Experimental evidence of time-delay-induced death in coupled limit-cycle oscillators

Analysis of bifurcation and explosive amplitude death in a pair of oscillators coupled via time-delay connection

Delay-induced amplitude death (AD) has received considerable research interest. Most studies on delay-induced AD investigated the local stability of equilibrium points. The present study examines the global dynamics of delay-induced AD in a pair of identical Stuart-Landau oscillators. Bifurcation diagrams consisting of synchronized periodic orbits and an equilibrium point are used to determine the mechanism of the emergence of delay-induced AD. It is shown that explosive delay-induced AD can occur via a Hopf bifurcation at the equilibrium point and a saddle-node bifurcation of synchronized periodic orbits when the delay time for the connection is continuously varied. The Hopf and saddle-node bifurcation curves in the coupling parameter space clarify the dependence of the coupling parameters on the global dynamics.

Synchronization of two coupled massive oscillators in the time-delayed Kuramoto model

We examine the impact of the time delay on two coupled massive oscillators within the second-order Kuramoto model, which is relevant to the operations of real-world networks that rely on signal transmission speed constraints. Our analytical and numerical exploration shows that time delay can cause multi-stability within phase-locked solutions, and the stability of these solutions decreases as the inertia increases. In addition to phase-locked solutions, we discovered non-phase-locked solutions that exhibit periodic and chaotic behaviors, depending on the amount of inertia and time delay.

Robust design against frequency variation for amplitude death in delay-coupled oscillators

Amplitude death has the potential to suppress unwanted oscillations in various engineering applications. However, in some engineering applications, such as dc microgrids, airfoil systems, and thermoacoustic systems, oscillation frequency is highly susceptible to external influences, leading to considerable variations. To maintain amplitude death amidst these frequency variations, we propose a design procedure that is robust against frequency variation for inducing amplitude death in delay-coupled oscillators. We first analytically derive the oscillator frequency band in which amplitude death can occur. The frequency bandwidth is maximized when the coupling strength is inversely proportional to the connection delay. Furthermore, our analysis reveals that the oscillator frequency band is influenced by the minimum eigenvalue of the normalized adjacency matrix (i.e., network topology) and that bipartite networks exhibit limited robustness to frequency variations. Our design procedure maintains the stability of amplitude death even under substantial frequency variations and is applicable to various network topologies. Numerical simulations confirm the validity of the proposed design.

Delay-induced amplitude death in multiplex oscillator network with frequency-mismatched layers

The present paper analytically investigates the stability of amplitude death in a multiplex Stuart-Landau oscillator network with a delayed interlayer connection. The network consists of two frequency-mismatched layers, and all oscillators in each layer have identical frequencies. We show that, if the matrices describing the network topologies of each layer commute, then the characteristic equation governing the stability can be reduced to a simple form. This form reveals that the stability of amplitude death in the multiplex network is equally or more conservative than that in a pair of frequency-mismatched oscillators coupled by a delayed connection. In addition, we provide a procedure for designing the delayed interlayer connection such that amplitude death is stable for any commuting matrices and for any intralayer coupling strength. These analytical results are verified through numerical examples. Moreover, we numerically discuss the results for the case in which the commutative property does not hold.

Quenching of oscillations via attenuated coupling for dissimilar electrochemical systems

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

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

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

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.

Quenching and amplification of thermoacoustic oscillations in two nonidentical Rijke tubes interacting via time-delay and dissipative coupling

We numerically explore the quenching and amplification of self-excited thermoacoustic oscillations in two nonidentical Rijke tubes interacting via time-delay and dissipative coupling. On applying either type of coupling separately, we find that the presence of nonidentical heater powers can shrink the regions of amplitude death in both oscillators, while producing new regions of amplitude amplification in the weaker oscillator. We find that the magnitude of amplitude amplification grows with the heater power mismatch and with the total power input. These effects are also present when both types of coupling are applied simultaneously. This study highlights the critical role that nonidentical thermal loads can play in determining the amplitude response of coupled thermoacoustic systems, facilitating the design of control strategies for coupled oscillatorlike devices such as gas turbines.

