Generalized Kuramoto Model Research Articles

Two-step and explosive synchronization in frequency-weighted Kuramoto model

We explore the dynamics of interacting phase oscillators in the generalized Kuramoto model with frequency-weighted couplings, focusing on the interplay of frequency distribution and network topology on the nature of transition to synchrony. We explore the impact of heterogeneity in the network topology and the frequency distribution. Our analysis includes unimodal (Gaussian, truncated Gaussian, and uniform) and bimodal frequency distributions. For a unimodal Gaussian distribution, we observe that in comparison to fully-connected network, the competition between topological and dynamical hubs hinders the transition to synchrony in the scale-free network, though explosive synchronization eventually happens. However, in the absence of very large frequencies, the transition is gradual. While uniform frequency distributions lead to explosive synchronization. In bimodal distributions, narrow distribution produce a two-step transition. In this case, central frequencies dominate the dynamics, overshadowing the topological features of the network. For wider bimodal distributions, scale-free network exhibits a gradual increase in the order parameter, whereas in fully-connected networks a first-order transition happens. These results specifically elucidate the mechanisms driving two-step and explosive synchronization in frequency-weighted Kuramoto models, offering new insights into managing synchronization phenomena in complex networks like power grids, neural systems, and social systems.

Averaging data on the unit hypersphere

Averaging data on the unit sphere S d (also called a unit hypersphere) is a common problem in computer vision, robotics and other fields, with applications ranging from motion planning to DNA modelling. In this paper, we introduce a new method for averaging data represented as points on the unit sphere S d−1 using the d-dimensional generalized Kuramoto model. Our method is verified on a range of benchmark data sets and compared with common data averaging algorithms. Also, we showcase the applicability of this method for solving rotation averaging problem.

Interpolation on the unit hypersphere using the n-dimensional generalized Kuramoto model

Computer graphics, robotics, and physics are one of the many domains where interpolation on the unit sphere S n (often called a unit hypersphere or unit n-sphere) plays a crucial role. In this paper, we introduce a novel approach for achieving smooth and precise interpolation on the unit sphere S n−1 using the n-dimensional generalized Kuramoto model. The proposed algorithm finds the shortest and most direct path between two points on that non-Euclidean manifold. Our simulation results demonstrate that it achieves performance comparable to that of a Spherical Linear Interpolation algorithm. Also, the paper proposes the application of our algorithm in the interpolation of rotations that are presented in the form of four-dimensional data.

Chimera dynamics of generalized Kuramoto–Sakaguchi oscillators in two-population networks

Chimera dynamics, an intriguing phenomenon of coupled oscillators, is characterized by the coexistence of coherence and incoherence, arising from a symmetry-breaking mechanism. Extensive research has been performed in various systems, focusing on a system of Kuramoto–Sakaguchi (KS) phase oscillators. In recent developments, the system has been extended to the so-called generalized Kuramoto model, wherein an oscillator is situated on the surface of an M-dimensional unit sphere, rather than being confined to a unit circle. In this paper, we exploit the model introduced in Tanaka (2014 New. J. Phys. 16 023016) where the macroscopic dynamics of the system was studied using the extended Watanabe–Strogatz transformation both for real and complex spaces. Considering two-population networks of the generalized KS oscillators in 2D complex spaces, we demonstrate the existence of chimera states and elucidate different motions of the order parameter vectors depending on the strength of intra-population coupling. Similar to the KS model on the unit circle, stationary and breathing chimeras are observed for comparatively strong intra-population coupling. Here, the breathing chimera changes their motion upon decreasing intra-population coupling strength via a global bifurcation involving the completely incoherent state. Beyond that, the system exhibits periodic alternation of the two order parameters with weaker coupling strength. Moreover, we observe that the chimera state transitions into a componentwise aperiodic dynamics when the coupling strength weakens even further. The aperiodic chimera dynamics emerges due to the breaking of conserved quantities that are preserved in the stationary, breathing and alternating chimera states. We provide a detailed explanation of this scenario in both the thermodynamic limit and for finite-sized ensembles. Furthermore, we note that an ensemble in 4D real spaces demonstrates similar behavior.

