Second-order Kuramoto Model Research Articles

Synchronization Transition of the Second-Order Kuramoto Model on Lattices

The second-order Kuramoto equation describes the synchronization of coupled oscillators with inertia, which occur, for example, in power grids. On the contrary to the first-order Kuramoto equation, its synchronization transition behavior is significantly less known. In the case of Gaussian self-frequencies, it is discontinuous, in contrast to the continuous transition for the first-order Kuramoto equation. Herein, we investigate this transition on large 2D and 3D lattices and provide numerical evidence of hybrid phase transitions, whereby the oscillator phases θi exhibit a crossover, while the frequency is spread over a real phase transition in 3D. Thus, a lower critical dimension dlO=2 is expected for the frequencies and dlR=4 for phases such as that in the massless case. We provide numerical estimates for the critical exponents, finding that the frequency spread decays as ∼t-d/2 in the case of an aligned initial state of the phases in agreement with the linear approximation. In 3D, however, in the case of the initially random distribution of θi, we find a faster decay, characterized by ∼t-1.8(1) as the consequence of enhanced nonlinearities which appear by the random phase fluctuations.

Reconstruction of cascading failures in dynamical models of power grids

Abstract In this article, we propose a method to reconstruct the active links of a power network described by a second-order Kuramoto model and subject to dynamically induced cascading failures. Starting from the assumption (realistic for power grids) that the structure of the network is known, our method reconstructs the active links from the evolution of the relevant dynamical quantities of the nodes of the system, that is, the node phases and angular velocities. We find that, to reconstruct the temporal sequence of the faults, it is crucial to use time series with a small number of samples, as the observation window should be smaller than the temporal distance between subsequent events. This requirement is in contrast with the need of using larger sets of data in the presence of noise, such that the number of samples to feed in the algorithm has to be selected as a trade-off between the prediction error and temporal resolution of the active link reconstruction.

Stability and Bifurcation of Mixing in the Kuramoto Model with Inertia

The Kuramoto model of coupled second order damped oscillators on convergent sequences of graphs is analyzed in this work. The oscillators in this model have random intrinsic frequencies and interact with each other via nonlinear coupling. The connectivity of the coupled system is assigned by a graph which may be random as well. In the thermodynamic limit the behavior of the system is captured by the Vlasov equation, a hyperbolic partial differential equation for the probability distribution of the oscillators in the phase space. We study stability of mixing, a steady state solution of the Vlasov equation, corresponding to the uniform distribution of phases. Specifically, we identify a critical value of the strength of coupling, at which the system undergoes a pitchfork bifurcation. It corresponds to the loss of stability of mixing and marks the onset of synchronization. As for the classical Kuramoto model, the presence of the continuous spectrum on the imaginary axis poses the main difficulty for the stability analysis. To overcome this problem, we use the methods from the generalized spectral theory developed for the original Kuramoto model. The analytical results are illustrated with numerical bifurcation diagrams computed for the Kuramoto model on Erd\H{o}s--R\'enyi and small-world graphs. Applications of the second-order Kuramoto model include power networks, coupled pendula, and various biological networks. The analysis in this paper provides a mathematical description of the onset of synchronization in these systems.

Partial synchronization in the second-order Kuramoto model: An auxiliary system method.

Partial synchronization emerges in an oscillator network when the network splits into clusters of coherent and incoherent oscillators. Here, we analyze the stability of partial synchronization in the second-order finite-dimensional Kuramoto model of heterogeneous oscillators with inertia. Toward this goal, we develop an auxiliary system method that is based on the analysis of a two-dimensional piecewise-smooth system whose trajectories govern oscillating dynamics of phase differences between oscillators in the coherent cluster. Through a qualitative bifurcation analysis of the auxiliary system, we derive explicit bounds that relate the maximum natural frequency mismatch, inertia, and the network size that can support stable partial synchronization. In particular, we predict threshold-like stability loss of partial synchronization caused by increasing inertia. Our auxiliary system method is potentially applicable to cluster synchronization with multiple coherent clusters and more complex network topology.

Vulnerability and stability of power grids modeled by second-order Kuramoto model: a mini review

Vulnerability and stability of power grids modeled by second-order Kuramoto model: a mini review

Stability of Clusters in the Second-Order Kuramoto Model on Random Graphs

The Kuramoto model of coupled phase oscillators with inertia on Erdos-Renyi graphs is analyzed in this work. For a system with intrinsic frequencies sampled from a bimodal distribution we identify a variety of two cluster patterns and study their stability. To this end, we decompose the description of the cluster dynamics into two systems: one governing the (macro) dynamics of the centers of mass of the two clusters and the second governing the (micro) dynamics of individual oscillators inside each cluster. The former is a low-dimensional ODE whereas the latter is a system of two coupled Vlasov PDEs. Stability of the cluster dynamics depends on the stability of the low-dimensional group motion and on coherence of the oscillators in each group. We show that the loss of coherence in one of the clusters leads to the loss of stability of a two-cluster state and to formation of chimera states. The analysis of this paper can be generalized to cover states with more than two clusters and to coupled systems on W-random graphs. Our results apply to a model of a power grid with fluctuating sources.

