Kuramoto Oscillators Research Articles

The Kuramoto model on dynamic random graphs

We propose a Kuramoto model of coupled oscillators on a time-varying graph, whose dynamics are dictated by a Markov process in the space of graphs. The simplest representative is considering a base graph and then the subgraph determined by N independent random walks on the underlying graph. We prove a synchronisation result for solutions starting from a phase-cohesive set independent of the speed of the random walkers, an averaging principle and a global synchronisation result with high probability for sufficiently fast processes. We also consider Kuramoto oscillators in a dynamical version of the random conductance model.

Phase transitions in an adaptive network with the global order parameter adaptation.

We consider an adaptive network of Kuramoto oscillators with purely dyadic coupling, where the adaption is proportional to the degree of the global order parameter. We find only the continuous transition to synchronization via the pitchfork bifurcation, an abrupt synchronization (desynchronization) transition via the pitchfork (saddle-node) bifurcation resulting in the bistable region R_{1}. This is a smooth continuous transition to a weakly synchronized state via the pitchfork bifurcation followed by a subsequent abrupt transition to a strongly synchronized state via a second saddle-node bifurcation along with an abrupt desynchronization transition via the first saddle-node bifurcation resulting in the bistable region R_{2} between the weak and strong synchronization. The transition goes from the bistable region R_{1} to the bistable region R_{2}, and transition from the incoherent state to the bistable region R_{2} as a function of the coupling strength for various ranges of the degree of the global order parameter and the adaptive coupling strength. We also find that the phase-lag parameter enlarges the spread of the weakly synchronized state and the bistable states R_{1} and R_{2} to a large region of the parameter space. We also derive the low-dimensional evolution equationsfor the global order parameters using the Ott-Antonsen ansatz. Further, we also deduce the pitchfork, first and second saddle-node bifurcation conditions, which is in agreement with the simulation results.

Individual differences of limitation to extract beat from Kuramoto coupled oscillators: Transition from beat-based tapping to frequent tapping with weaker coupling.

Musical performers synchronize to each other despite differences in sound-onset timings which reflect each musician's sense of the beat. A dynamical system of Kuramoto oscillators can simulate this spread of onsets at varying levels of temporal alignment with a variety of tempo and sound densities which also influence individual abilities for beat extraction. Here, we examined how people's sense of beat emerges when tapping with Kuramoto oscillators of varying coupling strengths which distribute onsets around periodic moments in time. We hypothesized that people tap regularly close to the sound onset density peaks when coupling is strong. When weaker coupling produces multiple inter-onset intervals that are more widely spread, people may interpret their variety and distributions differently in order to form a sense of beat. Experiment 1 with a small in-person cohort indeed showed a few individuals who responded with high frequency tapping to slightly weak coupled stimuli although the rest found regular beats. Experiment 2 with a larger on-line cohort revealed three groups based on characteristics of inter-tap-intervals analyzed by k-means clustering, namely a Regular group (about 1/3 of the final sample) with the most robust beat extraction, Fast group (1/6) who maintained frequent tapping except for the strongest coupling, and Hybrid group (1/2) who maintained beats except for the weakest coupling. Furthermore, the adaptation time course of tap interval variability was slowest in Regular group. We suggest that people's internal criterion for forming beats may involve different perceptual timescales where multiple stimulus intervals could be integrated or processed sequentially as is, and that the highly frequent tapping may reflect their approach in actively seeking synchronization. Our study provides the first documentation of the novel limits of sensorimotor synchronization and individual differences using coupled oscillator dynamics as a generative model of collective behavior.

Composed solutions of synchronized patterns in multiplex networks of Kuramoto oscillators.

Networks with different levels of interactions, including multilayer and multiplex networks, can display a rich diversity of dynamical behaviors and can be used to model and study a wide range of systems. Despite numerous efforts to investigate these networks, obtaining mathematical descriptions for the dynamics of multilayer and multiplex systems is still an open problem. Here, we combine ideas and concepts from linear algebra and graph theory with nonlinear dynamics to offer a novel approach to study multiplex networks of Kuramoto oscillators. Our approach allows us to study the dynamics of a large, multiplex network by decomposing it into two smaller systems: one representing the connection scheme within layers (intra-layer), and the other representing the connections between layers (inter-layer). Particularly, we use this approach to compose solutions for multiplex networks of Kuramoto oscillators. These solutions are given by a combination of solutions for the smaller systems given by the intra- and inter-layer systems, and in addition, our approach allows us to study the linear stability of these solutions.

