Nonidentical Oscillators Research Articles

Coarse graining the dynamics of delayed phase oscillators on Cayley trees by star networks

It is of interest in many natural and artificial networks to find simpler systems to understand spatio-temporal dynamics. For this, we consider an extended fragmentary networks by considering a system of phase oscillators with time delay and degree heterogeneity on Cayley trees to address it. The coupled oscillators on Cayley trees exhibit various dynamical regimes such as clusters synchronization, desynchronization, and phase-flip between synchronized clusters. The dynamical regimes and transitions of oscillators on the Cayley tree are found to be similar to the star network’s oscillator dynamics. We confirm the Cayley tree and star network’s functional equivalence by finding the collective frequencies of phase oscillators in both systems. Analytical estimates of linearly stable common frequencies of oscillators on star networks agree with oscillator’s numerical solutions on Cayley trees. Our results hold for identical and non-identical oscillators, and they provide a theoretical framework to understand the collective dynamics of delayed phase oscillators in extended systems.

Spatiotemporal Regimes in the Kuramoto–Battogtokh System of Nonidentical Oscillators

We consider the spatiotemporal states of an ensemble of nonlocally coupled nonidentical phase oscillators, which correspond to different regimes of the long-term evolution of such a system. We have obtained homogeneous, twisted, and nonhomogeneous stationary solutions to the Ott–Antonsen equations corresponding to key variants of the realized collective rotational motion of elements of the medium in question with nonzero mesoscopic characteristics determining the degree of coherence of the dynamics of neighboring particles. We have described the procedures of the search for the class of nonhomogeneous solutions as stationary points of the auxiliary point map and of determining the stability based on analysis of the eigenvalue spectrum of the composite operator. Static and breather cluster regimes have been demonstrated and described, as well as the regimes with an irregular behavior of averaged complex fields including, in particular, the local order parameter.

Proportional–Integral Synchronization for Nonidentical Wireless Packet-Coupled Oscillators With Delays

Precise timing among wireless sensor nodes is a key enabling technology for time-sensitive industrial wireless sensor networks (WSNs). However, the accuracy of timing is degraded by manufacturing tolerance, aging of crystal oscillators, and communication delays. This article develops a framework of packet-coupled oscillators (PkCOs) to characterize the dynamics of communication and time synchronization of clocks in WSNs. The nonidentical clock is derived to describe the embedded clock's behavior accurately. A proportional-integral (PI) packet coupling scheme is proposed for synchronizing networked embedded clocks, while, scheduling wireless Sync packets to different slots for transmission. It also possesses the feature of automatically eliminating the effects of unknown processing delay, which further improves the synchronization performance. The rigorous theoretical analysis of PI-based PkCOs is presented via studying a closed-loop time synchronization system. The performance of PI-based PkCOs is evaluated on a hardware testbed of IEEE 802.15.4 WSN. The experimental results show that the precision of the proportional-integral PkCOs protocol is as high as 60 μs (i.e., 2 ticks) for 32.768 kHz crystal oscillator-based clocks.

Emergence of order from chaos: A phenomenological model of coupled oscillators

Abstract This paper aims to study the transition to order from chaos using a mathematical model of coupled non-identical oscillators. In the course of our analysis, we adopt a different control parameter other than the conventional parameters: coupling strength and frequency-mismatch, generally used in literature to obtain the aforementioned transition. Both the interacting oscillators become periodic, and they lead to the synchronized state as the control parameter changes monotonically. Initially, the participating oscillators, with low amplitude of oscillations, are in the chaotic desynchronized state. Besides, during the periodic oscillations, the oscillators exhibit high amplitude of oscillations. We illustrate the corresponding results using two examples of coupled oscillators. Such emergence of order from chaos with an accompanying increase in amplitude is ascertained in various numerical simulations and experimental observations.