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.

Effects of frequency mismatch on amplitude death in delay-coupled oscillators

The present paper analytically reveals the effects of frequency mismatch on the stability of an equilibrium point within a pair of Stuart-Landau oscillators coupled by a delay connection. By analyzing the roots of the characteristic function governing the stability, we find that there exist four types of boundary curves of stability in a coupling parameters space. These four types depend only on the frequency mismatch. The analytical results allow us to design coupling parameters and frequency mismatch such that the equilibrium point is locally stable. We show that, if we choose appropriate frequency mismatches and delay times, then it is possible to induce amplitude death with strong stability, even by weak coupling. In addition, we show that parts of these analytical results are valid for oscillator networks with complete bipartite topologies.

Low-order modeling of the mutual synchronization between two turbulent thermoacoustic oscillators

We use a low-order oscillator model to investigate the mutual synchronization of a thermoacoustic system consisting of two turbulent lean-premixed combustors coupled via a cross-talk tube. The model consists of two Van der Pol (VDP) oscillators coupled via dissipative and time-delay terms. We show that, despite its simplicity, the model can reproduce many of the synchronization phenomena observed experimentally, such as amplitude death, desynchronization (quasiperiodicity), synchronization (phase locking), and nonlinear energy pumping from a limit-cycle mode to a damped mode. This study shows that the mutual synchronization dynamics of a turbulent thermoacoustic system can be reproduced with just a simple coupled VDP model. This suggests that such a model could be used to identify new strategies for quenching limit-cycle oscillations in turbulent thermoacoustic systems, such as gas turbines and rocket engines.

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.

Stochastic resonance in deformable potential with time-delayed feedback

This paper reports the stochastic resonance (SR) phenomenon with memory effects for a Brownian particle in a potential whose shape is subjected to deformation. We model the deformation in the system by the Remoissenet-Peyrard potential and the memory effects by the time-delayed feedback. The question of the possible influence of time-delayed feedback on the occurrence of SR is then of our interest. We examine numerically the effect of feedback strength as well as time delay on SR phenomenon in terms of hysteresis loop area. It is found that time-delayed feedback has a significant effect on SR and can induce double resonances in the system. We show that the properties of SR are varying, depending on interdependence between feedback strength, time delay and shape parameter. This article is part of the theme issue 'Vibrational and stochastic resonance in driven nonlinear systems (part 1)'.

Detecting coexisting oscillatory patterns in delay coupled Lur'e systems

This work addresses the problem of pattern analysis in networks consisting of delay-coupled identical Lur'e systems. We study a class of nonlinear systems, which, being isolated, are globally asymptotically stable. Assembling such systems into a network via time-delayed coupling may result in the change of network equilibrium stability under parameter variation in the coupling. In this work, we focus on cases where a Hopf bifurcation causes the change of stability of the network equilibrium and leads to the occurrence of oscillatory modes (patterns). Moreover, some of these patterns can co-exist for the same set of coupling parameters, which makes the analysis by means of common methods, such as the Lyapunov-Krasovskii method or the analysis of Poincaré maps, cumbersome. A numerically efficient algorithm, aiming at the computation of the oscillatory patterns occurring in such networks, is presented. Moreover, we show that our approach is able to deal with co-existing patterns, and both stable and unstable regimes can be simultaneously computed, which gives deep insight into the network dynamics. In order to illustrate the efficiency of the method, we present two examples in which the instability of the network equilibria is caused by a subcritical and a supercritical Hopf bifurcation. In addition, a bifurcation analysis of the subcritical case is performed in order to further explain the occurrence of the detected coexisting modes.