Diverse phase transitions in Kuramoto model with adaptive mean-field coupling breaking the rotational symmetry

We consider the paradigmatic Kuramoto oscillators coupled via an adaptive mean-field variable that breaks the rotational symmetry. The evolution equation for the mean-field variable is governed by the linear ordinary differential equation corresponding to a low-pass filter with a time-scale parameter, which attenuates the mean-field signal by filtering its high frequency components. We find distinct phase transitions among the incoherent state, standing wave state and synchronized stationary state in both forward and reverse traces in the entire range of the time-scale parameter. In particular, one can observe two distinct continuous transitions, a continuous transition followed by a discontinuous transition, or a single discontinuous transition among the distinct dynamical states in the forward trace depending on the range of the time-scale parameter, which determines the degree of attenuation. Further, one can also observe two distinct continuous transitions or a single discontinuous desynchronization transition in the reverse trace depending on the time-scale parameter. We derive the evolution equations for the macroscopic order parameters governing the dynamics of the discrete set of coupled phase oscillators. We also deduce the analytical conditions for the stability of the Hopf, pitchfork and saddle–node bifurcations, and find them to be in consistent with the dynamical transitions observed using the simulation.

Higher-order interactions in Kuramoto oscillators with inertia.

How do higher-order interactions influence the dynamical landscape of a network of the second-order phase oscillators? We address this question using three coupled Kuramoto phase oscillators with inertia under pairwise and higher-order interactions, in search of various collective states, and new states, if any, that show marginal presence or absence under pairwise interactions. We explore this small network for varying phase lag in the coupling and over a range of negative to positive coupling strength of pairwise as well as higher-order or group interactions. In the extended coupling parameter plane of the network we record several well-known states such as synchronization, frequency chimera states, and rotating waves that appear with distinct boundaries. In the parameter space, we also find states generated by the influence of higher-order interactions: The 2+1 antipodal point and the 2+1 phase-locked states. Our results demonstrate the importantance of the choices of the phase lag and the sign of the higher-order coupling strength for the emergent dynamics of the network. We provide analytical support to our numerical results.

Mean-Field Approximations With Adaptive Coupling for Networks With Spike-Timing-Dependent Plasticity.

Understanding the effect of spike-timing-dependent plasticity (STDP) is key to elucidating how neural networks change over long timescales and to design interventions aimed at modulating such networks in neurological disorders. However, progress is restricted by the significant computational cost associated with simulating neural network models with STDP and by the lack of low-dimensional description that could provide analytical insights. Phase-difference-dependent plasticity (PDDP) rules approximate STDP in phase oscillator networks, which prescribe synaptic changes based on phase differences of neuron pairs rather than differences in spike timing. Here we construct mean-field approximations for phase oscillator networks with STDP to describe part of the phase space for this very high-dimensional system. We first show that single-harmonic PDDP rules can approximate a simple form of symmetric STDP, while multiharmonic rules are required to accurately approximate causal STDP. We then derive exact expressions for the evolution of the average PDDP coupling weight in terms of network synchrony. For adaptive networks of Kuramoto oscillators that form clusters, we formulate a family of low-dimensional descriptions based on the mean-field dynamics of each cluster and average coupling weights between and within clusters. Finally, we show that such a two-cluster mean-field model can be fitted to synthetic data to provide a low-dimensional approximation of a full adaptive network with symmetric STDP. Our framework represents a step toward a low-dimensional description of adaptive networks with STDP, and could for example inform the development of new therapies aimed at maximizing the long-lasting effects of brain stimulation.

An analytical approach for the stability analysis of power networks through Kuramoto oscillators model

In recent years, ensuring stability in power networks has posed a significant challenge for network control engineers, primarily due to the substantial growth in power demand and the complexities involved in integrating renewable energy sources. This study endeavours to augment the existing research by presenting theoretical findings in a practical manner through the introduction of a novel Lyapunov function and the application of LaSalle’s invariance principle to establish an unequivocal and accurate synchronization condition in power networks. The study’s salient contributions include (I) the Fresh approach of utilizing a second-order configuration of asymmetrically networked, dissimilar oscillators; (II) the incorporation of lossy power networks, culminating in non-identical phase delays in the analytical model of the oscillating network; and (III) the introduction of an entirely original Lyapunov function to provide a precise and unambiguous estimation for the region of attraction in power networks. Analytical findings substantiate that network synchronization is assured by the algebraic connectivity of the network graph exceeding a particular threshold value. The study’s transient stability conditions have been authenticated via numerical simulations of the IEEE 39-bus system.

