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

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.

Quantum Turing bifurcation: Transition from quantum amplitude death to quantum oscillation death

An important transition from a homogeneous steady state to an inhomogeneous steady state via the Turing bifurcation in coupled oscillators was reported recently [Phys. Rev. Lett. 111, 024103 (2013)PRLTAO0031-900710.1103/PhysRevLett.111.024103]. However, the same in the quantum domain is yet to be observed. In this paper, we discover the quantum analog of the Turing bifurcation in coupled quantum oscillators. We show that a homogeneous steady state is transformed into an inhomogeneous steady state through this bifurcation in coupled quantum van der Pol oscillators. We demonstrate our results by a direct simulation of the quantum master equation in the Lindblad form. We further support our observations through an analytical treatment of the noisy classical model. Our study explores the paradigmatic Turing bifurcation at the quantum-classical interface and opens up the door toward its broader understanding.

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.

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.

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.

Amplitude suppression of oscillators with delay connections and slow switching topology

The present paper shows that the amplitudes of oscillators in delay-coupled oscillator networks can be suppressed by switching the network topology at a rate much lower than the oscillator frequencies. The mechanism of suppression was clarified numerically, and a procedure for determining the connection parameters to induce suppression is presented. The analytical and numerical results were obtained with Stuart-Landau oscillators and were experimentally validated using double-scroll chaotic circuits.

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.

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.

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.

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.

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.

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.

Synchronization of cells with activator-inhibitor pathways through adaptive environment-mediated coupling

In this paper, we report the synchronized dynamics of cells with activator-inhibitor pathways via an adaptive environment-mediated coupling scheme with feedbacks and control mechanisms. The adaptive character of the extracellular medium is modeled via its damping parameter as a physiological response aiming at annihilating the cellular differentiation existing between the chaotic biochemical pathways of the cells, in order to preserve homeostasis. We perform an investigation on the existence and stability of the synchronization manifold of the coupled system under the proposed coupling pattern. Both mathematical and computational tools suggest the accessibility of conducive prerequisites (conditions) for the emergence of a robust synchronous regime. The relevance of a phase-synchronized dynamics is appraised and several numerical indicators advocate for the prevalence of this fascinating phenomenon among the interacting cells in the phase space.

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.

Revival of oscillation from mean-field-induced death: Theory and experiment

The revival of oscillation and maintaining rhythmicity in a network of coupled oscillators offer an open challenge to researchers as the cessation of oscillation often leads to a fatal system degradation and an irrecoverable malfunctioning in many physical, biological, and physiological systems. Recently a general technique of restoration of rhythmicity in diffusively coupled networks of nonlinear oscillators has been proposed in Zou et al. [Nat. Commun. 6, 7709 (2015)], where it is shown that a proper feedback parameter that controls the rate of diffusion can effectively revive oscillation from an oscillation suppressed state. In this paper we show that the mean-field diffusive coupling, which can suppress oscillation even in a network of identical oscillators, can be modified in order to revoke the cessation of oscillation induced by it. Using a rigorous bifurcation analysis we show that, unlike other diffusive coupling schemes, here one has two control parameters, namely the density of the mean-field and the feedback parameter that can be controlled to revive oscillation from a death state. We demonstrate that an appropriate choice of density of the mean field is capable of inducing rhythmicity even in the presence of complete diffusion, which is a unique feature of this mean-field coupling that is not available in other coupling schemes. Finally, we report the experimental observation of revival of oscillation from the mean-field-induced oscillation suppression state that supports our theoretical results.

Spatial coexistence of synchronized oscillation and death: A chimeralike state

We report an interesting spatiotemporal state, namely the chimeralike incongruous coexistence of synchronized oscillation and stable steady state (CSOD) in a network of nonlocally coupled oscillators. Unlike the chimera and chimera death state, in the CSOD state identical oscillators are self-organized into two coexisting spatially separated domains: In one domain neighboring oscillators show synchronized oscillation and in another domain the neighboring oscillators randomly populate either a synchronized oscillating state or a stable steady state (we call it a death state). We consider a realistic ecological network and show that the interplay of nonlocality and coupling strength results in two routes to the CSOD state: One is from a coexisting mixed state of amplitude chimera and death, and another one is from a globally synchronized state. We provide a qualitative explanation of the origin of this state. We further explore the importance of this study in ecology that gives insight into the relationship between spatial synchrony and global extinction of species. We believe this study will improve our understanding of chimera and chimeralike states.

Dispersal-induced synchrony, temporal stability, and clustering in a mean-field coupled Rosenzweig-MacArthur model

In spatial ecology, dispersal among a set of spatially separated habitats, named as metapopulation, preserves the diversity and persistence by interconnecting the local populations. Understanding the effects of several variants of dispersion in metapopulation dynamics and to identify the factors which promote population synchrony and population stability are important in ecology. In this paper, we consider the mean-field dispersion among the habitats in a network and study the collective dynamics of the spatially extended system. Using the Rosenzweig-MacArthur model for individual patches, we show that the population synchrony and temporal stability, which are believed to be of conflicting outcomes of dispersion, can be simultaneously achieved by oscillation quenching mechanisms. Particularly, we explore the more natural coupling configuration where the rates of dispersal of different habitats are disparate. We show that asymmetry in dispersal rate plays a crucial role in determining inhomogeneity in an otherwise homogeneous metapopulation. We further identify an unusual emergent state in the network, namely, a multi-branch clustered inhomogeneous steady state, which arises due to the intrinsic parameter mismatch among the patches. We believe that the present study will shed light on the cooperative behavior of spatially structured ecosystems.

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.