Oscillations of Delta-like1 regulate the balance between differentiation and maintenance of muscle stem cells

Cell-cell interactions mediated by Notch are critical for the maintenance of skeletal muscle stem cells. However, dynamics, cellular source and identity of functional Notch ligands during expansion of the stem cell pool in muscle growth and regeneration remain poorly characterized. Here we demonstrate that oscillating Delta-like 1 (Dll1) produced by myogenic cells is an indispensable Notch ligand for self-renewal of muscle stem cells in mice. Dll1 expression is controlled by the Notch target Hes1 and the muscle regulatory factor MyoD. Consistent with our mathematical model, our experimental analyses show that Hes1 acts as the oscillatory pacemaker, whereas MyoD regulates robust Dll1 expression. Interfering with Dll1 oscillations without changing its overall expression level impairs self-renewal, resulting in premature differentiation of muscle stem cells during muscle growth and regeneration. We conclude that the oscillatory Dll1 input into Notch signaling ensures the equilibrium between self-renewal and differentiation in myogenic cell communities.

Oscillation behavior driven by processing delay in diffusively coupled inactive systems: Cluster synchronization and multistability

Couplings involving time delay play a relevant role in the dynamical behavior of complex systems. In this work, we address the effect of processing delay, which is a specific kind of coupling delay, on the steady state of general nonlinear systems and prove that it may drive the system to Hopf bifurcation and, in turn, to a rich oscillatory behavior. Additionally, one may observe multistable states and size-dependent cluster synchronization. We derive the analytic conditions to obtain an oscillatory regime and confirm the result by numerically simulated experiments on different oscillator networks. Our results demonstrate the importance of processing delay for complex systems and pave the way for a better understanding of dynamical control and synchronization in oscillatory networks.

Traveling amplitude death in coupled pendula

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

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

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

Effect of filtered feedback on birhythmicity: Suppression of birhythmic oscillation

The birhythmic oscillation, generally known as birhythmicity, arises in a plethora of physical, chemical, and biological systems. In this paper we investigate the effect of filtered feedback on birhythmicity as both are relevant in many living and engineering systems. We show that the presence of a low-pass filter in the feedback path of a birhythmic system suppresses birhythmicity and supports monorhythmic oscillations depending on the filtering parameter. Using harmonic decomposition and energy balance methods we determine the conditions for which birhythmicity is removed. We carry out a detailed bifurcation analysis to unveil the mechanism behind the quenching of birhythmic oscillations. Finally, we demonstrate our theoretical findings in analog simulation with electronic circuit. This study may have practical applications in quenching birhythmicity in several biochemical and physical systems.

Collective dynamics of globally delay-coupled complex Ginzburg-Landau oscillators

The effect of time-delayed coupling on the collective behavior of a population of globally coupled complex Ginzburg-Landau oscillators is investigated. A detailed numerical study is carried out to study the impact of time delay on various collective states that include synchronous states, multicluster states, chaos, amplitude-mediated chimeras, and incoherent states. It is found that time delay can bring about significant changes in the dynamical properties of these states including their regions of existence and stability. In general, an increase in time delay is seen to lower the threshold value of the coupling strength for the occurrence of such states and to shift the existence domain toward more negative values of the linear dispersion parameter. Further insights into the numerical findings are provided, wherever possible, by exact equilibrium and stability analysis of these states in the presence of time delay.

Frequency and Phase Characteristics of Candle Flame Oscillation

The combustion of candles exhibits a variety of dynamical behaviors. Binding several candles together will result in flickering of candle flames, which is generally described as a nonlinear oscillator. The impact on the frequency of the flame by several factors, such as the arrangement, the number and the asymmetry of the oscillators, is discussed. Experimental results show that the frequency gradually decreases as the number of candles increases in the case of an isolated oscillator, while alternation between the in-phase and the anti-phase synchronization appears in a coupled system of two oscillators. Moreover, envelopes in the amplitude of the oscillatory luminance are displayed when candles are coupled asymmetrically. Since the coupling between oscillators is dominated by thermal radiation, a “overlapped peaks model” is proposed to phenomenologically explain the relationship between temperature distribution, coupling strength and the collective behavior in coupled system of candle oscillators in both symmetric and asymmetric cases.

