Ensemble Of Phase Oscillators Research Articles

Mean-field model of synchronization for open-loop, swirl controlled thermoacoustic system.

Open-loop control is known to be an effective strategy for controlling self-excited periodic oscillations, known as thermoacoustic instability, in turbulent combustors. Here, we present experimental observations and a synchronization model for the suppression of thermoacoustic instability achieved by rotating the otherwise static swirler in a lab-scale turbulent combustor. Starting with the state of thermoacoustic instability in the combustor, we find that a progressive increase in the swirler rotation rate leads to a transition from the state of limit cycle oscillations to the low-amplitude aperiodic oscillations through a state of intermittency. To model such a transition while also quantifying the underlying synchronization characteristics, we extend the model of Dutta et al. [Phys. Rev. E 99, 032215 (2019)] by introducing a feedback between the ensemble of phase oscillators and the acoustic. The coupling strength in the model is determined by considering the effect of the acoustic and swirl frequencies. The link between the model and experimental results is quantitatively established by implementing an optimization algorithm for model parameter estimation. We show that the model is capable of replicating the bifurcation characteristics, nonlinear features of time series, probability density function, and amplitude spectrum of acoustic pressure and heat release rate fluctuations at various dynamical states observed during the transition to the state of suppression. Most importantly, we discuss the flame dynamics and demonstrate that the model without any spatial inputs qualitatively captures the characteristics of the spatiotemporal synchronization between the local heat release rate fluctuations and the acoustic pressure that underpins a transition to the state of suppression. As a result, the model emerges as a powerful tool for explaining and controlling instabilities in thermoacoustic and other extended fluid dynamical systems, where spatiotemporal interactions lead to rich dynamical phenomena.

Heterogeneity-induced synchronization in delay-coupled electronic oscillators.

We study synchronization in networks of delay-coupled electronic oscillators, so-called phase-locked loops (PLLs). Using a phase-model description, we study the collective dynamics of mutually coupled PLLs and report the phenomenon of heterogeneity-induced synchronization. This phenomenon refers to the observation that heterogeneity in the system's parameters can induce synchronization by stabilizing the states which are unstable without such heterogeneity. In systems where component heterogeneity can be tuned and controlled, we show how the complex collective self-organized dynamics can be guided towards synchronized states with specific operational frequencies and phase relations. This is of importance for the technical applicability of self-organized dynamics. In electrical engineering, for example, where components can be strongly heterogeneous, our theoretical framework can inform the design process for networks of spatially distributed PLLs. The results presented here are also useful in understanding the collective dynamics in ensembles of phase oscillators with time-delayed interactions, inertia, and heterogeneity.

Hierarchy of Exact Low-Dimensional Reductions for Populations of Coupled Oscillators.

We consider an ensemble of phase oscillators in the thermodynamic limit, where it is described by a kinetic equation for the phase distribution density. We propose an Ansatz for the circular moments of the distribution (Kuramoto-Daido order parameters) that allows for an exact truncation at an arbitrary number of modes. In the simplest case of one mode, the Ansatz coincides with that of Ott and Antonsen [Chaos 18, 037113 (2008)CHAOEH1054-150010.1063/1.2930766]. Dynamics on the extended manifolds facilitate higher-dimensional behavior such as chaos, which we demonstrate with a simulation of a Josephson junction array. The findings are generalized for oscillators with a Cauchy-Lorentzian distribution of natural frequencies.

Chimera states for directed networks.

We demonstrate that chimera behavior can be observed in ensembles of phase oscillators with unidirectional coupling. For a small network consisting of only three identical oscillators (cyclic triple), tiny chimera islands arise in the parameter space. They are surrounded by developed chaotic switching behavior caused by a collision of rotating waves propagating in opposite directions. For larger networks, as we show for a hundred oscillators (cyclic century), the islands merge into a single chimera continent, which incorporates the world of chimeras of different configurations. The phenomenon inherits from networks with intermediate ranges of the unidirectional coupling and it diminishes as the coupling range decreases.

