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

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.

Collective behaviors of mean-field coupled Stuart-Landau limit-cycle oscillators under additional repulsive links

We study the collective behaviors of a large population of Stuart-Landau limit-cycle oscillators that coupled diffusively and equally with all of the others via the conjugate of the mean field, where the underlying interaction is shown to break the rotational symmetry of the coupled system. In the model, an ensemble of Stuart-Landau oscillators are in fact diffusively coupled via the mean field in the real parts, whereas additional repulsive links are present in the imaginary parts. All the oscillators are linked via the similar state variables, which distinctly differs from the conjugate coupling through dissimilar variables in the previous studies. We show that depending on the strength of coupling and the distribution of natural frequencies, the coupled system exhibits three qualitatively different types of collective stationary behaviors: amplitude death (AD), oscillation death (OD), and incoherent state. Our goal is to analytically characterize the onset of the above three typical macrostates by performing the rigorous linear stability analyses of the corresponding mean-field coupled system. We prove that AD is able to occur in the coupled system with identical frequencies, where the stable coupling interval of AD is found to be independent on the system's size N. When the natural frequencies are distributed according to a general density function, we obtain the analytic equations that govern the exact stability boundaries of AD, OD, and the incoherence for a coupled system in the thermodynamic limit N→∞. All the theoretical predictions are well confirmed via numerical simulations of the coupled system with a specific Lorentzian frequency distribution.

Repulsive inter-layer coupling induces anti-phase synchronization

We present numerical results for the synchronization phenomena in a bilayer network of repulsively coupled 2D lattices of van der Pol oscillators. We consider the cases when the network layers have either different or the same types of intra-layer coupling topology. When the layers are uncoupled, the lattice of van der Pol oscillators with a repulsive interaction typically demonstrates a labyrinth-like pattern, while the lattice with attractively coupled van der Pol oscillators shows a regular spiral wave structure. We reveal for the first time that repulsive inter-layer coupling leads to anti-phase synchronization of spatiotemporal structures for all considered combinations of intra-layer coupling. As a synchronization measure, we use the correlation coefficient between the symmetrical pairs of network nodes, which is always close to -1 in the case of anti-phase synchronization. We also study how the form of synchronous structures depends on the intra-layer coupling strengths when the repulsive inter-layer coupling is varied.

Spatiotemporal patterns in a 2D lattice with linear repulsive and nonlinear attractive coupling

We explore the emergence of a variety of different spatiotemporal patterns in a 2D lattice of self-sustained oscillators, which interact nonlocally through an active nonlinear element. A basic element is a van der Pol oscillator in a regime of relaxation oscillations. The active nonlinear coupling can be implemented by a radiophysical element with negative resistance in its current-voltage curve taking into account nonlinear characteristics (for example, a tunnel diode). We show that such coupling consists of two parts, namely, a repulsive linear term and an attractive nonlinear term. This interaction leads to the emergence of only standing waves with periodic dynamics in time and absence of any propagating wave processes. At the same time, many different spatiotemporal patterns occur when the coupling parameters are varied, namely, regular and complex cluster structures, such as chimera states. This effect is associated with the appearance of new periodic states of individual oscillators by the repulsive part of coupling, while the attractive term attenuates this effect. We also show influence of the coupling nonlinearity on the spatiotemporal dynamics.

Antiphase synchronization in multiplex networks with attractive and repulsive interactions

A series of recent publications, within the framework of network science, have focused on the coexistence of mixed attractive and repulsive (excitatory and inhibitory) interactions among the units within the same system, motivated by the analogies with spin glasses as well as to neural networks, or ecological systems. However, most of these investigations have been restricted to single layer networks, requiring further analysis of the complex dynamics and particular equilibrium states that emerge in multilayer configurations. This article investigates the synchronization properties of dynamical systems connected through multiplex architectures in the presence of attractive intralayer and repulsive interlayer connections. This setting enables the emergence of antisynchronization, i.e., intralayer synchronization coexisting with antiphase dynamics between coupled systems of different layers. We demonstrate the existence of a transition from interlayer antisynchronization to antiphase synchrony in any connected bipartite multiplex architecture when the repulsive coupling is introduced through any spanning tree of a single layer. We identify, analytically, the required graph topologies for interlayer antisynchronization and its interplay with intralayer and antiphase synchronization. Next, we analytically derive the invariance of intralayer synchronization manifold and calculate the attractor size of each oscillator exhibiting interlayer antisynchronization together with intralayer synchronization. The necessary conditions for the existence of interlayer antisynchronization along with intralayer synchronization are given and numerically validated by considering Stuart-Landau oscillators. Finally, we also analytically derive the local stability condition of the interlayer antisynchronization state using the master stability function approach.

Synchronization of wave structures in a heterogeneous multiplex network of 2D lattices with attractive and repulsive intra-layer coupling

We explore numerically the synchronization effects in a heterogeneous two-layer network of two-dimensional (2D) lattices of van der Pol oscillators. The inter-layer coupling of the multiplex network has an attractive character. One layer of 2D lattices is characterized by attractive coupling of oscillators and demonstrates a spiral wave regime for both local and nonlocal interactions. The oscillators in the second layer are coupled through active elements and the interaction between them has repulsive character. We show that the lattice with the repulsive type of coupling demonstrates complex spatiotemporal cluster structures, which can be called labyrinth-like structures. We show for the first time that this multiplex network with fundamentally various types of intra-layer coupling demonstrates mutual synchronization and a competition between two types of structures. Our numerical study indicates that the synchronization threshold and the type of spatiotemporal patterns in both layers strongly depend on the ratio of the intra-layer coupling strength of the two lattices. We also analyze the impact of intra-layer coupling ranges on the synchronization effects.