Targeted synchronization in an externally driven population of mechanical oscillators

We experimentally investigate the synchronization of driven metronomes using a servo motor to impose external control. We show that a driven metronome will only synchronize in a narrow range near its own frequency; when we introduce coupling between metronomes, we can widen the range of frequencies over which a metronome will synchronize to the external input. Using these features, we design a signal to synchronize a population of dissimilar metronomes; separately we design a signal to selectively synchronize a subpopulation of metronomes within a heterogeneous population.

Experimental Evidence of Amplitude Death and Phase-Flip Bifurcation between In-Phase and Anti-Phase Synchronization

Nonlinear phenomena emerging from the coupled behaviour of a pair of oscillators have attracted considerable research attention over the years, of which, amplitude death (AD) and phase-flip bifurcation (PFB) are two noteworthy examples. Although theoretical research has postulated the coexistence of AD and PFB upon variation of different control parameters, such an occurrence has not been reported in practical systems. Here, we provide the first experimental evidence of the coexistence of AD and PFB in a physical system, comprising of a coupled pair of candle-flame oscillators. As the strength of coupling between the oscillators is increased, we report a decrease in the span of AD region between the states of in-phase and anti-phase oscillations, leading up to a point of PFB. Understanding such a switching of phenomena between AD and PFB helps us to evade their undesirable occurrences such as AD in neuron and brain cells, oscillatory state in prey-predator systems, oscillatory spread of epidemics and so forth.

Synchronization in networks with heterogeneous coupling delays

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

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.

Effect of time-delay and dissipative coupling on amplitude death in coupled thermoacoustic oscillators

We here systematically investigate amplitude death (AD) phenomenon in a thermoacoustic system using a mathematical model of coupled prototypical thermoacoustic oscillators, the horizontal Rijke tubes. AD has recently been identified as a relatively simple phenomenon, which can be exploited to stop the unwanted high amplitude pressure oscillations resulting from the occurrence of thermoacoustic instability. We examine the effect of time-delay and dissipative couplings on a system of two Rijke tubes when they are symmetrically and asymmetrically coupled. The regions where appropriate combinations of delay time, detuning, and the strengths of time-delay and dissipative coupling lead to AD are identified. The relative ease of attaining AD when both the couplings are applied simultaneously is inferred from the model. In the presence of strong enough coupling, AD is observed even when the oscillators of dissimilar amplitudes are coupled, while a significant reduction in the amplitudes of both the oscillators is observed when the coupling strength is not enough to attain AD.

Anticipated and zero-lag synchronization in motifs of delay-coupled systems

Anticipated and zero-lag synchronization have been observed in different scientific fields. In the brain, they might play a fundamental role in information processing, temporal coding and spatial attention. Recent numerical work on anticipated and zero-lag synchronization studied the role of delays. However, an analytical understanding of the conditions for these phenomena remains elusive. In this paper, we study both phenomena in systems with small delays. By performing a phase reduction and studying phase locked solutions, we uncover the functional relation between the delay, excitation and inhibition for the onset of anticipated synchronization in a sender-receiver-interneuron motif. In the case of zero-lag synchronization in a chain motif, we determine the stability conditions. These analytical solutions provide an excellent prediction of the phase-locked regimes of Hodgkin-Huxley models and Roessler oscillators.

Aging transition in systems of oscillators with global distributed-delay coupling

We consider a globally coupled network of active (oscillatory) and inactive (nonoscillatory) oscillators with distributed-delay coupling. Conditions for aging transition, associated with suppression of oscillations, are derived for uniform and gamma delay distributions in terms of coupling parameters and the proportion of inactive oscillators. The results suggest that for the uniform distribution increasing the width of distribution for the same mean delay allows aging transition to happen for a smaller coupling strength and a smaller proportion of inactive elements. For gamma distribution with sufficiently large mean time delay, it may be possible to achieve aging transition for an arbitrary proportion of inactive oscillators, as long as the coupling strength lies in a certain range.