Improved numerical scheme for the generalized Kuramoto model

We present an improved and more accurate numerical scheme for a generalization of the Kuramoto model of coupled phase oscillators to the three-dimensional space. The present numerical scheme relies crucially on our observation that the generalized Kuramoto model corresponds to particles on the unit sphere undergoing rigid body rotations with position-dependent angular velocities. We demonstrate that our improved scheme is able to reproduce known analytic results and capture the expected behavior of the three-dimensional oscillators in various cases. On the other hand, we find that the conventional numerical method, which amounts to a direct numerical integration with the constraint that forces the particles to be on the unit sphere at each time step, may result in inaccurate and misleading behavior especially in the long time limit. We analyze in detail the origin of the discrepancy between the two methods and present the effectiveness of our method in studying the limit cycle of the Kuramoto oscillators.

Generalization of the Kuramoto model to the Winfree model by a symmetry breaking coupling

We construct a nontrivial generalization of the paradigmatic Kuramoto model by using an additional coupling term that explicitly breaks its rotational symmetry resulting in a variant of the Winfree model. Consequently, we observe the characteristic features of the phase diagrams of both the Kuramoto model and the Winfree model depending on the degree of the symmetry breaking coupling strength for unimodal frequency distribution. The phase diagrams of both the Kuramoto and the Winfree models resemble each other for symmetric bimodal frequency distribution for a range of the symmetry breaking coupling strength except for region shift and difference in the degree of spread of the macroscopic dynamical states and bistable regions. The dynamical transitions in the bistable states are characterized by an abrupt (first-order) transition in both the forward and reverse traces. For asymmetric bimodal frequency distribution, the onset of bistable regions depends on the degree of the asymmetry. Large degree of the symmetry breaking coupling strength promotes the synchronized stationary state, while a large degree of heterogeneity, proportional to the separation between the two central frequencies, facilitates the spread of the incoherent and standing wave states in the phase diagram for a low strength of the symmetry breaking coupling. We deduce the low-dimensional equations of motion for the complex order parameters using the Ott-Antonsen ansatz for both unimodal and bimodal frequency distributions. We also deduce the Hopf, pitchfork, and saddle-node bifurcation curves from the evolution equations for the complex order parameters mediating the dynamical transitions. Simulation results of the original discrete set of equations of the generalized Kuramoto model agree well with the analytical bifurcation curves.

Dynamics of a Kuramoto Model with Two-Body and Three-Body Interactions

In this paper, in order to study the dynamic behavior of the three-body interaction, the generalized Kuramoto model with bimodal frequency distribution under the joint interaction of two-body and three-body is proposed. The comparative numerical results of the phase synchronization paths of the three-body interaction under different coupling strengths show that the three-body interaction can transform the continuous transition process into the first-order transition process. Interestingly, the change from continuous to discontinuous transition due to the variation of the coupling strength of the three-body interaction is similar to the shape of the bimodal distribution of the natural frequency. The critical coupling strength of the two-body interaction of synchronous transition is derived from the Ott–Antonsen–Ansatz method. The numerical results are consistent with the theoretical ones. The findings help our understanding of the transformation process from being continuous to discontinuous.