Unified approach for applications of oscillatory associative-memory networks with error-free retrieval.

Given a set of standard binary patterns and a defective pattern, the binary pattern retrieval task is to find the closest pattern to the defective one among these standard patterns. The associative-memory network of Kuramoto oscillators consisting of a Hebbian coupling term and a second-order Fourier term can be applied to this task. When the memorized patterns stored in the Hebbian coupling are mutually orthogonal, recent studies show that the network is capable of distinguishing the memorized patterns from most other patterns. However, the orthogonality usually fails in real situations. In this paper, we present a unified approach for the application of this model in pattern retrieval problems with any general set of standard patterns. By subgrouping the standard patterns and employing an orthogonal lift of each subgroup, this approach makes use of the theory in the case of mutually orthogonal memorized patterns. In particular, the error-free retrieval can be guaranteed, which requires that the retrieved pattern must coincide with one of the standard patterns. As illustrative simulations, pattern retrieval tests for partly sheltered Arabic number symbols are presented.

Exponential synchronization for nonidentical high-dimensional Kuramoto model

This paper investigates exponential synchronization for a group of nonidentical high-dimensional Kuramoto oscillators interconnected by a digraph. A set of matrix differential equations for synchronization errors is established. Under the assumption of strong connectivity of the digraph, the exponential synchronization is proved, and the synchronization rate is calculated.

Transient Phase Clusters in a Two-Population Network of Kuramoto Oscillators with Heterogeneous Adaptive Interaction.

Adaptive interactions are an important property of many real-word network systems. A feature of such networks is the change in their connectivity depending on the current states of the interacting elements. In this work, we study the question of how the heterogeneous character of adaptive couplings influences the emergence of new scenarios in the collective behavior of networks. Within the framework of a two-population network of coupled phase oscillators, we analyze the role of various factors of heterogeneous interaction, such as the rules of coupling adaptation and the rate of their change in the formation of various types of coherent behavior of the network. We show that various schemes of heterogeneous adaptation lead to the formation of transient phase clusters of various types.

Equilibria in Kuramoto Oscillator Networks: An Algebraic Approach

Equilibria in Kuramoto Oscillator Networks: An Algebraic Approach

Tiered synchronization in Kuramoto oscillators with adaptive higher-order interactions.

Phase transitions widely occur in natural systems. Incorporation of higher-order interactions in coupled dynamics is known to cause first-order phase transition to synchronization in an otherwise smooth second-order in the presence of only pairwise interactions. Here, we discover that adaptation in higher-order interactions restores the second-order phase transition in the former setup and notably produces additional bifurcation referred as tiered synchronization as a consequence of combination of super-critical pitchfork and two saddle node bifurcations. The Ott-Antonsen manifold underlines the interplay of higher-order interactions and adaptation in instigating tiered synchronization, as well as provides complete description of all (un)stable states. These results would be important in comprehending dynamics of real-world systems with inherent higher-order interactions and adaptation through feedback coupling.

Frequency Synchronization of Heterogeneous Second-Order Forced Kuramoto Oscillator Networks: A Differential Inequality Approach

This paper is concerned with frequency synchronization of heterogeneous second-order forced Kuramoto oscillator networks. Under this model, each second-order oscillator is driven by a periodic external force and its dynamics will be affected by the inertia and the damping coefficient. By introducing the second-order differential inequality, we first show that this model exhibits frequency synchronization for any initial data if the amplitude of force is large and the inertia is small. In case that the amplitude of force is not large, we prove that this model still can drive frequency synchronization provided that the coupling strength is large and the initial configuration is confined to a sector. Comparison studies with existing synchronization conditions are made to indicate that these proposed criteria seem to be less conservative. A desynchronization for large inertia is shown in our numerical experiment.