Amplitude death in a ring of nonidentical nonlinear oscillators with unidirectional coupling

We study the collective behaviors in a ring of coupled nonidentical nonlinear oscillators with unidirectional coupling, of which natural frequencies are distributed in a random way. We find the amplitude death phenomena in the case of unidirectional couplings and discuss the differences between the cases of bidirectional and unidirectional couplings. There are three main differences; there exists neither partial amplitude death nor local clustering behavior but an oblique line structure which represents directional signal flow on the spatio-temporal patterns in the unidirectional coupling case. The unidirectional coupling has the advantage of easily obtaining global amplitude death in a ring of coupled oscillators with randomly distributed natural frequency. Finally, we explain the results using the eigenvalue analysis of the Jacobian matrix at the origin and also discuss the transition of dynamical behavior coming from connection structure as the coupling strength increases.

Optogenetic perturbation and bioluminescence imaging to analyze cell-to-cell transfer of oscillatory information

Isomura et al. developed an integrated approach that combines optogenetic perturbations and single-cell bioluminescence imaging to visualize and reconstitute synchronized oscillatory gene expression in signal-sending and signal-receiving processes.

Cells communicate with each other to coordinate their gene activities at the population level through signaling pathways. It has been shown that many gene activities are oscillatory and that the frequency and phase of oscillatory gene expression encode various types of information. However, whether or how such oscillatory information is transmitted from cell to cell remains unknown. Here, we developed an integrated approach that combines optogenetic perturbations and single-cell bioluminescence imaging to visualize and reconstitute synchronized oscillatory gene expression in signal-sending and signal-receiving processes. We found that intracellular and intercellular periodic inputs of Notch signaling entrain intrinsic oscillations by frequency tuning and phase shifting at the single-cell level. In this way, the oscillation dynamics are transmitted through Notch signaling, thereby synchronizing the population of oscillators. Thus, this approach enabled us to control and monitor dynamic cell-to-cell transfer of oscillatory information to coordinate gene expression patterns at the population level.

Experimental observation of phase-flip transitions in two inductively coupled glow discharge plasmas

We report an experimental observation of a phase-flip transition in the frequency synchronization of two dc glow discharge plasma sources that are coupled in a noninvasive fashion. When the fundamental oscillation frequency of the potential fluctuations of one of the sources is progressively increased, by raising its discharge voltage, a frequency pulling regime is observed, followed by a synchronized regime that shows a frequency jump phenomenon. The jump is associated with a phase-flip transition that takes the synchronized state from an in-phase to an antiphase state. When the process is reversed, the transition takes place at a different frequency, thereby exhibiting a hysteresis effect. A heuristic model, consisting of two van der Pol oscillators that are coupled to each other through a dynamic common medium, eminently captures the essential features of our experimental observations.

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.

Experimental and modeling analysis of asymmetrical on-off oscillation in coupled non-identical inverted bottle oscillators

Upside-down bottles containing water which are common in our daily life exhibit rich vibration dynamics. Rich dynamic regimes are observed in bottle oscillators by directly measuring the pressure difference between inside and outside of a bottle with the aid of pressure sensors. We observe experimentally that an asymmetrical oscillation process between the outflow of water and the inflow of air is formed in a single bottle oscillator and, in addition, a kind of 2:1 frequency synchronization occurs in a coupled system of two non-identical bottle oscillators. The peak values of the oscillation of pressure differences between inside and outside of the bottle decease as the height of the liquid surface steps down, while the oscillation period increases gradually. The theoretical model of the oscillator is amended to understand the regimes in the experiment by introducing time-dependent parameters related to the asymmetrical oscillation processes. Our numerical results based on the model fit well with the experimental ones.

Nonlinear behavior of the tarka flute's distinctive sounds

The Andean tarka flute generates multiphonic sounds. Using spectral techniques, we verify two distinctive musical behaviors and the nonlinear nature of the tarka. Through nonlinear time series analysis, we determine chaotic and hyperchaotic behavior. Experimentally, we observe that by increasing the blow pressure on different fingerings, peculiar changes from linear to nonlinear patterns are produced, leading ultimately to quenching.