Inverse Problem in the Kuramoto Model with a Phase Lag: Application to the Sun

We solve the inverse problem in the Kuramoto model with three nonidentical oscillators and a phase lag. The model represents a three-cell-in-depth radial profile of the solar meridional circulation in each solar hemisphere and describes the synchronization between the two components (toroidal and poloidal) of the solar magnetic field. We reconstruct natural frequencies of the top and the bottom oscillators from the evolution of their phases when the oscillators are phase-locked. The phase-locking allows to solve the inverse problem when the phase of the middle oscillator is not available. We present the exact solution and investigate its stability. The model reveals a crucial role of the phase lag in the inverse problem solution. We apply the model to the reconstruction of the deep meridional circulation of the Sun and discuss peculiarities of its evolution in terms of solar dynamo.

Coupled Möbius maps as a tool to model Kuramoto phase synchronization.

We propose Möbius maps as a tool to model synchronization phenomena in coupled phase oscillators. Not only does the map provide fast computation of phase synchronization, it also reflects the underlying group structure of the sinusoidally coupled continuous phase dynamics. We study map versions of various known continuous-time collective dynamics, such as the synchronization transition in the Kuramoto-Sakaguchi model of nonidentical oscillators, chimeras in two coupled populations of identical phase oscillators, and Kuramoto-Battogtokh chimeras on a ring, and demonstrate similarities and differences between the iterated map models and their known continuous-time counterparts.

Entangled chimeras in nonlocally coupled bicomponent phase oscillators: From synchronous to asynchronous chimeras

Chimera states, a symmetry-breaking spatiotemporal pattern in nonlocally coupled identical dynamical units, have been identified in various systems and generalized to coupled nonidentical oscillators. It has been shown that strong heterogeneity in the frequencies of nonidentical oscillators might be harmful to chimera states. In this work, we consider a ring of nonlocally coupled bicomponent phase oscillators in which two types of oscillators are randomly distributed along the ring: some oscillators with natural frequency ω1 and others with ω2. In this model, the heterogeneity in frequency is measured by frequency mismatch ∣ω1 − ω2∣ between the oscillators in these two subpopulations. We report that the nonlocally coupled bicomponent phase oscillators allow for chimera states no matter how large the frequency mismatch is. The bicomponent oscillators are composed of two chimera states, one supported by oscillators with natural frequency ω1 and the other by oscillators with natural frequency ω2. The two chimera states in two subpopulations are synchronized at weak frequency mismatch, in which the coherent oscillators in them share similar mean phase velocity, and are desynchronized at large frequency mismatch, in which the coherent oscillators in different subpopulations have distinct mean phase velocities. The synchronization-desynchronization transition between chimera states in these two subpopulations is observed with the increase in the frequency mismatch. The observed phenomena are theoretically analyzed by passing to the continuum limit and using the Ott-Antonsen approach.

On mathematical analysis of synchronization to bidirectionally coupled Kuramoto oscillators

This article is concerned about the synchronization problem of the bidirectionally coupled Kuramoto model. In this model, each oscillator only interacts with the oscillators with adjacent labeling numbers. Namely, the oscillator θi only interacts with θi+1 and θi−1. In real applications, this is a typical setting of concatenation connection. We first prove the global convergence of frequency synchronization for the identical oscillators. Also, we present two results of phase synchronization for the identical case in relatively wide initial configuration regimes. In the case that the coupling strength is sufficiently large (equivalent to the difference of the natural frequencies is sufficiently small), we show that for non-identical oscillators within a suitable initial configuration regime, the bidirectionally coupled Kuramoto model achieves a frequency synchronization. The supportive numerical simulations are presented as well.