Sufficient conditions for asymptotic phase-locking to the generalized Kuramoto model

<p style='text-indent:20px;'>In this paper, we introduce a generalized Kuramoto model and provide several sufficient conditions leading to asymptotic phase-locking. The proposed generalized Kuramoto model incorporates relativistic Kuramoto type models which can be derived from the relativistic Cucker-Smale (RCS) on the unit sphere via suitable approximations. For asymptotic phase-locking, we present several sufficient frameworks leading to complete synchronization in terms of initial data and system parameters. For the relativistic Kuramoto model, we show that it reduces to the Kuramoto model in a finite time interval, as the speed of light tends to infinity. Moreover, for some admissible initial data, nonrelativistic limit can be made uniformly in time. We also provide several numerical examples for two approximations of the relativistic Kuramoto model, and compare them with analytical results.</p>

Synchronization of the generalized Kuramoto model with time delay and frustration

<abstract><p>We studied the collective behaviors of the time-delayed Kuramoto model with frustration under general network topology. For the generalized Kuramoto model with the graph diameter no greater than two and under a sufficient regime in terms of small time delay and frustration and large coupling strength, we showed that the complete frequency synchronization occurs exponentially fast when the initial configuration is distributed in a half circle. We also studied a complete network, which is a small perturbation of all-to-all coupling, as well as presented sufficient frameworks leading to the exponential emergence of frequency synchronization for the initial data confined in a half circle.</p></abstract>

Abrupt desynchronization and abrupt transition to [formula omitted]-state in globally coupled oscillator simplexes with contrarians and conformists

We study the phase transitions in a generalized Kuramoto model with contrarians and conformists along with a higher order coupling encoded simplex. Incoherent state persists for a large fraction of contrarians, while there is a continuous transition from the former to π-state and traveling wave state as the fraction of conformists increases under the high proportion of the pair-wise coupling and for a higher coupling strength of the conformists than contrarians. Hong and Strogatz (2011) showed the abrupt transition to the π-state when the contrarians coupling strength supersede in a purely pair-wise coupling. A large proportion of higher order coupling induces the abrupt transition to the π-state even when the contrarians coupling strength is feeble than that of the conformists, destabilizes the traveling wave state, and stabilizes the incoherent state in the entire phase diagram despite the presence of a large fraction of the conformists, which coexists with the π-state. Consequently, one can observe an abrupt desynchronization transition to the incoherent state during the backward trace, while the system remains in the incoherent state during the forward trace. The range of the bistable region between the incoherent state and the π-state increases with the proportion of the higher order coupling to a maximum until the abrupt transition to the π-state ceases to exist. We deduce the pitchfork and saddle–node bifurcation curves across which the incoherent state and the π-state lose their stability from the evolution equation for the low-dimensional dynamics corresponding to the macroscopic order parameters.

Comment on “Low-dimensional behavior of generalized Kuramoto model” by S. Ameli and K. A. Samani

Comment on “Low-dimensional behavior of generalized Kuramoto model” by S. Ameli and K. A. Samani

Synchronization dynamics of phase oscillator populations with generalized heterogeneous coupling

The Kuramoto model, serving as a paradigmatic tool, has been used to shed light on the collective behaviors in large ensembles of coupled dynamic agents. It is well known that the model displays a second-order(continuous) phase transition towards synchrony by increasing the homogeneous(uniform) global coupling strength. Recently, there is a great interest in investigating the effects of the heterogeneous patterns on the collective dynamics of coupled oscillator systems. Here, we consider a generalized Kuramoto model of globally coupled phase oscillators with quenched disorder in their natural frequencies and coupling strength. By correlating these two types of inhomogeneity, we systematically explore the impacts of heterogeneous structure, the correlation exponent, and the intrinsic frequency distribution on the synchronized dynamics. We develop an analytic framework for capturing the essential dynamic properties involved in synchronization transition. In particular, we demonstrate that the forward critical threshold describing the onset of synchronization is unaffected by the location of heterogeneity, which, however, does depend crucially on the correlation exponent and the form of frequency distribution. Furthermore, we reveal that the backward critical points featuring the desynchronization transition are significantly shaped by all the considered effects, thereby enhancing or weakening the ability of synchronization transition (bifurcation). Our investigation is a step forward in highlighting the importance of heterogeneous pattern presented in the complex systems, and could, thus, provide significant insights for designing the strategy of inducing and controlling synchronization.