Restoring oscillatory behavior from amplitude death with anti-phase synchronization patterns in networks of electrochemical oscillations

The dynamical behavior of delay-coupled networks of electrochemical reactions is investigated to explore the formation of amplitude death (AD) and the synchronization states in a parameter region around the amplitude death region. It is shown that difference coupling with odd and even numbered ring and random networks can produce the AD phenomenon. Furthermore, this AD can be restored by changing the coupling type from difference to direct coupling. The restored oscillations tend to create synchronization patterns in which neighboring elements are in nearly anti-phase configuration. The ring networks produce frozen and rotating phase waves, while the random network exhibits a complex synchronization pattern with interwoven frozen and propagating phase waves. The experimental results are interpreted with a coupled Stuart-Landau oscillator model. The experimental and theoretical results reveal that AD behavior is a robust feature of delayed coupled networks of chemical units; if an oscillatory behavior is required again, even a small amount of direct coupling could be sufficient to restore the oscillations. The restored nearly anti-phase oscillatory patterns, which, to a certain extent, reflect the symmetry of the network, represent an effective means to overcome the AD phenomenon.

Oscillatory control of Delta-like1 in cell interactions regulates dynamic gene expression and tissue morphogenesis

Shimojo et al. developed a live-imaging system and found that Notch ligand Delta-like1 (Dll1) protein expression oscillates in neural progenitors and presomitic mesoderm cells, and this regulates dynamic gene expression and tissue morphogenesis.

Notch signaling regulates tissue morphogenesis through cell–cell interactions. The Notch effectors Hes1 and Hes7 are expressed in an oscillatory manner and regulate developmental processes such as neurogenesis and somitogenesis, respectively. Expression of the mRNA for the mouse Notch ligand Delta-like1 (Dll1) is also oscillatory. However, the dynamics of Dll1 protein expression are controversial, and their functional significance is unknown. Here, we developed a live-imaging system and found that Dll1 protein expression oscillated in neural progenitors and presomitic mesoderm cells. Notably, when Dll1 expression was accelerated or delayed by shortening or elongating the Dll1 gene, Dll1 oscillations became severely dampened or quenched at intermediate levels, as modeled mathematically. Under this condition, Hes1 and Hes7 oscillations were also dampened. In the presomitic mesoderm, steady Dll1 expression led to severe fusion of somites and their derivatives, such as vertebrae and ribs. In the developing brain, steady Dll1 expression inhibited proliferation of neural progenitors and accelerated neurogenesis, whereas optogenetic induction of Dll1 oscillation efficiently maintained neural progenitors. These results indicate that the appropriate timing of Dll1 expression is critical for the oscillatory networks and suggest the functional significance of oscillatory cell–cell interactions in tissue morphogenesis.

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.

Linear Augmentation for Stabilizing Stationary Solutions: Potential Pitfalls and Their Application

Linear augmentation has recently been shown to be effective in targeting desired stationary solutions, suppressing bistablity, in regulating the dynamics of drive response systems and in controlling the dynamics of hidden attractors. The simplicity of the procedure is the main highlight of this scheme but questions related to its general applicability still need to be addressed. Focusing on the issue of targeting stationary solutions, this work demonstrates instances where the scheme fails to stabilize the required solutions and leads to other complicated dynamical scenarios. Examples from conservative as well as dissipative systems are presented in this regard and important applications in dissipative predator-prey systems are discussed, which include preventative measures to avoid potentially catastrophic dynamical transitions in these systems.

Delay- and topology-independent design for inducing amplitude death on networks with time-varying delay connections

We present a procedure to systematically design the connection parameters that will induce amplitude death in oscillator networks with time-varying delay connections. The parameters designed by the procedure are valid in oscillator networks with any network topology and with any connection delay. The validity of the design procedure is confirmed by numerical simulation. We also consider a partial time-varying delay connection, which has both time-invariant and time-varying delays. The effectiveness of the partial connection is shown theoretically and numerically.