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.

Infection spreading and recovery in a square lattice

We investigate spreading and recovery of disease in a square lattice, and, in particular, emphasize the role of the initial distribution of infected patches in the network on the progression of an endemic and initiation of a recovery process, if any, due to migration of both the susceptible and infected hosts. The disease starts in the lattice with three possible initial distribution patterns of infected and infection-free sites, viz., infected core patches (ICP), infected peripheral patches (IPP), and randomly distributed infected patches (RDIP). Our results show that infection spreads monotonically in the lattice with increasing migration without showing any sign of recovery in the ICP case. In the IPP case, it follows a similar monotonic progression with increasing migration; however, a self-organized healing process starts for higher migration, leading the lattice to full recovery at a critical rate of migration. Encouragingly, for the initial RDIP arrangement, chances of recovery are much higher with a lower rate of critical migration. An eigenvalue-based semianalytical study is made to determine the critical migration rate for realizing a stable infection-free lattice. The initial fraction of infected patches and the force of infection play significant roles in the self-organized recovery. They follow an exponential law, for the RDIP case, that governs the recovery process. For the frustrating case of ICP arrangement, we propose a random rewiring of links in the lattice allowing long-distance migratory paths that effectively initiate a recovery process. Global prevalence of infection thereby declines and progressively improves with the rewiring probability that follows a power law with the critical migration and leads to the birth of emergent infection-free networks.

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

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

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.

Effect of repulsive links on frustration in attractively coupled networks

We investigate the impact of attractive-repulsive interaction in networks of limit cycle oscillators. Mainly we focus on the design principle for generating an antiphase state between adjacent nodes in a complex network. We establish that a partial negative control throughout the branches of a spanning tree inside the positively coupled limit cycle oscillators works efficiently well in comparison with randomly chosen negative links to establish zero frustration (antiphase synchronization) in bipartite graphs. Based on the emergence of zero frustration, we develop a universal 0-π rule to understand the antiphase synchronization in a bipartite graph. Further, this rule is used to construct a nonbipartite graph for a given nonzero frustrated value. We finally show the generality of 0-π rule by implementing it in arbitrary undirected nonbipartite graphs of attractive-repulsively coupled limit cycle oscillators and successfully calculate the nonzero frustration value, which matches with numerical data. The validation of the rule is checked through the bifurcation analysis of small networks. Our work may unveil the underlying mechanism of several synchronization phenomena that exist in a network of oscillators having a mixed type of coupling.

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.

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.

Asymmetry in initial cluster size favors symmetry in a network of oscillators

Counterintuitive to the common notion of symmetry breaking, asymmetry favors synchrony in a network of oscillators. Our observations on an ensemble of identical Stuart-Landau systems under a symmetry breaking coupling support our conjecture. As usual, for a complete deterministic and the symmetric choice of initial clusters, a variety of asymptotic states, namely, multicluster oscillation death (1-OD, 3-OD, and m -OD), chimera states, and traveling waves emerge. Alternatively, multiple chimera death (1-CD, 3-CD, and m -CD) and completely synchronous states emerge in the network whenever some randomness is added to the symmetric initial states. However, in both the cases, an increasing asymmetry in the initial cluster size restores symmetry in the network, leading to the most favorable complete synchronization state for a broad range of coupling parameters. We are able to reduce the network model using the mean-field approximation that reproduces the dynamical features of the original network.

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.

Multicluster oscillation death and chimeralike states in globally coupled Josephson Junctions

We observe the multiclustered oscillation death and chimeralike states in an array of Josephson junctions under a combination of self-repulsive and cross-attractive mean-field interaction when each isolated junction is in a bistable state, a coexisting fixed point and an oscillatory state. We locate the parameter landscape of the multiclustered oscillation death and chimeralike states. Alternatively, a purely repulsive mean-field interaction in an array of all oscillatory junctions produces chimeralike states with signatures of metastability in the incoherent subpopulation of junctions.

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.

Experimental demonstration of revival of oscillations from death in coupled nonlinear oscillators

We experimentally demonstrate that a processing delay, a finite response time, in the coupling can revoke the stability of the stable steady states, thereby facilitating the revival of oscillations in the same parameter space where the coupled oscillators suffered the quenching of oscillation. This phenomenon of reviving of oscillations is demonstrated using two different prototype electronic circuits. Further, the analytical critical curves corroborate that the spread of the parameter space with stable steady state is diminished continuously by increasing the processing delay. Finally, the death state is completely wiped off above a threshold value by switching the stability of the stable steady state to retrieve sustained oscillations in the same parameter space. The underlying dynamical mechanism responsible for the decrease in the spread of the stable steady states and the eventual reviving of oscillation as a function of the processing delay is explained using analytical results.

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.

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.

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.

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.

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.

Emergence of amplitude and oscillation death in identical coupled oscillators

We deduce rigorous conditions for the onset of amplitude death (AD) and oscillation death (OD) in a system of identical coupled paradigmatic Stuart-Landau oscillators. A nonscalar coupling and high frequency are beneficial for the onset of AD. In strong contrast, scalar diffusive coupling and low intrinsic frequency are in favor of the emergence of OD. Our finding contributes to clearly distinguish intrinsic geneses for AD and OD, and further substantially corroborates that AD and OD are indeed two dynamically distinct oscillation quenching phenomena due to distinctly different mechanisms.

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.