We study the synchronization of oscillators with inertias and phase shifts, namely the second-order Kuramoto-Sakaguchi model. Using the self-consistent method, we find that the effect of inertia is the introduction of effective phase shifts. The discontinuous synchronization transition of the Kuramoto-Sakaguchi model changes to a continuous one when the value of inertia is small. In addition, we find a new synchronization process, in which with increasing coupling strength the system reaches an oscillating state instead of complete synchronization due to the cross-effect of phase shifts and inertias. Through numerical simulations, the same type of synchronization process is also found for oscillators in complex networks, including scale-free, small-world and random networks.
The Ott--Antonsen ansatz is a powerful tool to extract the behaviors of coupled phase oscillators, but it imposes a strong restriction on the initial condition. Herein, a systematic extension of the Ott--Antonsen ansatz is proposed to relax the restriction, enabling the systematic approximation of the behavior of a globally coupled phase oscillator system with an arbitrary initial condition. The proposed method is validated on the Kuramoto--Sakaguchi model of identical phase oscillators. The method yields cluster and chimera-like solutions that are not obtained by the conventional ansatz.
We derive the Kuramoto-Sivashinsky-type phase equation from the Kuramoto-Sakaguchi-type phase model via the Ott-Antonsen-type complex amplitude equation and demonstrate heterogeneity-induced collective-phase turbulence in nonlocally coupled individual-phase oscillators.
We study a variant of Kuramoto-Sakaguchi model in which oscillators are divided into two groups, each characterized by its coupling constant and phase lag. Specifically, we consider the case that one coupling constant is positive and the other negative, and calculate numerically the traveling speed of two clusters emerging in the system and average separation between them as well as the order parameters for positive and negative oscillators, as the two coupling constants, phase lags, and the fraction of positive oscillators are varied. An expression explaining the dependence of the traveling s…
We investigate transition to synchrony in degree-frequency correlated Sakaguchi-Kuramoto (SK) model on complex networks both analytically and numerically. We analytically derive self-consistent equations for group angular velocity and order parameter for the model in the thermodynamic limit. Using the self-consistent equations we investigate transition to synchronization in SK model on uncorrelated scale-free (SF) and Erdős-Rényi (ER) networks in detail. Depending on the degree distribution exponent ($γ$) of SF networks and phase-frustration parameter, the population undergoes from first order…
We have studied the dynamics of the paradigmatic Kuramoto-Sakaguchi model of identical coupled phase oscilla- tors with various kinds of time-dependent connectivity using Eulerian discretization. We first explore the parameter spaces for various types of collective states using the phase plots of the two statistical quantities, namely the strength of incoherence and the discontinity measure. In the quasi-static limit of the changing of coupling range, we have observed how the system relaxes from one state to another and have identified a few interesting collective dynamical states along the wa…
The Kuramoto--Sakaguchi model is a modification of the well-known Kuramoto model that adds a phase-lag paramater, or "frustration" to a network of phase-coupled oscillators. The Kuramoto model is a flow of gradient type, but adding a phase-lag breaks the gradient structure, significantly complicating the analysis of the model. We present several results determining the stability of phase-locked configurations: the first of these gives a sufficient condition for stability, and the second a sufficient condition for instability. (In fact, the instability criterion gives a count, modulo 2, of the …
We explore the collective phase dynamics of Wien-bridge oscillators coupled resistively. We carefully analyze the behavior of two coupled oscillators, obtaining a transformation from voltage to effective phase. From the phase dynamics we show that the coupling can be quantitatively described by Sakaguchi's modification to the Kuramoto model. We also examine an ensemble of oscillators whose frequencies are taken from a flat distribution within a fixed frequency interval. We characterize in detail the synchronized cluster, its initial formation, as well as its effect on unsynchronized oscillator…
The Kuramoto-Sakaguchi model for coupled phase oscillators with phase-frustration is often studied in the thermodynamic limit of infinitely many oscillators. Here we extend a model reduction method based on collective coordinates to capture the collective dynamics of finite size Kuramoto-Sakaguchi models. We find that the inclusion of the effects of rogue oscillators is essential to obtain an accurate description, in contrast to the original Kuramoto model where we show that their effects can be ignored. We further introduce a more accurate ansatz function to describe the shape of synchronized…
Cortical regions without direct neuronal connections have been observed to exhibit synchronized dynamics. A recent empirical study has further revealed that such regions that share more common neighbors are more likely to behave coherently. To analytically investigate the underlying mechanisms, we consider that a set of n oscillators, which have no direct connections, are linked through m intermediate oscillators (called mediators), forming a complete bipartite network structure. Modeling the oscillators by the Kuramoto-Sakaguchi model, we rigorously prove that mediated remote synchronization,…