Взаимная синхронизация сложных структур во взаимодействующих ансамблях нелокально-связанных ротаторов

Background and Objectives: One of the actual problems in nonlinear dynamics is the formation and interaction of complex spatial structures such as chimeras and solitary states arising in multicomponent systems. Chimera states are typical for ensembles of identical oscillators with regular, chaotic, and even stochastic behavior in a case of nonlocal interaction of the elements. They represent cluster structures, including groups of elements with synchronous and non-synchronous oscillations. Chimeras were discovered and investigated in real experiments, that indicates the possibility of observing such regimes in complex systems in living nature and in technology. Solitary states are less studied today. The regime of solitary states is characterized by the synchronous behavior of most elements of the ensemble, while individual oscillators behave in a “special state”. In the present work, an ensemble of phase oscillators with inertia (rotators) is chosen as the basic model for investigation. Such ensembles with a specific coupling topology are widely used in modeling the operation of energy networks. Ensembles of rotators with nonlocal coupling are characterized by both chimera states and solitary state regimes. The problem of interaction of ensembles of rotators with nonlocal coupling and synchronization of complex spatial structures (chimeras and solitary states) formed in them has not been studied yet. Materials and Methods: A two-layer multiplex network of rotators with a nonlocal character of intralayer interactions is considered. Each layer consists of 100 elements with the same value of the coupling coefficient and coupling phase shift for each element within one layer. The interlayer coupling is symmetric. At the initial stage, with a random choice of initial conditions, steady regimes (chimeras or solitary states) in non-interacting layers were found. Next, the interlayer coupling was introduced and the evolution of the layer dynamics in the selected initial regimes was studied. Four cases of interaction with various initial states of the layers were considered. In the first case, the two layers are completely identical and demonstrate slightly different chimera structures without interlayer coupling. Their evolution with the introduction and growth of the interlayer coupling is considered for two values of the coupling phase shift. It is shown that, starting from a certain threshold value of the interlayer coupling coefficient, the complete synchronization regime is established in the layers, and the coupling phase shift significantly affects the value of the synchronization threshold. In the second case, the previous experiment is reproduced for the two layers with a frequency mismatch. Chimera states established without interlayer interaction are characterized by significantly different average frequencies of the elements in the two layers. In the presence of non-identity of the layers (in this case, frequency mismatch), the regime of complete synchronization of spatial structures is impossible. However, with an increase in the interlayer coupling coefficient, effective synchronization can be obtained which corresponds to a slight difference in the phases of rotators in the interacting layers with full frequency synchronization. In the third case, we consider the interaction between the layers in the solitary state regimes with different spatial structures. In this case, a frequency mismatch is also introduced for the elements of the two layers. For solitary states, the effective synchronization regime with an increase in the interlayer coupling is also established. In both layers the same configurations of solitary states are realized and frequency synchronization is observed. In the fourth case, a heterogeneous multiplex network is considered, in which one layer is in the chimera state, the second layer shows the solitary state mode. With a certain strength of the interlayer coupling the complex structures are destroyed in both layers of the network and a spatially uniform regimes are established. In this case, all the rotators of the two layers rotate at the same frequency, and the difference in the regimes in the layers reduces to a small phase shift, the same for all pairs of coupled rotators of the two layers. Conclusion: The effects of synchronization in the multiplex network were established for two layers in the regimes of complex spatio-temporal dynamics, such as chimera states and solitary states. The influence of the frequency mismatch of the network elements and the phase shift in the interlayer coupling on the synchronization phenomena was studied.

Energy exchange in globally coupled mechanical phase oscillators

We study the stationary dynamics of energy exchange in an ensemble of phase oscillators, coupled through a mean-field mechanical interaction and added with friction and an external periodic excitation. The degree of entrainment between different parts of the ensemble and the external forcing determines three dynamical regimes, each of them characterized by specific rates of energy exchange. Using suitable approximations, we are able to obtain analytical expressions for those rates, which are in satisfactory agreement with results from numerical integration of the equations of motion. In some of the dynamical regimes, the rates of energy exchange show nontrivial dependence on the friction coefficients-in particular, nonmonotonic behavior and sign switching. This suggests that, even in this kind of stylized model, power transfer between different parts of the ensemble and to the environment can be manipulated by a convenient choice of the individual oscillator parameters.

Synchronization of phase oscillators under asymmetric and bimodal distributions of natural frequencies

Abstract We report on the setting of synchronization in ensembles of phase oscillators under an asymmetric and bimodal distribution of initial frequencies. In particular, by both the Ott–Antonsen reduction and direct numerical simulations we give evidence that the system exhibits four different transition routes towards synchronization, including continuous, discontinuous, and hybrid transitions, depending on the choice of the parameters of the frequency distribution. As the coupling strength increases, several coherent states emerge, such as two types of travelling wave states and the Bellerophon state, which are carefully characterized.