Синхронизация и несимметрия в модели Курамото из трех неидентичных осцилляторов: особенности моделирования меридионального потока Солнца

Модели Курамото нелинейно связанных осцилляторов позволяют достаточно просто описывать фазовую синхронизацию в сложных системах. В данной работе мы рассматриваем частный случай модели Курамото с тремя осцилляторами, возникший в процессе исследования и моделирования меридионального потока в конвективной зоне Солнца. В рассматриваемой модели крайние осцилляторы связаны только со средним, а прямая связь между ними отсутствует. В отличие от классических моделей Курамото рассматриваемая система предполагает существенную асимметрию в связях каждого из осцилляторов с двумя другими. Мы исследуем, какое влияние на синхронизацию оказывает коэффициент связи, характеризующий асимметрию связей среднего осциллятора. Необходимое и достаточное условия синхронизации в этой работе выписываются аналитически и получаются отличными от достаточных условий синхронизации в классической (симметричной) модели. Мы формулируем обратную задачу восстановления коэффициентов связи из фазовой разницы крайних осцилляторов при известных естественных частотах. Восстановление проводится в предположении синхронизации. Получено, что коэффициенты связи с точностью до знака восстанавливаются для любого значения коэффициента несимметрии среднего осциллятора. Мы исследуем, как меняется график зависимости суммарной связи от коэффициента несимметрии при изменении разности фаз крайних осцилляторов, а также в особых случаях совпадающих или сильно отличающихся естественных частот. В случае общего положения, при разности фаз крайних осцилляторов, близких к $\pi$, суммарная связь, соответствующая сильной асимметрии связей среднего осциллятора, оказывается меньше, чем в симметричном случае. Мы рассматриваем значения естественных частот, пересчитанные из скоростей меридионального потока Солнца. В зависимости от интерпретации данных гелиосейсмологии мы получаем два случая: случай общего положения, соответствующий наблюдениям средней ячейки, и особый случай, соответствующий наблюдениям нижней ячейки. Однозначное (с точностью до знака) восстановление коэффициентов связи в случае слабой суммарной связи возможно только в случае общего положения. В заключении делаются выводы о возможности использования курамотовских моделей с асимметрией связей, относящихся к одному осциллятору, для моделирования слабо связанных систем, к каким, по всей видимости, относится солнечная меридиональная циркуляция.

On the generic complete synchronization of the discrete Kuramoto model

We study the emergent behavior of discrete-time approximation of the finite-dimensional Kuramoto model. Compared to Zhang and Zhu's recent work in [ 38 ], we do not rely on the consistency of one-step foward Euler scheme but analyze the discrete model directly to obtain sharper and more explicit result. More precisely, we present the optimal condition for the convergence and order preserving for identical oscilators with generic initial data. Then, we give the exact convergence rate of the identical oscillators to their limit under the reasonable assuption on time step. Finally, we provide an alternative proof of the asymptotic phase-locking of nonidentical oscillators which can be applied whenever the given Lyapunov functional is continuous and all zeros are isolated.

Synchronization of a Kuramoto-like model for power grids with frustration

We discuss the complete synchronization for a Kuramoto-like model for power grids with frustration. For identical oscillators without frustration, it will converge to complete phase and frequency synchronization exponentially fast if the initial phases are distributed in a half circle. For nonidentical oscillators with frustration, we present a framework leading to complete frequency synchronization where the initial phase configurations are located inside the half of a circle. Our estimates are based on the monotonicity arguments of extremal phase and frequency.

Quantum description of linearly coupled harmonic oscillator systems using oblique coordinates

In this article we extend our previous quantum-mechanical treatment of the system of identical harmonic oscillators linearly coupled in the kinetic and potential energies using oblique coordinates (Zúñiga et al 2019 J. Phys. B: At. Mol. Opt. Phys. 52 055101) to the general system of coupled non-identical harmonic oscillators. Oblique coordinates are obtained by making non-orthogonal rotations of the original coordinates that convert the matrix representation of the quadratic Hamiltonian operator into a block diagonal matrix. Accordingly, we derive the analytical formula for the oblique rotation angles, and obtain the expressions, also analytical, for the energy levels and eigenfunctions of the system in terms of the oblique parameters. We also show that oblique coordinates are in fact dependent on one of the rotational angles whose value can be freely chosen, so they form, in fact, a continuous set of coordinates that are especially flexible in dealing with more complex vibrational systems. To illustrate this, we make a numerical application to a system of kinetically coupled Barbanis oscillators, which clearly shows the advantages of using oblique coordinates to determine the energy levels and wave functions of the system variationally.