In this numerical work we have systematically studied the dynamical phase transitions in the Kuramoto- Sakaguchi model of synchronizing phase oscillators controlled by disorder in the Sakaguchi phases. We find out the numerical steady state phase diagrams for quenched and annealed kinds of disorder in the Sakaguchi parameters using the conventional order parameter and other statistical quantities like strength of incoherence and discontinuity measures. We have also considered the correlation profile of the local order parameter fluctuations in the various identified phases. The phase diagrams …
The celebrated Kuramoto model provides an analytically tractable framework to study spontaneous collective synchronization and comprises globally coupled limit-cycle oscillators interacting symmetrically with one another. The Sakaguchi-Kuramoto model is a generalization of the basic model that considers the presence of a phase-lag parameter in the interaction, thereby making it asymmetric between oscillator pairs. Here, we consider a further generalization, by adding an interaction that breaks the phase-shift symmetry of the model. The highlight of our study is the unveiling of a very rich bif…
We consider the Kuramoto-Sakaguchi model of identical coupled phase oscillators with a common noisy forcing. While common noise always tends to synchronize the oscillators, a strong repulsive coupling prevents the fully synchronous state and leads to a nontrivial distribution of oscillator phases. In previous numerical simulations, a formation of stable multicluster states has been observed in this regime. However we argue here, that because identical phase oscillators in the Kuramoto-Sakaguchi model form a partially integrable system according to the Watanabe-Strogatz theory, the formation of…
We consider chimera states of coupled identical phase oscillators where some oscillators are phase synchronized while others are desynchronized. It is known that chimera states of non-locally coupled Kuramoto--Sakaguchi oscillators in arrays of finite size are chaotic transients when the phase lag parameter $α\in (0, π/2)$; after a transient time, all the oscillators are phase synchronized, with the transient time increasing exponentially with the number of oscillators. In this work, we consider a small array of six non-locally coupled oscillators with the phase lag parameter $α\in (π/2, π)$ i…
We study the dynamics of a generalized version of the famous Kuramoto-Sakaguchi coupled oscillator model. In the classic version of this system, all oscillators are governed by the same ODE, which depends on the order parameter of the oscillator configuration. The order parameter is the arithmetic mean of the configuration of complex oscillator phases, multiplied by some constant complex coupling factor. In the generalized model we consider, the order parameter is allowed to be any complex linear combination of the complex oscillator phases, so the oscillators are no longer necessarily weighte…
We investigate transition to synchronization in Sakaguchi-Kuramoto (SK) model on complex networks analytically as well as numerically. Natural frequencies of a percentage ($f$) of higher degree nodes of the network are assumed to be correlated with their degrees and that of the remaining nodes are drawn from some standard distribution namely Lorenz distribution. The effects of variation of $f$ and phase frustration parameter $α$ on transition to synchronization are investigated in detail. Self-consistent equations involving critical coupling strength ($λ_c$) and group angular velocity ($Ω_c$) …
We consider a model of three interacting sets of decision-making agents, labeled Blue, Green and Red, represented as coupled phased oscillators subject to frustrated synchronisation dynamics. The agents are coupled on three networks of differing topologies, with interactions modulated by different cross-population frustrations, internal and cross-network couplings. The intent of the dynamic model is to examine the degree to which two of the groups of decision-makers, Blue and Red, are able to realise a strategy of being ahead of each others' decision-making cycle while internally seeking synch…
We examine a model of two interacting populations of phase oscillators labelled `Blue' and `Red'. To this we apply tempered stable Lévy noise, a generalisation of Gaussian noise where the heaviness of the tails parametrised by a power law exponent $α$ can be controlled by a tempering parameter $λ$. This system models competitive dynamics, where each population seeks both internal phase synchronisation and a phase advantage with respect to the other population, subject to exogenous stochastic shocks. We study the system from an analytic and numerical point of view to understand how the phase la…
A recently proposed dimensional reduction approach for studying synchronization in the Kuramoto model is employed to build optimal network topologies to favor or to suppress synchronization. The approach is based in the introduction of a collective coordinate for the time evolution of the phase locked oscillators, in the spirit of the Ott-Antonsen ansatz. We show that the optimal synchronization of a Kuramoto network demands the maximization of the quadratic function $ω^T L ω$, where $ω$ stands for the vector of the natural frequencies of the oscillators, and $L$ for the network Laplacian matr…
We study two groups of nonidentical Kuramoto oscillators with differing frequency distributions. Coupling between the groups is repulsive, while coupling between oscillators of the same group is attractive. This asymmetry of interactions leads to an interesting synchronization behavior. For small coupling strength, the mean-fields of both groups resemble a more weakly coupled Kuramoto model. After increasing the coupling strength beyond a threshold, they reach the Bellerophon state of multiple clusters of averagely entrained oscillators. A further increase in the coupling strength then asympto…