Two-Bunch Solutions for the Dynamics of Ott–Antonsen Phase Ensembles

We have developed a method for deriving systems of closed equations for the dynamics of order parameters in the ensembles of phase oscillators. The Ott–Antonsen equation for the complex order parameter is a particular case of such equations. The simplest nontrivial extension of the Ott–Antonsen equation corresponds to two-bunch states of the ensemble. Based on the equations obtained, we study the dynamics of multi-bunch chimera states in coupled Kuramoto–Sakaguchi ensembles. We show an increase in the dimensionality of the system dynamics for two-bunch chimeras in the case of identical phase elements and a transition to one-bunch “Abrams chimeras” for imperfect identity (in the latter case, the one-bunch chimeras become attractive).

Global Stability of a Synchronous Regime in Hub Clusters of the Power Networks

We study stability of a synchronous regime in hub clusters of the power networks, which are simulated by ensembles of phase oscillators. An approach allowing one to estimate the regions in the parameter space, which correspond to the global asymptotic stability of this regime, is presented. The method is illustrated by an example of a hub cluster consisting of one generator and two consumers.

Competing influence of common noise and desynchronizing coupling on synchronization in the Kuramoto-Sakaguchi ensemble

We describe analytically synchronization and desynchronization effects in an ensemble of phase oscillators driven by common noise and by global coupling. Adopting the Ott-Antonsen ansatz, we reduce the dynamics to closed stochastic equations for the order parameters, and study these equations for the cases of populations of identical and nonidentical oscillators. For nonidentical oscillators we demonstrate a counterintuitive effect of divergence of individual frequencies for moderate repulsive coupling, while the order parameter remains large.

СИНХРОНИЗАЦИЯ В АНСАМБЛЯХ КУРАМОТО–САКАГУЧИ ПРИ КОНКУРИРУЮЩЕМ ВЛИЯНИЯ ОБЩЕГО ШУМА И ГЛОБАЛЬНОЙ СВЯЗИ

We study the effects of synchronization and desynchronization in ensembles of phase oscillators with the global Kuramoto-Sakaguchi coupling under common noise driving. Since the mechanisms of synchronization by coupling and by common noise are essentially different, their interplay is of interest. In the thermodynamic limit of large number of oscillators, employing the Ott-Antonsen approach, we derive stochastic equations for the order parameters and consider their dynamics for two cases: (i) identical oscillators and (ii) small natural frequency mismatch. For identical oscillators, the stability of the perfect synchrony state is studied; a strong enough common noise is revealed to prevail over a moderate negative (repelling) coupling and to synchronize the ensemble. An inequality between the states of maximal asynchrony (zero-value of the order parameter) and perfect synchrony; the former can be only weakly stable, while the latter can become adsorbing (the transition to the synchrony becomes unidirectional). The dependence of the temporal dynamics of the transition on the system parameters is investigated. For nonidentical oscillators the perfect synchrony state becomes impossible and an absorbing state disappears; on its place, only a weakly stable state of imperfect synchrony remains. A nontrivial effect of the divergence of individual frequencies of oscillators with different natural frequencies is revealed and studied for moderate repelling coupling; meanwhile, the order parameter remains non-small for this case. In Appendix we provide an introduction to the theories of Ott-Antonsen and Watanabe-Strogatz.

Dynamics of phase oscillators with generalized frequency-weighted coupling.

Heterogeneous coupling patterns among interacting elements are ubiquitous in real systems ranging from physics, chemistry to biology communities, which have attracted much attention during recent years. In this paper, we extend the Kuramoto model by considering a particular heterogeneous coupling scheme in an ensemble of phase oscillators, where each oscillator pair interacts with different coupling strength that is weighted by a general function of the natural frequency. The Kuramoto theory for the transition to synchronization can be explicitly generalized, such as the expression for the critical coupling strength. Also, a self-consistency approach is developed to predict the stationary states in the thermodynamic limit. Moreover, Landau damping effects are further revealed by means of linear stability analysis and resonance poles theory below the critical threshold, which turns to be far more generic. Our theoretical analysis and numerical results are consistent with each other, which can help us understand the synchronization transition in general networks with heterogenous couplings.

Effects of nonresonant interaction in ensembles of phase oscillators

We consider general properties of groups of interacting oscillators, for which the natural frequencies are not in resonance. Such groups interact via nonoscillating collective variables like the amplitudes of the order parameters defined for each group. We treat the phase dynamics of the groups using the Ott-Antonsen ansatz and reduce it to a system of coupled equations for the order parameters. We describe different regimes of cosynchrony in the groups. For a large number of groups, heteroclinic cycles, corresponding to a sequential synchronous activity of groups and chaotic states where the order parameters oscillate irregularly, are possible.

