Networks Of Identical Oscillators Research Articles

On coupled oscillators modeling bio-inspired acoustic sensors: Bifurcation analysis toward tunability enhancement.

Oscillators exhibiting an Andronov-Hopf bifurcation are candidates to mimic the functionality of the cochlea, since the transfer response of these oscillators is compressive and frequency selective. The former implies that small stimuli are amplified and strong stimuli are attenuated, while the latter means that the oscillator only reacts in a (small) frequency band. However, this implies that many oscillators are needed to cover a relevant frequency band. By introducing the notion of tunable characteristic frequencies, i.e., the characteristic frequency can be adjusted by a controllable input, the number of oscillators can be eventually reduced. Subsequently, the tunability enhancement of coupled oscillators is investigated by analyzing the local dynamics of a network of oscillators. For this, necessary conditions for the emergence of Andronov-Hopf bifurcations are determined for networks consisting of two groups, i.e., a group is a network of identical oscillators. By choosing the eigenvalues of the product of the cross-coupling matrix as bifurcation parameters and exploiting the structure of the transfer matrix of this network, the critical points and, thus, the characteristic frequency at this point can be derived. Tunability of the characteristic frequency is then enabled by controlling the asymmetry between the groups of oscillators.

Chimera states in a network of identical oscillators with symmetric coexisting attractors

Chimera states in a network of identical oscillators with symmetric coexisting attractors

Adaptive rewiring in nonuniform coupled oscillators.

Structural plasticity of the brain can be represented in a highly simplified form as adaptive rewiring, the relay of connections according to the spontaneous dynamic synchronization in network activity. Adaptive rewiring, over time, leads from initial random networks to brain-like complex networks, that is, networks with modular small-world structures and a rich-club effect. Adaptive rewiring has only been studied, however, in networks of identical oscillators with uniform or random coupling strengths. To implement information-processing functions (e.g., stimulus selection or memory storage), it is necessary to consider symmetry-breaking perturbations of oscillator amplitudes and coupling strengths. We studied whether nonuniformities in amplitude or connection strength could operate in tandem with adaptive rewiring. Throughout network evolution, either amplitude or connection strength of a subset of oscillators was kept different from the rest. In these extreme conditions, subsets might become isolated from the rest of the network or otherwise interfere with the development of network complexity. However, whereas these subsets form distinctive structural and functional communities, they generally maintain connectivity with the rest of the network and allow the development of network complexity. Pathological development was observed only in a small proportion of the models. These results suggest that adaptive rewiring can robustly operate alongside information processing in biological and artificial neural networks.

Transient chimera-like states for forced oscillators.

Chimera states occur widely in networks of identical oscillators as has been shown in the recent extensive theoretical and experimental research. In such a state, different groups of oscillators can exhibit coexisting synchronous and incoherent behaviors despite homogeneous coupling. Here, we consider a star network, in which N identical peripheral end nodes are connected to the central hub node. We find that if a single node exhibits transient chaotic behavior in the whole network, the pattern of transient chimeralike state, which persists for a significant amount of time, is created. As a proof of the concept, we examine the system of N double pendula (peripheral end nodes) located on the periodically oscillating platform (central hub). We show that such transient chimeralike states can be observed in simple experiments with mechanical oscillators, which are controlled by elementary dynamical equations. Our finding suggests that transient chimeralike states are observable in networks relevant to various real-world systems.

The emergence of chimera states in a network of nephrons

The kidney plays an essential role in our body, mainly by controlling secretion and reabsorption of water and salts. The kidneys consist of a large number of nephrons which are the functional units of the kidney. The interactions between these nephrons induce different behaviors which can be considered by a dynamical model. In this paper, a network of coupled nephron models and its dynamics is investigated. Numerical simulations of the network reveal various types of dynamical patterns depending on the coupling function and strength. One of the observed phenomenon is the emergence of chimera state. A chimera state is defined by the coexistence of coherent and incoherent groups in a network of identical oscillators. The occurrence of the chimera state can be related to the situation of disturbed synchronous oscillation of the TGF-mediated proximal pressure.

Chimera states in a ring of map-based neurons

Chimera state is a simultaneous combination of synchronous and asynchronous environments in a network of identical oscillators, attracts the attention of researchers in the different fields of science. Neuroscientists believe that existing both the synchronous and asynchronous behavior in the brain plays a vital role in brain neuronal activity. As a result, in this paper, we investigate the ability of a neural network to produce chimera. Our focus has been on using a discrete neuronal model which resolve the curse of dimensionality by retraining the complex behavior at the same time. Each neuron is coupled to its neighboring with limited scope. The interdependency of the spatiotemporal dynamic on the coupling strength in a fixed range made the switching process visible, which rises with the appearance of thin bands of incoherence, occupying the whole space ultimately.