Remote synchronization in human cerebral cortex network with identical oscillators

Remote synchronization (RS) has recently received an increasing interest in the systems of nonidentical oscillators, but little attention has been paid to the systems of identical oscillators such as the case of human cerebral cortex where RS takes a key role in the functions of brain. Based on the real network of human cerebral cortex, we here show that RS can be also observed in the systems of identical oscillators, provided that appropriate time delay is considered. We interestingly find that RS may appear in different local places of cerebral cortex and thus results in a diversity of patterns, i.e., a new framework of multiple starlike graphs connected by common leaf nodes. To understand its mechanism, we present a model of two starlike graphs connected by common leaf nodes. We further show that the common leaf nodes take a key role for the emergence of RS in the new framework. A theoretical analysis is given. These findings may be helpful to understand the segregation and integration processes of information transmission in brain.

Fixed-time event-triggered synchronization of a multilayer Kuramoto-oscillator network

This paper investigates the synchronization problem of the Kuramoto-oscillator network with non-identical oscillators. The fixed-time event-triggered synchronization control strategies are developed for phase agreement and frequency synchronization under both continuous and intermittent communication. With the developed fixed-time controller, the synchronization can be achieved within a pre-defined time for any initial phase of each oscillator. The event-triggered mechanism avoids continuous controller update and data transmission, which significantly saves the computation and communication resources. Furthermore, theoretical analysis shows that the fixed-time convergence can be guaranteed and the Zeno behavior is avoided by the proposed methods. The numerical simulations of each situation also verify the effectiveness of the proposed synchronization control strategies.

Emergence of synchronous behaviors for the Schrödinger–Lohe model with frustration

We introduce the Schrödinger–Lohe model (S–L) with interaction frustration which is a toy model for quantum synchronization, and study its emergent collective dynamics. Our proposed model might not capture quantum entanglement which is one of genuine quantum phenomenon as it is, however it exhibits collective synchronous behaviors and can be reduced to the Kuramoto model with frustration in the case of spatial homogeneity and unit mass. For the emergent dynamics of the model, we present two sufficient frameworks leading to the complete and practical synchronizations for identical and non-identical oscillators, respectively, in terms of frustration function, system parameters and initial data. When the frustration functions admit real diagonal perturbations, complete synchronization emerges for identical ensemble with some class of initial data, in contrast, for non-identical ensemble, we can expect practical synchronization in which the diameter for wave functions can be controlled by tuning the coupling strength. For purely imaginary valued frustrations, we analytically show that complete or practical synchronization cannot occur for generic initial data. This illustrates that the S–L model exhibits some interesting dynamic patterns different from the Kuramoto synchronization. We provide several numerical examples which manifest periodic behaviors, reminiscent of the Kuramoto model with cosine coupling.

Cluster synchronization in networked nonidentical chaotic oscillators.

In exploring oscillator synchronization, a general observation is that as the oscillators become nonidentical, e.g., introducing parameter mismatch among the oscillators, the propensity for synchronization will be deteriorated. Yet in realistic systems, parameter mismatch is unavoidable and even worse in some circumstances, the oscillators might follow different types of dynamics. Considering the significance of synchronization to the functioning of many realistic systems, it is natural to ask the following question: Can synchronization be achieved in networked oscillators of clearly different parameters or dynamics? Here, by the model of networked chaotic oscillators, we are able to demonstrate and argue that, despite the presence of parameter mismatch (or different dynamics), stable synchronization can still be achieved on symmetric complex networks. Specifically, we find that when the oscillators are configured on the network in such a way that the symmetric nodes have similar parameters (or follow the same type of dynamics), cluster synchronization can be generated. The stabilities of the cluster synchronization states are analyzed by the method of symmetry-based stability analysis, with the theoretical predictions in good agreement with the numerical results. Our study sheds light on the interplay between symmetry and cluster synchronization in complex networks and give insights into the functionalities of realistic systems where nonidentical nonlinear oscillators are presented and cluster synchronization is crucial.

