of generic identical oscillators. The proposed design of dynamic coupling facilitates the onset of a plethora of asymptotic states including synchronous states, amplitude death states, oscillation death states, a mixed state (complete synchronized cluster and small amplitude desynchronized domain), and bistable states (coexistence of two attractors). The dynamical transitions from the oscillatory to the death state are characterized using an average temporal interaction approximation, which agrees with the numerical results in temporal interaction. A first-order phase transition behavior may change into a second-order transition in spatial dynamic interaction solely depending on the choice of initial conditions in the bistable regime. However, this possible abrupt first-order like transition is completely non-existent in the case of temporal dynamic interaction. Besides the study on periodic Stuart-Landau systems, we present results for the paradigmatic chaotic model of Rössler oscillators and the MacArthur ecological model.

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.

Static and dynamic attractive-repulsive interactions in two coupled nonlinear oscillators

Many systems exhibit both attractive and repulsive types of interactions, which may be dynamic or static. A detailed understanding of the dynamical properties of a system under the influence of dynamically switching attractive or repulsive interactions is of practical significance. However, it can also be effectively modeled with two coexisting competing interactions. In this work, we investigate the effect of time-varying attractive-repulsive interactions as well as the hybrid model of coexisting attractive-repulsive interactions in two coupled nonlinear oscillators. The dynamics of two coupled nonlinear oscillators, specifically limit cycles as well as chaotic oscillators, are studied in detail for various dynamical transitions for both cases. Here, we show that dynamic or static attractive-repulsive interactions can induce an important transition from the oscillatory to steady state in identical nonlinear oscillators due to competitive effects. The analytical condition for the stable steady state in dynamic interactions at the low switching time period and static coexisting interactions are calculated using linear stability analysis, which is found to be in good agreement with the numerical results. In the case of a high switching time period, oscillations are revived for higher interaction strength.

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.

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.

Hindmarsh-Rose neuron model with memristors

We analyze single and coupled Hindmarsh-Rose neurons in the presence of a time varying electromagnetic field which results from the exchange of ions across the membrane. Memristors are used to model the relation between magnetic flux of the electromagnetic field and the membrane potential of interacting neurons. The bifurcation analysis of Hindmarsh-Rose neurons has been carried out by varying the modulation intensity of induced current on the membrane potential. Many important dynamical behaviors such as synchrony, desynchrony, amplitude death, anti-phase oscillations, coexistence of resting and spiking state, and near death rare spikes are observed when the neurons are coupled using electrical and chemical synapses. In all cases the transverse Lyapunov exponents are plotted to observe the point of transition from desynchrony to synchrony. The memristor based analysis on neural networks can contribute to biological system modeling and can be used as a synapse in hardware of artificial neural networks.

Enhancing coherence via tuning coupling range in nonlocally coupled Stuart-Landau oscillators

Nonlocal coupling, as an important connection topology among nonlinear oscillators, has attracted increasing attention recently with the research boom of chimera states. So far, most previous investigations have focused on nonlocally coupled systems interacted via similar variables. In this work, we report the evolutions of dynamical behaviors in the nonlocally coupled Stuart-Landau oscillators by applying conjugate variables feedback. Through rigorous analysis, we find that the oscillation death (OD) can convert into the amplitude death (AD) via the cluster state with the increasing of coupling range, making the AD regions to be expanded infinitely along two directions of both the natural frequency and the coupling strength. Moreover, the limit cycle oscillation (OS) region and the mixed region of OD and OS will turn to anti-synchronization state through amplitude-mediated chimera. Therefore, the procedure from local coupling to nonlocal one implies indeed the continuous enhancement of coherence among neighboring oscillators in coupled systems.