Nonautonomous driving induces stability in network of identical oscillators.

Nonautonomous driving of an oscillator has been shown to enlarge the Arnold tongue in parameter space, but little is known about the analogous effect for a network of oscillators. To test the hypothesis that deterministic nonautonomous perturbation is a good candidate for stabilizing complex dynamics, we consider a network of identical phase oscillators driven by an oscillator with a slowly time-varying frequency. We investigate both the short- and long-term stability of the synchronous solutions of this nonautonomous system. For attractive couplings we show that the region of stability grows as the amplitude of the frequency modulation is increased, through the birth of an intermittent synchronization regime. For repulsive couplings, we propose a control strategy to stabilize the dynamics by altering very slightly the network topology. We also show how, without changing the topology, time-variability in the driving frequency can itself stabilize the dynamics. As a byproduct of the analysis, we observe chimeralike states. We conclude that time-variability-induced stability phenomena are also present in networks, reinforcing the idea that this is a quite realistic scenario for living systems to use in maintaining their functioning in the face of ongoing perturbations.

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.

Solitary states for coupled oscillators with inertia.

Networks of identical oscillators with inertia can display remarkable spatiotemporal patterns in which one or a few oscillators split off from the main synchronized cluster and oscillate with different averaged frequency. Such "solitary states" are impossible for the classical Kuramoto model with sinusoidal coupling. However, if inertia is introduced, these states represent a solid part of the system dynamics, where each solitary state is characterized by the number of isolated oscillators and their disposition in space. We present system parameter regions for the existence of solitary states in the case of local, non-local, and global network couplings and show that they preserve in both thermodynamic and conservative limits. We give evidence that solitary states arise in a homoclinic bifurcation of a saddle-type synchronized state and die eventually in a crisis bifurcation after essential variation of the parameters.

Non-identical multiplexing promotes chimera states

We present the emergence of chimeras, a state referring to coexistence of partly coherent, partly incoherent dynamics in networks of identical oscillators, in a multiplex network consisting of two non-identical layers which are interconnected. We demonstrate that the parameter range displaying the chimera state in the homogeneous first layer of the multiplex networks can be tuned by changing the link density or connection architecture of the same nodes in the second layer. We focus on the impact of the interconnected second layer on the enlargement or shrinking of the coupling regime for which chimeras are displayed in the homogeneous first layer. We find that a denser homogeneous second layer promotes chimera in a sparse first layer, where chimeras do not occur in isolation. Furthermore, while a dense connection density is required for the second layer if it is homogeneous, this is not true if the second layer is inhomogeneous. We demonstrate that a sparse inhomogeneous second layer which is common in real-world complex systems, can promote chimera states in a sparse homogeneous first layer.

Chimera states in complex networks: interplay of fractal topology and delay

Chimera states are an example of intriguing partial synchronization patterns emerging in networks of identical oscillators. They consist of spatially coexisting domains of coherent (synchronized) and incoherent (desynchronized) dynamics. We analyze chimera states in networks of Van der Pol oscillators with hierarchical connectivities, and elaborate the role of time delay introduced in the coupling term. In the parameter plane of coupling strength and delay time we find tongue-like regions of existence of chimera states alternating with regions of existence of coherent travelling waves. We demonstrate that by varying the time delay one can deliberately stabilize desired spatio-temporal patterns in the system.

Synchronization patterns and chimera states in complex networks: Interplay of topology and dynamics

We review chimera patterns, which consist of coexisting spatial domains of coherent (synchronized) and incoherent (desynchronized) dynamics in networks of identical oscillators. We focus on chimera states involving amplitude as well as phase dynamics, complex topologies like small-world or hierarchical (fractal), noise, and delay. We show that a plethora of novel chimera patterns arise if one goes beyond the Kuramoto phase oscillator model. For the FitzHugh-Nagumo system, the Van der Pol oscillator, and the Stuart-Landau oscillator with symmetry-breaking coupling various multi-chimera patterns including amplitude chimeras and chimera death occur. To test the robustness of chimera patterns with respect to changes in the structure of the network, regular rings with coupling range R, small-world, and fractal topologies are studied. We also address the robustness of amplitude chimera states in the presence of noise. If delay is added, the lifetime of transient chimeras can be drastically increased.