An incorporation of higher-order interactions is known to lead an abrupt first-order transition to synchronization in otherwise smooth second-order one for pair-wise coupled systems. Here, we show that adaptation in higher-order coupling strength may yield completely different phenomena, notably, second-order transition to synchronization and tiered synchronization. Using the Ott-Antonsen approach, we perform rigorous theoretical calculations illustrating the origin of these emerging phenomena along with a complete description of all (un)stable states. Numerical simulations for limited-size ne…
The concept of "remote synchronization" (RS) was introduced in [Phys. Rev. E 85, 026208 (2012)], where synchronization in a star network of Stuart-Landau oscillators was investigated. In the RS regime therein described, the central hub served as a transmitter of information between peripheral nodes, while maintaining independent dynamics that were asynchronous with the rest of the network. One of the key conclusions of that paper was that RS cannot occur in pure phase-oscillator networks. Here, we show that the RS regime can exist in networks of Kuramoto oscillators, and that hub nodes can act…
We study the synchronization of Kuramoto oscillators on networks where only a fraction of them is subjected to a periodic external force. When all oscillators receive the external drive the system always synchronize with the periodic force if its intensity is sufficiently large. Our goal is to understand the conditions for global synchronization as a function of the fraction of nodes being forced and how these conditions depend on network topology, strength of internal couplings and intensity of external forcing. Numerical simulations show that the force required to synchronize the network wit…
We propose Moebius 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 non-identical oscillators, chimeras in two coupled populations of identical phase oscillators, and Kuramoto-Battogtokh chimeras on a ring, and demonstrate similarities and dif…
A fundamental understanding of synchronized behavior in multi-agent systems can be acquired by studying analytically tractable Kuramoto models. However, such models typically diverge from many real systems whose dynamics evolve under non-negligible resource constraints. Here we construct a system of coupled Kuramoto oscillators that consume or produce resources as a function of their oscillation frequency. At high coupling, we observe strongly synchronized dynamics, whereas at low coupling we observe independent oscillator dynamics, as expected from standard Kuramoto models. For intermediate c…
We numerically study the synchronization of an identical population of Kuramoto-Sakaguchi phase oscillators in Watts-Strogatz networks. We find that, unlike random networks, phase-shift could enhance the synchronization in small-world networks. We also observe abrupt phase transition with hysteresis at some values of phase shifts in small-world networks, signs of an explosive phase transition. Moreover, we report the emergence of Chimera states at some values of phase-shift close to the transition points, which consist of spatially coexisting synchronized and desynchronized domains.
We establish the existence of chimera states, simultaneously supporting synchronous and asynchronous dynamics, in a network consisting of two symmetrically linked star subnetworks consisting of identical oscillators with shear and Kuramoto--Sakaguchi coupling. We show that the chimera states may be metastable or asymptotically stable. If the intra-star coupling strength is of order $\varepsilon$, the chimera states persist on time scales at least of order $1/\varepsilon$ in general, and on time-scales at least of order $1/\varepsilon^2$ if the intra-star coupling is of Kuramoto--Sakaguchi type…
We show that an introduction of a phase parameter ($α$), with $0 \le α\le π/2$, in the interlayer coupling terms of multiplex networks of Kuramoto oscillators can induce explosive synchronization (ES) in the multiplexed layers. Along with the α values, the hysteresis width is determined by the interlayer coupling strength and the frequency mismatch between the mirror (inter-connected) nodes. A mean-field analysis is performed to support the numerical results. Similar to the earlier works, we find that the suppression of synchronization is accountable for the origin of ES. The robustness of ES …
We consider the problem of global synchronization in a large random network of Kuramoto oscillators where some of them are subject to an external periodically driven force. We explore a recently proposed dimensional reduction approach and introduce an effective two-dimensional description for the problem. From the dimensionally reduced model, we obtain analytical predictions for some critical parameters necessary for the onset of a globally synchronized state in the system. Moreover, the low dimensional model also allows us to introduce an optimization scheme for the problem. Our main conclusi…
This paper presents new methods and results on almost global synchronization of coupled Hopf nonlinear oscillators, which are commonly used as the dynamic model of engineered central pattern generators (CPGs). On balanced graphs, any positive coupling gain is proven to induce almost global asymptotic synchronization, and a threshold value for truly global exponential synchronization is also computed. Furthermore, a hierarchical connection between coupled Hopf oscillators and Kuramoto oscillators is identified. Finally, a new result on the synchronization of Kuramoto oscillators with arbitrary …
The cluster synchronization (CS) is a very important characteristic for the higher harmonic cou- pling Kuramoto system. A novel method from the symmetry transformation is provided, and it gives CS a profoundly mathematical explanation and clear physical annotation. Detailed numerical studies for the order parameters in various conditions confirm the theoretical predictions from this new view of the symmetry transformation. The work is very beneficial to the further study on CS in various systems.