Interplay of inertia and adaptive couplings in the emergent dynamics of Kuramoto ensemble

We study the emergent dynamics of Kuramoto oscillators under the interplay between inertia and adaptive couplings. For the phase dynamics, we use the inertial Kuramoto model with time-dependent mutual coupling strengths. For the constant and uniform coupling strength, the inertial Kuramoto model can exhibit the slow relaxation dynamics toward phase-locked state under suitable conditions on system parameters and initial data. To model the time-evolution of mutual coupling strengths, we employ two types of coupling functions, namely “Hebbian coupling” and “anti-Hebbian coupling”. With these modeling spirit, the resulting coupled dynamics for the phase and mutual coupling strength becomes the coupled second-order and first-order systems of ordinary differential equations. For the proposed coupled system, we provide several sufficient frameworks for phase and frequency synchronization in terms of system parameters and initial data for Hebbian and anti-Hebbian coupling functions.

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.

Chimera states in coupled pendulum with higher-order interaction

The study of chimera states in non-pairwise interaction networks is one of the challenging issues in current research. In recent work [S. Kundu and D. Ghosh (2022)], it was discovered that higher-order interactions promote chimera states in nonlocally coupled identical Kuramoto oscillators. In this work, we investigate a higher-order interaction network of a nonlocally coupled pendulum with inertia. By studying pairwise and non-pairwise interaction strengths, we observe different collective states, like synchronization, coherent traveling waves, single-head, multi-head, imperfect traveling chimera states, and incoherent states. In particular, we discover a novel non-stationary chimera state, namely a penetrable traveling chimera state, where the oscillators in the coherent domain of the network travel regularly while others drift randomly in the incoherent domain. We make a map of all the spatiotemporal behaviors in the parameter space of interactive coupling and identify the transition from non-stationary chimeras to coherent states passing through stationary chimeras. As higher-order coupling strength increases, collective dynamics eventually transit to coherence since higher-order interactions are conducive to the emergence of a multi-stable state even without non-pairwise interactions, as demonstrated by the basin stability simulations. After analyzing the damping effects, we consolidate the generality of damping in eradicating dynamical behavior. The abundant dynamics appear, then deteriorate, and even disappear in the corresponding model with inertia. The study of rich dynamic behavior is essential for facilitating an understanding of the impact of higher-order interactions and damping effects on the dynamics of complex real-world networks.

Cyclops States in Repulsive Kuramoto Networks: The Role of Higher-Order Coupling.

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 rotators with higher-order Fourier modes in the 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 reminiscent of the Cyclops' eye. As their mythological counterparts, the cyclops states are giants that dominate the system's phase space in weakly repulsive networks with first-order coupling. Astonishingly, the addition of the second or third harmonics to the Kuramoto coupling function makes the cyclops states global attractors practically across the full range of coupling's repulsion. Beyond the Kuramoto oscillators, we show that this effect is robustly present in networks of canonical theta neurons with adaptive coupling. At a more general level, our results suggest clues for finding dominant rhythms in repulsive physical and biological networks.

Explosive synchronization dependence on initial conditions: The minimal Kuramoto model

Transitions from incoherent to coherent dynamical states can be observed in various real-world networks, ranging from neurons to power-grids. These transitions can be explosive or continuous, with far-reaching implications for the functioning of the affected system. It is therefore of the utmost importance to determine the conditions under which such transitions occur. While a lot of studies in literature focused on the dynamical and/or structural network properties that may generate explosive synchronization, here we report on the effects of different initial conditions. To this purpose, we consider the minimal network of Kuramoto oscillator that may display explosive synchronization, and we show that the nature of the transition changes from continuous to discontinuous as phases are differently initialized. We also determine the critical coupling strength for explosive synchronization, which also depends on the initial conditions.

Broadcasting solutions on networked systems of phase oscillators

Networked systems have been used to model and investigate the dynamical behavior of a variety of systems. For these systems, different levels of complexity can be considered in the modeling procedure. On one hand, this can offer a more realistic and rich modeling option. On the other hand, it can lead to intrinsic difficulty in analyzing the system. Here, we present an approach to investigate the dynamics of Kuramoto oscillators on networks with different levels of connections: a network of networks. To do so, we utilize a construction in network theory known as the join of networks, which represents “intra-area” and “inter-area” connections. This approach provides a reduced representation of the original, multi-level system, where both systems have equivalent dynamics. Then, we can find solutions for the reduced system and broadcast them to the original network of networks. Moreover, using the same idea we can investigate the stability of these states, where we can obtain information on the Jacobian of the multi-level system by analyzing the reduced one. This approach is general for arbitrary connection schemes between nodes within the same area. Finally, our work opens the possibility of studying the dynamics of networked systems using a simpler representation, thus leading to a better understanding of the dynamical behavior of these systems.