The role of mRNA and protein stability in the function of coupled positive and negative feedback systems in eukaryotic cells

Oscillators and switches are important elements of regulation in biological systems. These are composed of coupling negative feedback loops, which cause oscillations when delayed, and positive feedback loops, which lead to memory formation. Here, we examine the behavior of a coupled feedback system, the Negative Autoregulated Frustrated bistability motif (NAF). This motif is a combination of two previously explored motifs, the frustrated bistability motif (FBM) and the negative auto regulation motif (NAR), which both can produce oscillations. The NAF motif was previously suggested to govern long term memory formation in animals, and was used as a synthetic oscillator in bacteria. We build a mathematical model to analyze the dynamics of the NAF motif. We show analytically that the NAF motif requires an asymmetry in the strengths of activation and repression links in order to produce oscillations. We show that the effect of time delays in eukaryotic cells, originating from mRNA export and protein import, are negligible in this system. Based on the reported protein and mRNA half-lives in eukaryotic cells, we find that even though the NAF motif possesses the ability for oscillations, it mostly promotes constant protein expression at the biologically relevant parameter regimes.

Restoration of rhythmicity in diffusively coupled dynamical networks

Oscillatory behaviour is essential for proper functioning of various physical and biological processes. However, diffusive coupling is capable of suppressing intrinsic oscillations due to the manifestation of the phenomena of amplitude and oscillation deaths. Here we present a scheme to revoke these quenching states in diffusively coupled dynamical networks, and demonstrate the approach in experiments with an oscillatory chemical reaction. By introducing a simple feedback factor in the diffusive coupling, we show that the stable (in)homogeneous steady states can be effectively destabilized to restore dynamic behaviours of coupled systems. Even a feeble deviation from the normal diffusive coupling drastically shrinks the death regions in the parameter space. The generality of our method is corroborated in diverse non-linear systems of diffusively coupled paradigmatic models with various death scenarios. Our study provides a general framework to strengthen the robustness of dynamic activity in diffusively coupled dynamical networks.

Exceptional points in coupled dissipative dynamical systems

We study the transient behavior in coupled dissipative dynamical systems based on the linear analysis around the steady state. We find that the transient time is minimized at a specific set of system parameters and show that at this parameter set, two eigenvalues and two eigenvectors of the Jacobian matrix coalesce at the same time; this degenerate point is called the exceptional point. For the case of coupled limit-cycle oscillators, we investigate the transient behavior into the amplitude death state, and clarify that the exceptional point is associated with a critical point of frequency locking, as well as the transition of the envelope oscillation.

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.

Delay-induced patterns in a two-dimensional lattice of coupled oscillators

We show how a variety of stable spatio-temporal periodic patterns can be created in 2D-lattices of coupled oscillators with non-homogeneous coupling delays. The results are illustrated using the FitzHugh-Nagumo coupled neurons as well as coupled limit cycle (Stuart-Landau) oscillators. A “hybrid dispersion relation” is introduced, which describes the stability of the patterns in spatially extended systems with large time-delay.

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

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

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

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

Design of time-delayed connection parameters for inducing amplitude death in high-dimensional oscillator networks

The present paper studies time-delayed-connection induced amplitude death in high-dimensional oscillator networks. We provide two procedures for design of a coupling strength and a transmission delay: these procedures do not depend on the topology of oscillator networks (i.e., network structure and number of oscillators). A graphical procedure based on the Nyquist criterion is proposed and then is numerically confirmed for the case of five-dimensional oscillators, called generalized Rössler oscillators, which have two pairs of complex conjugate unstable roots. In addition, for the case of high-dimensional oscillators having two unstable roots, the procedure can be systematically carried out using only a simple algebraic calculation. This systematic procedure is numerically confirmed for the case of three-dimensional oscillators, called Moore-Spiegel oscillators, which have two positive real unstable roots.

Phase locking of two limit cycle oscillators with delay coupling

Mutual phase locking in the system of two limit cycle oscillators with delay coupling is studied. Conditions of phase locking are derived as a result of analysis of a generalized Adler equation. The analytical results are compared with numerical simulation. Depending on the phase shift of the coupling signal propagating between the two oscillators, either in-phase or anti-phase mode of synchronization may arise. The number of possible modes of synchronization increases with the delay time.

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.