We analyze two classes of Kuramoto models on spheres that have been introduced in previous studies. Our analysis is restricted to ensembles of identical oscillators with the global coupling. In such a setup, with an additional assumption that the initial distribution of oscillators is uniform on the sphere, one can derive equations for order parameters in closed form. The rate of synchronization in a real Kuramoto model depends on the dimension of the sphere. Specifically, synchronization is faster on higher-dimensional spheres. On the other side, real order parameter in complex Kuramoto model…
A system of nearest neighbors Kuramoto-like coupled oscillators placed in a ring is studied above the critical synchronization transition. We find a richness of solutions when the coupling increases, which exists only within a solvability region (SR). We also find that they posses different characteristics, depending on the section of the boundary of the SR where the solutions appear. We study the birth of these solutions and how they evolve when {K} increases, and determine the diagram of solutions in phase space.
We explore both analytically and numerically an ensemble of coupled phase-oscillators governed by a Kuramoto-type system of differential equations. However, we have included the effects of time-delay (due to finite signal-propagation speeds) and network plasticity (via dynamic coupling constants) inspired by the Hebbian learning rule in neuroscience. When time-delay and learning effects combine, novel synchronization phenomena are observed. We investigate the formation of spatio-temporal patterns in both one- and two-dimensional oscillator lattices with periodic boundary conditions and comment…
In this work, we study the synchronization of coupled phase oscillators on the underlying topology of scale-free networks. In particular, we assume that each network's component is an oscillator and that each interacts with the others following the Kuramoto model. We then study the onset of global phase synchronization and fully characterize the system's dynamics. We also found that the resynchronization time of a perturbed node decays as a power law of its connectivity, providing a simple analytical explanation to this interesting behavior.
A new collective behavior of resonant synchronization is discovered and the ability to retrieve information from brain memory is proposed based on this mechanism. We use modified Kuramoto phase oscillator to simulate the dynamics of a single neuron in self-oscillation state, and investigate the collective responses of a neural network, which is composed of $N$ globally coupled Kuramoto oscillators, to the external stimulus signals in a critical state just below the synchronization threshold of Kuramoto model. The input signals at different driving frequencies, which are used to denote differen…
The cluster synchronization is a very important characteristic for the higher harmonic coupling Kuramoto system. A novel transformation is provided, and it gives cluster synchronization by the periodic properties of the density function. The periodic properties of the density function also make the cluster sections' boundaries barrier-like, which helps to explain the sensitiveness of cluster synchronization on the initial conditions of the oscillators. Detailed numerical studies confirm the theoretical predictions from this new view of the symmetry transformation. The work is very beneficial t…
In his classical work, Kuramoto analytically described the onset of synchronization in all-to-all coupled networks of phase oscillators with random intrinsic frequencies. Specifically, he identified a critical value of the coupling strength, at which the incoherent state loses stability and a gradual build-up of coherence begins. Recently, Kuramoto's scenario was shown to hold for a large class of coupled systems on convergent families of deterministic and random graphs. Guided by these results, in the present work, we study several model problems illustrating the link between network topology…
Repulsive oscillator networks can exhibit multiple cooperative rhythms, including chimera and cluster splay states. Yet, understanding which rhythm prevails remains challenging. Here, we address this fundamental question in the context of Kuramoto-Sakaguchi networks of identical rotators with higher-order coupling. Through analysis and numerics, we show that three-cluster splay states with two distinct coherent clusters and a solitary oscillator are the prevalent rhythms in networks with an odd number of units. We denote such tripod patterns cyclops states with the solitary oscillator reminisc…
A modified Kuramoto model of synchronization in a finite discrete system of locally coupled oscillators is studied. The model consists of N oscillators with random natural frequencies arranged on a ring. It is shown analytically and numerically that finite-size systems may have many different synchronized stable solutions which are characterised by different values of the winding number. The lower bound for the critical coupling is given, as well as an algorithm for its exact calculation. It is shown that in general phase-locking does not lead to phase coherence in 1D.
Synchronization commonly occurs in many natural and man-made systems, from neurons in the brain to cardiac cells to power grids to Josephson junction arrays. Transitions to or out of synchrony for coupled oscillators depend on several factors, such as individual frequencies, coupling, interaction time delays and network structure-function relation. Here, using a generalized Kuramoto model of time-delay coupled phase oscillators with frequency-weighted coupling, we study the stability of incoherent and coherent states and the transitions to or out of explosive (abrupt, first-order like) phase s…
Using a recently introduced linear reformulation of the Kuramoto model of self-synchronizing oscillator systems (arXiv:0704.1166), we study a new class of analytically solvable oscillator systems defined by a particular coupling scheme. We show that these systems have a logarithimic scaling law in the vicinity of the critical point, which may be seen as anomalous with respect to the usual power-law behavior exhibited by the standard Kuramoto model.