Explosive death of conjugate coupled Van der Pol oscillators on networks

Explosive death phenomenon has been gradually gaining attention of researchers due to the research boom of explosive synchronization, and it has been observed recently for the identical or nonidentical coupled systems in all-to-all network. In this work, we investigate the emergence of explosive death in networked Van der Pol (VdP) oscillators with conjugate variables coupling. It is demonstrated that the network structures play a crucial role in identifying the types of explosive death behaviors. We also observe that the damping coefficient of the VdP system not only can determine whether the explosive death state is generated but also can adjust the forward transition point. We further show that the backward transition point is independent of the network topologies and the damping coefficient, which is well confirmed by theoretical analysis. Our results reveal the generality of explosive death phenomenon in different network topologies and are propitious to promote a better comprehension for the oscillation quenching behaviors.

Networked-oscillator-based modeling and control of unsteady wake flows

A networked-oscillator-based analysis is performed to examine and control the transfer of kinetic energy for periodic bluff body flows. The dynamics of energy fluctuations in the flow field are described by a set of oscillators defined by conjugate pairs of spatial proper orthogonal decomposition (POD) modes. To extract the network of interactions among oscillators, impulse responses of the oscillators to amplitude and phase perturbations are tracked. Tracking small energy inputs and using linear regression, a networked-oscillator model is constructed that reveals energy exchange among the modes. The model captures the nonlinear interactions among the modal oscillators through a linear approximation. A large collection of system responses is aggregated to capture the general network structure of oscillator interactions. The present networked-oscillator model describes the modal perturbation dynamics more accurately than the empirical Galerkin reduced-order model. The linear network model for nonlinear dynamics is subsequently utilized to design a model-based feedback controller. The controller suppresses the modal amplitudes that result in wake unsteadiness leading to drag reduction. The strength of the proposed approach is demonstrated for a canonical example of two-dimensional unsteady flow over a circular cylinder. The present formulation enables the characterization of modal interactions to control fundamental energy transfers in unsteady bluff body flows.

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.

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.

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.

Explosive death induced by mean-field diffusion in identical oscillators

We report the occurrence of an explosive death transition for the first time in an ensemble of identical limit cycle and chaotic oscillators coupled via mean–field diffusion. In both systems, the variation of the normalized amplitude with the coupling strength exhibits an abrupt and irreversible transition to death state from an oscillatory state and this first order phase transition to death state is independent of the size of the system. This transition is quite general and has been found in all the coupled systems where in–phase oscillations co–exist with a coupling dependent homogeneous steady state. The backward transition point for this phase transition has been calculated using linear stability analysis which is in complete agreement with the numerics.

Phase-flip and oscillation-quenching-state transitions through environmental diffusive coupling

We study the dynamics of nonlinear oscillators coupled through environmental diffusive coupling. The interaction between the dynamical systems is maintained through its agents which, in turn, interact globally with each other in the common dynamical environment. We show that this form of coupling scheme can induce an important transition like phase-flip transition as well transitions among oscillation quenching states in identical limit-cycle oscillators. This behavior is analyzed in the parameter plane by analytical and numerical studies of specific cases of the Stuart-Landau oscillator and van der Pol oscillator. Experimental evidences of the phase-flip transition and quenching states are shown using an electronic version of the van der Pol oscillators.

Oscillation quenching in third order phase locked loop coupled by mean field diffusive coupling

We explored analytically the oscillation quenching phenomena (amplitude death and parameter dependent inhomogeneous steady state) in a coupled third order phase locked loop (PLL) both in periodic and chaotic mode. The phase locked loops were coupled through mean field diffusive coupling. The lower and upper limits of the quenched state were identified in the parameter space of the coupled PLL using the Routh-Hurwitz technique. We further observed that the ability of convergence to the quenched state of coupled PLLs depends on the design parameters. For identical systems, both the systems converge to the homogeneous steady state, whereas for non-identical parameter values they converge to an inhomogeneous steady state. It was also observed that for identical systems, the quenched state is wider than the non-identical case. When the system parameters are so chosen that each isolated loop is chaotic in nature, we observe narrowing down of the quenched state. All these phenomena were also demonstrated through numerical simulations.