Restoration of oscillation in network of oscillators in presence of direct and indirect interactions

The suppression of oscillations in coupled systems may lead to several unwanted situations, which requires a suitable treatment to overcome the suppression. In this paper, we show that the environmental coupling in the presence of direct interaction, which can suppress oscillation even in a network of identical oscillators, can be modified by introducing a feedback factor in the coupling scheme in order to restore the oscillation. We inspect how the introduction of the feedback factor helps to resurrect oscillation from various kinds of death states. We numerically verify the resurrection of oscillations for two paradigmatic limit cycle systems, namely Landau–Stuart and Van der Pol oscillators and also in generic chaotic Lorenz oscillator. We also study the effect of parameter mismatch in the process of restoring oscillation for coupled oscillators.

Quantum signatures of chimera states.

Chimera states are complex spatiotemporal patterns in networks of identical oscillators, characterized by the coexistence of synchronized and desynchronized dynamics. Here we propose to extend the phenomenon of chimera states to the quantum regime, and uncover intriguing quantum signatures of these states. We calculate the quantum fluctuations about semiclassical trajectories and demonstrate that chimera states in the quantum regime can be characterized by bosonic squeezing, weighted quantum correlations, and measures of mutual information. Our findings reveal the relation of chimera states to quantum information theory, and give promising directions for experimental realization of chimera states in quantum systems.

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.

Spatially structured networks of pulse-coupled phase oscillators on metric spaces

The Winfree model describes finite networks of phase oscillators. Oscillators interact by broadcasting pulses that modulate the frequencies of connected oscillators. We study a generalization of the model and its fluid-dynamical limit for networks, where oscillators are distributed on some abstract $\sigma$-finite Borel measure space over a separable metric space. We give existence and uniqueness statements for solutions to the continuity equation for the oscillator phase densities. We further show that synchrony in networks of identical oscillators is locally asymptotically stable for finite, strictly positive measures and under suitable conditions on the oscillator response function and the coupling kernel of the network. The conditions on the latter are a generalization of the strong connectivity of finite graphs to abstract coupling kernels.

Chimera states in heterogeneous networks

Chimera states in networks of coupled oscillators occur when some fraction of the oscillators synchronize with one another, while the remaining oscillators are incoherent. Several groups have studied chimerae in networks of identical oscillators, but here we study these states in heterogeneous models for which the natural frequencies of the oscillators are chosen from a distribution. For a model consisting of two subnetworks, we obtain exact results by reduction to a finite set of differential equations, and for a network of oscillators in a ring, we generalize known results. We find that heterogeneity can destroy chimerae, destroy all states except chimerae, or destabilize chimerae in Hopf bifurcations, depending on the form of the heterogeneity.

The size of the sync basin

We suggest a new line of research that we hope will appeal to the nonlinear dynamics community, especially the readers of this Focus Issue. Consider a network of identical oscillators. Suppose the synchronous state is locally stable but not globally stable; it competes with other attractors for the available phase space. How likely is the system to synchronize, starting from a random initial condition? And how does the probability of synchronization depend on the way the network is connected? On the one hand, such questions are inherently difficult because they require calculation of a global geometric quantity, the size of the "sync basin" (or, more formally, the measure of the basin of attraction for the synchronous state). On the other hand, these questions are wide open, important in many real-world settings, and approachable by numerical experiments on various combinations of dynamical systems and network topologies. To give a case study in this direction, we report results on the sync basin for a ring of n >> 1 identical phase oscillators with sinusoidal coupling. Each oscillator interacts equally with its k nearest neighbors on either side. For k/n greater than a critical value (approximately 0.34, obtained analytically), we show that the sync basin is the whole phase space, except for a set of measure zero. As k/n passes below this critical value, coexisting attractors are born in a well-defined sequence. These take the form of uniformly twisted waves, each characterized by an integer winding number q, the number of complete phase twists in one circuit around the ring. The maximum stable twist is proportional to n/k; the constant of proportionality is also obtained analytically. For large values of n/k, corresponding to large rings or short-range coupling, many different twisted states compete for their share of phase space. Our simulations reveal that their basin sizes obey a tantalizingly simple statistical law: the probability that the final state has q twists follows a Gaussian distribution with respect to q. Furthermore, as n/k increases, the standard deviation of this distribution grows linearly with square root of n/k. We have been unable to explain either of these last two results by anything beyond a hand-waving argument.

Global analysis of limit cycles in networks of oscillators

This paper is concerned with the global stability of limit cycle oscillations for a particular class of systems and networks. In previous work, we defined a class of parameter-dependent nonlinear systems exhibiting an almost globally asymptotically stable limit cycle. The results were proven for values of the parameter in the vicinity of a bifurcation value. In the present paper we restrict ourselves to a piecewise linear version of this class of systems and adapt numerical tools recently proposed in the literature to prove global stability of the limit cycle for a fixed value of the parameter above the bifurcation value. Furthermore, we show how the global stability results for one isolated oscillator are useful to prove the existence of a globally synchrone oscillation in particular networks of identical oscillators.