Synchronization in ensembles of delay-coupled nonidentical neuronlike oscillators

We study both numerically and experimentally the synchronization in an ensemble of nonidentical neuronlike oscillators described by the FitzHugh–Nagumo equations. The cases of constant values of time-delayed couplings between the oscillators and adaptively controlled values of time-delayed couplings are considered. For the experimental study of the ensemble of neuronlike oscillators, we construct a radio engineering setup, in which the ability to specify both constant values and adaptively tuned values of couplings between the oscillators is implemented. Moreover, it is possible to specify an arbitrary architecture and type of dynamical couplings between oscillators in the setup. By the example of a system of two bidirectionally coupled nonidentical oscillators and a ring consisting of ten unidirectionally coupled nonidentical FitzHugh–Nagumo systems, it is shown that the using of an adaptively controlled time-delayed coupling allows one to achieve the in-phase synchronization of all oscillators in the ensemble even in the case of a large parameter mismatch. The results obtained in the physical experiment are in good agreement with the results of the numerical simulation.

Synchronization in alternating linear chains of vortex-based spin-torque oscillators

Abstract The dynamic properties of magnetic vortices in single nanopillars, such as the gyrotropic orbital radius and frequency, have been shown to be well-modeled by the Thiele equation. Similarly, a set of coupled Thiele equations also describe synchronization of two or more interacting free layer vortices. Here the Thiele equations are used to model vortex dynamics in linear chains of alternating radius (where nearest-neighbor free layer have different radii) free layer oscillators, which is a system of interacting non-identical oscillators that can synchronize and lock onto a common frequency. For vortices interacting as magnetic dipoles it is shown that there are critical nanopillar separations below which the oscillator chain will synchronize. As the nanopillar separation increases, it is noticed that there is a narrow region of parameter space where synchronization is temporarily lost owing to phase slips and at larger separation synchronization is lost completely. The relation between the critical separation and the free layer radii difference is also estimated analytically.

Finite-Time and Fixed-Time Synchronization of Kuramoto-Oscillator Network With Multiplex Control

In this paper, the finite-time and fixed-time synchronization of a generalized Kuramoto model with identical and nonidentical oscillators are investigated, respectively. Given a (weak) coupling strength or initial phases dispersed over a circle, the Kuramoto-oscillator network can achieve phase-frequency synchronization within a finite time, and the upper bound of synchronization time can be estimated. Compared with the finite-time synchronization where the convergence rate relies on the initial conditions, the settling time of fixed-time synchronization can be adjusted to desired values in advance regardless of initial configurations. Deploying a novel multiplex control, the local and global connectivity conditions for realizing finite-time and fixed-time phase-frequency synchronization are derived successively based on the Lyapunov stability theory, and numerical examples verify their effectiveness and feasibility.

A graphical approach to estimate the critical coupling strength for Kuramoto networks.

The Kuramoto model is an archetypal model for studying synchronization in groups of nonidentical oscillators. Each oscillator is imbued with its own personal inherent driving frequency and experiences attractive coupling forces toward all the other oscillators in the system. As the coupling increases, there exists a minimal coupling strength called the critical coupling beyond which the system moves in a collective rhythm. A unified approach for creating approximations of the critical coupling is created. It is based on an interpretation of a measurement of phase synchronization among the oscillators (the order parameter) as a function of the coupling strength. The approach allows a graphical way to develop new approximations that are provably, strict lower bounds. It is shown that several of the critical coupling bounds that have been previously studied can be interpreted in this unified framework. In addition, a process based on fixed point sampling is introduced that converts upper bounds for the critical coupling into associated lower bounds.