Chimera states and collective chaos in pulse-coupled neural networks

Understanding the collective motion of networks of oscillators is crucial in many contexts, starting from neuronal circuits [1]. So far, most of the efforts have been devoted to the characterization of strong forms of synchronization. However, more subtle phenomena, like the onset of coherent oscillations in an ensemble of neurons can also play a relevant role for information coding. A peculiar coherent state, termed Chimera, appears in two symmetrically coupled populations of oscillators, where a population fully synchronizes while the other exhibit an asynchronous dynamics [2]. This can represents an idealized mathematical representation of the so-called unihemispheric sleep. Many creatures sleep with only half their brain at a time [3]. Such phenomenon was first reported in dolphins and other sea mammals and, recently, in birds; when brain waves are recorded, the awake side of the brain shows desynchronized electrical activity, corresponding to millions of neurons oscillating out of phase, whereas the sleeping side is highly synchronized. In our study we have investigated the dynamics of two symmetrically coupled populations of identical leaky integrate-and-fire neurons characterized by excitatory coupling [4]. Upon varying the coupling strengths between and within the two populations, we found symmetry-breaking transitions that lead to the onset of various chimera states. To be more specific, in the observed chimera states one population is fully synchronized, while the other one is in the so-called Partially Synchronous (PS) regime. This regime is characterized by a coherent periodic activity at a collective level (somehow corresponding to to the Local Field Potential), while the single neurons behaves quasi-periodically [5]. To our knowledge this represents the first evidence of chimera states in pulse-coupled neural networks. Furthermore a new regime, where the two populations are both PS but with a different degree of synchronization have been also observed. Even more interesting is the identification of a regions of parameters where the coherent activity of the network, characterized by collective variables, becomes chaotic. Collective chaos, meant as irregular dynamics of coarse-grained observables, has been found in ensembles of fully coupled one-dimensional maps as well as in two-dimensional continuous-time oscillators. In both classes of models, the single dynamical unit can behave chaotically under the action of a periodic forcing. Only a few examples of low-dimensional chaotic collective motion have been found in ensembles of phase oscillators because in this setup there is little space for a high-dimensional dynamics [6]. Our results reveal for the first time chaos at a macroscopic level in a neural network. The computation of the the finite-amplitude Lyapunov exponent allows us to firmly establish the chaoticity of the (collective) dynamics in a finite region of the phase plane. The further numerical study of the standard Lyapunov spectrum reveals the presence of several positive exponents, indicating that the microscopic dynamics is high-dimensional.

Interplay of noise and coupling in heterogeneous ensembles of phase oscillators

We study the effects of noise on the collective dynamics of an ensemble of coupled phase oscillators whose natural frequencies are all identical, but whose coupling strengths are not the same all over the ensemble. The intensity of noise can also be heterogeneous, representing diversity in the individual responses to external fluctuations. We show that the desynchronization transition induced by noise may be completely suppressed, even for arbitrarily large noise intensities, is the distribution of coupling strengths decays slowly enough for large couplings. Equivalently, if the response to noise of a sufficiently large fraction of the ensemble is weak enough, desynchronization cannot occur. The two effects combine with each other when the response to noise and the coupling strength of each oscillator are correlated. This combination is quantitatively characterized and illustrated with explicit examples.

Demand-controlled desynchronization of oscillatory networks by means of a multisite delayed feedback stimulation

We study the potential of delayed feedback stimulation administered via multiple sites to control the synchronization and the spatio-temporal activity pattern within a network of oscillators. Two models with different levels of complexity are investigated: an ensemble of phase oscillators is used to study and analyze the effects of the stimulation in detail and a microscopic model is presented to exemplify the effectiveness of the method in terms of a stable desynchronization continuously maintained under realistic conditions. The investigation shows that depending on the spatial stimulation profile a multi-cluster state or a fully desynchronized state results. Both states are characterized by a strong suppression of the first order parameter indicating a strong desynchronization. Multisite delayed feedback stimulation realizes a demand-controlled stimulation which stabilizes the desired desynchronized state while the used stimulation current is minimized. We propose the method to be tested as a therapy of diseases characterized by pathological synchronization and for the therapy of defective neuronal pattern generators.