Low-dimensional behavior of generalized Kuramoto model

Low-dimensional behavior of generalized Kuramoto model

Almost global convergence to practical synchronization in the generalized Kuramoto model on networks over the n-sphere

From the flashing of fireflies to autonomous robot swarms, synchronization phenomena are ubiquitous in nature and technology. They are commonly described by the Kuramoto model that, in this paper, we generalise to networks over n-dimensional spheres. We show that, for almost all initial conditions, the sphere model converges to a set with small diameter if the model parameters satisfy a given bound. Moreover, for even n, a special case of the generalized model can achieve phase synchronization with nonidentical frequency parameters. These results contrast with the standard n = 1 Kuramoto model, which is multistable (i.e., has multiple equilibria), and converges to phase synchronization only if the frequency parameters are identical. Hence, this paper shows that the generalized network Kuramoto models for n ≥ 2 displays more coherent and predictable behavior than the standard n = 1 model, a desirable property both in flocks of animals and for robot control.

Spatiotemporal Patterns of Oscillatory Electrochemical Reactions in the Presence of a Low-Pass Filter

In many electrochemical measurements, the measured signals are often filtered to remove experimental noise, for example, with a low-pass filter (LPF). At the same time, various electrical coupling (potential drop in the electrolyte) and feedback (galvanostatic constraints) effects can generate spatiotemporal patterns. In this contribution, we study the synchronization patterns of electrochemical oscillators induced by feedback in the presence of a low-pass filter. Close to a Hopf bifurcation, without the filter, as expected[1], the synchronization transition took place above a critical coupling strength following a second-order phase (Kuramoto) transition. With LPF, the transition occurred through explosive synchronization, which corresponds to a first-order phase transition and multistability between synchronized and desynchronized states. These experimental results confirm recent theoretical development on the impact of LPF on the synchronization of Stuart-Landau oscillators[2]. Further away from Hopf bifurcation, the LPF can generate patterns with one- and two-cluster states from a desynchronized state. The cluster formation and stability can be understood using a theory that estimates the LPF induced amplitude attenuation and phase shift of the coupling functions in the phase model of the oscillatory process. Our study points to the importance of using filtering in engineering synchronization states of oscillatory electrochemical reactions. Kiss, I.Z., Y. Zhai, and J.L. Hudson, Emerging Coherence in a Population of Chemical Oscillators. Science, 2002. 296(5573): p. 1676.Zou, W., M. Zhan, and J. Kurths, Phase transition to synchronization in generalized Kuramoto model with low-pass filter. Physical Review E, 2019. 100(1): p. 012209.

Frequency-Resolved Functional Connectivity: Role of Delay and the Strength of Connections.

The brain functional network extracted from the BOLD signals reveals the correlated activity of the different brain regions, which is hypothesized to underlie the integration of the information across functionally specialized areas. Functional networks are not static and change over time and in different brain states, enabling the nervous system to engage and disengage different local areas in specific tasks on demand. Due to the low temporal resolution, however, BOLD signals do not allow the exploration of spectral properties of the brain dynamics over different frequency bands which are known to be important in cognitive processes. Recent studies using imaging tools with a high temporal resolution has made it possible to explore the correlation between the regions at multiple frequency bands. These studies introduce the frequency as a new dimension over which the functional networks change, enabling brain networks to transmit multiplex of information at any time. In this computational study, we explore the functional connectivity at different frequency ranges and highlight the role of the distance between the nodes in their correlation. We run the generalized Kuramoto model with delayed interactions on top of the brain's connectome and show that how the transmission delay and the strength of the connections, affect the correlation between the pair of nodes over different frequency bands.

Stability and bifurcation of collective dynamics in phase oscillator populations with general coupling.