Reviving oscillation with optimal spatial period of frequency distribution in coupled oscillators

The spatial distributions of system's frequencies have significant influences on the critical coupling strengths for amplitude death (AD) in coupled oscillators. We find that the left and right critical coupling strengths for AD have quite different relations to the increasing spatial period m of the frequency distribution in coupled oscillators. The left one has a negative linear relationship with m in log-log axis for small initial frequency mismatches while remains constant for large initial frequency mismatches. The right one is in quadratic function relation with spatial period m of the frequency distribution in log-log axis. There is an optimal spatial period m₀ of frequency distribution with which the coupled system has a minimal critical strength to transit from an AD regime to reviving oscillation. Moreover, the optimal spatial period m₀ of the frequency distribution is found to be related to the system size N. Numerical examples are explored to reveal the inner regimes of effects of the spatial frequency distribution on AD.

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.

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.

Mean-field dispersion-induced spatial synchrony, oscillation and amplitude death, and temporal stability in an ecological model

One of the most important issues in spatial ecology is to understand how spatial synchrony and dispersal-induced stability interact. In the existing studies it is shown that dispersion among identical patches results in spatial synchrony; on the other hand, the combination of spatial heterogeneity and dispersion is necessary for dispersal-induced stability (or temporal stability). Population synchrony and temporal stability are thus often thought of as conflicting outcomes of dispersion. In contrast to the general belief, in this present study we show that mean-field dispersion is conducive to both spatial synchrony and dispersal-induced stability even in identical patches. This simultaneous occurrence of rather conflicting phenomena is governed by the suppression of oscillation states, namely amplitude death (AD) and oscillation death (OD). These states emerge through spatial synchrony of the oscillating patches in the strong-coupling strength. We present an interpretation of the mean-field diffusive coupling in the context of ecology and identify that, with increasing mean-field density, an open ecosystem transforms into a closed ecosystem. We report on the occurrence of OD in an ecological model and explain its significance. Using a detailed bifurcation analysis we show that, depending on the mortality rate and carrying capacity, the system shows either AD or both AD and OD. We also show that the results remain qualitatively the same for a network of oscillators. We identify a new transition scenario between the same type of oscillation suppression states whose geneses differ. In the parameter-mismatched case, we further report on the direct transition from OD to AD through a transcritical bifurcation. We believe that this study will lead to a proper interpretation of AD and OD in ecology, which may be important for the conservation and management of several communities in ecosystems.

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.

Experimental observation of a transition from amplitude to oscillation death in coupled oscillators

We report the experimental evidence of an important transition scenario, namely the transition from amplitude death (AD) to oscillation death (OD) state in coupled limit cycle oscillators. We consider two Van der Pol oscillators coupled through mean-field diffusion and show that this system exhibits a transition from AD to OD, which was earlier shown for Stuart-Landau oscillators under the same coupling scheme [T. Banerjee and D. Ghosh, Phys. Rev. E 89, 052912 (2014)]. We show that the AD-OD transition is governed by the density of mean-field and beyond a critical value this transition is destroyed; further, we show the existence of a nontrivial AD state that coexists with OD. Next, we implement the system in an electronic circuit and experimentally confirm the transition from AD to OD state. We further characterize the experimental parameter zone where this transition occurs. The present study may stimulate the search for the practical systems where this important transition scenario can be observed experimentally.

Transition from amplitude to oscillation death under mean-field diffusive coupling

We study the transition from the amplitude death (AD) to the oscillation death (OD) state in limit-cycle oscillators coupled through mean-field diffusion. We show that this coupling scheme can induce an important transition from AD to OD even in identical limit cycle oscillators. We identify a parameter region where OD and a nontrivial AD (NTAD) state coexist. This NTAD state is unique in comparison with AD owing to the fact that it is created by a subcritical pitchfork bifurcation and parameter mismatch does not support this state, but destroys it. We extend our study to a network of mean-field coupled oscillators to show that the transition scenario is preserved and the oscillators form a two-cluster state.

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.

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.