The Kuramoto model serves as an illustrative paradigm for studying the synchronization transitions and collective behaviors in large ensembles of coupled dynamical units. In this paper, we present a general framework for analytically capturing the stability and bifurcation of the collective dynamics in oscillator populations by extending the global coupling to depend on an arbitrary function of the Kuramoto order parameter. In this generalized Kuramoto model with rotation and reflection symmetry, we show that all steady states characterizing the long-term macroscopic dynamics can be expressed in a universal profile given by the frequency-dependent version of the Ott-Antonsen reduction, and the introduced empirical stability criterion for each steady state degenerates to a remarkably simple expression described by the self-consistent equation [Iatsenko et al., Phys. Rev. Lett. 110, 064101 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.064101]. Here, we provide a detailed description of the spectrum structure in the complex plane by performing a rigorous stability analysis of various steady states in the reduced system. More importantly, we uncover that the empirical stability criterion for each steady state involved in the system is completely equivalent to its linear stability condition that is determined by the nontrivial eigenvalues (discrete spectrum) of the linearization. Our study provides a new and widely applicable approach for exploring the stability properties of collective synchronization, which we believe improves the understanding of the underlying mechanisms of phase transitions and bifurcations in coupled dynamical networks.

Dynamics of the generalized Kuramoto model with nonlinear coupling: Bifurcation and stability.

We systematically study dynamics of a generalized Kuramoto model of globally coupled phase oscillators. The coupling of modified model depends on the fraction of phase-locked oscillators via a power-law function of the Kuramoto order parameter r through an exponent α, such that α=1 corresponds to the standard Kuramoto model, α<1 strengthens the global coupling, and the global coupling is weakened if α>1. With a self-consistency approach, we demonstrate that bifurcation diagrams of synchronization for different values of α are thoroughly constructed from two parametric equations. In contrast to the case of α=1 with a typical second-order phase transition to synchronization, no phase transition to synchronization is predicted for α<1, as the onset of partial locking takes place once the coupling strength K>0. For α>1, we establish an abrupt desynchronization transition from the partially (fully) locked state to the incoherent state, whereas there is no counterpart of abrupt synchronization transition from incoherence to coherence due to that the incoherent state remains linearly neutrally stable for all K>0. For each case of α, by performing a standard linear stability analysis for the reduced system with Ott-Antonsen ansatz, we analytically derive the continuous and discrete spectra of both the incoherent state and the partially (fully) locked states. All our theoretical results are obtained in the thermodynamic limit, which have been well validated by extensive numerical simulations of the phase-model with a sufficiently large number of oscillators.

Hysteretic behavior of spatially coupled phase-oscillators

Motivated by phenomena related to biological systems such as the synchronously flashing swarms of fireflies, we investigate a network of phase oscillators evolving under the generalized Kuramoto model with inertia. A distance-dependent, spatial coupling between the oscillators is considered. Zeroth and first order kernel functions with finite kernel radii were chosen to investigate the effect of local interactions. The hysteretic dynamics of the synchronization depending on the coupling parameter was analyzed for different kernel radii. Numerical investigations demonstrate that (1) locally locked clusters develop for small coupling strength values, (2) the hysteretic behavior vanishes for small kernel radii, (3) the ratio of the kernel radius and the maximal distance between the oscillators characterizes the behavior of the network.

High-dimensional Kuramoto models on Stiefel manifolds synchronize complex networks almost globally

The Kuramoto model of coupled phase oscillators is often used to describe synchronization phenomena in nature. Some applications, e.g., quantum synchronization and rigid-body attitude synchronization, involve high-dimensional Kuramoto models where each oscillator lives on the n-sphere or SO(n). These manifolds are special cases of the compact, real Stiefel manifold St(p,n). Using tools from optimization and control theory, we prove that the generalized Kuramoto model on St(p,n) converges to a synchronized state for any connected graph and from almost all initial conditions provided (p,n) satisfies p≤23n−1 and all oscillator frequencies are equal. This result could not have been predicted based on knowledge of the Kuramoto model in complex networks over the circle. In that case, almost global synchronization is graph dependent; it applies if the network is acyclic or sufficiently dense. This paper hence identifies a property that distinguishes many high-dimensional generalizations of the Kuramoto models from the original model.