Decreased resilience in power grids under dynamically induced vulnerabilities

In this paper, a methodology inspired on bond and site percolation methods is applied to the estimation of the resilience against failures in power grids. Our approach includes vulnerability measures with both dynamical and structural foundations as an attempt to find more insights about the relationships between topology and dynamics in the second-order Kuramoto model on complex networks. As test cases for numerical simulations, we use the real-world topology of the Colombian power transmission system, as well as randomly generated networks with spatial embedding. It is observed that, by focusing the attacks on those dynamical vulnerabilities, the power grid becomes, in general, more prone to reach a state of total blackout, which in the case of node removal procedures it is conditioned by the homogeneity of power distribution in the network.

Synchronization of random coupling delayed complex networks with random and adaptive coupling strength

Our paper devotes to synchronization issue of random coupling delayed complex networks with adaptive coupling strength as well as random coupling strength via intermittent control. Different from related literature, intermittent control in this paper is aperiodic. Furthermore, we assume coupling delay is a random variable and coupling function is nonlinear. Meanwhile, adaptive and random coupling strength are, respectively, taken into account in our model compared with most existing literature. Some novel sufficient conditions are derived by utilizing Lyapunov method and graph theory. What is more, synchronization of a second-order Kuramoto model, as an application of our theoretical results, is investigated. Meanwhile, some sufficient conditions are given as well. Eventually, some numerical simulations are given to show the effectiveness of results.

Emergent Behavior of a Second-Order Lohe Matrix Model on the Unitary Group

We study a second-order extension to the first-order Lohe matrix model on the unitary group which can be reduced to the second-order Kuramoto model with inertia as a special case. For the proposed second-order model, we present several sufficient frameworks leading to the emergence of the complete and practical synchronizations in terms of the initial data and the system parameters. For the identical hamiltonians, we show that the complete synchronization emerges asymptotically. In contrast, for the non-identical hamiltonians, the practical synchronization occurs for some class of initial data when the product of the coupling strength and inertia is sufficiently small.

Convergence of analytic gradient-type systems with periodicity and its applications in Kuramoto models

We consider the convergence of gradient-type systems with periodic and analytic potentials. The main tool is the celebrated Łojasiewicz inequality which is valid for any analytic function. Our results show that the convergence of such systems with periodic and analytic potentials is unconditional to the initial data; in other words, any trajectory converges to some equilibrium. As direct applications, we can show that any trajectory converges to phase-locked state for the first- and second-order Kuramoto models on a symmetric network with attractive–repulsive forces and identical natural frequencies. In particular, the inertial Kuramoto model with identical oscillators converges to phase-locked state for any initial configuration.

Generalized quantized intermittent control with adaptive strategy on finite-time synchronization of delayed coupled systems and applications

This paper is concerned with the finite-time synchronization problem of coupled drive-response systems with time-varying delays via quantized adaptive aperiodically intermittent control. Compared with previous research results, the intermittent control is aperiodic and coupling functions are nonlinear. Furthermore, both internal time-varying delays and coupling time-varying delays are considered. By employing Lyapunov–Krasovskii functional method and Kirchhoff’s Matrix Tree Theorem, some novel sufficient conditions are derived to guarantee that drive-response systems can achieve synchronization in finite time. Specially, the convergence time is discussed. It is worth pointing out that the convergence time is closely related to topological structure of networks and the maximum ratio of the rest width to the aperiodical time span. Finally, finite-time synchronization of a typical second-order Kuramoto model with time-varying delays and Chua’s circuits model with time-varying delays are studied by using our theoretical results. Meanwhile, the numerical simulations are given to illustrate the effectiveness of the theoretical results we obtained.

Stochastic basin stability in complex networks

In this paper, we propose a novel concept, stochastic basin stability, to investigate the stability of dynamical systems under random noises. By this concept, we investigate synchronization of multistable second-order Kuramoto models under continuously acting perturbations in complex networks. By a mean-field approach and the Fokker-Planck equation, we derive formulas of the density functions of both phase and frequency. Based on them, we provide an analytical treatment of stochastic basin stability and illustrate that the theoretical results are in good agreement with numerical simulations. This proposed concept integrates the perturbations of both initial condition and random intervention, and paves a general and efficient approach for analytically and numerically investigating stability of stochastic dynamical systems.

Inertial effect on frequency synchronization for the second-order Kuramoto model with local coupling

In this paper, we study the influence of the inertial effect on frequency synchronization in an ensemble of Kuramoto oscillators with finite inertia and symmetric and connected interactions. We present sufficient conditions in terms of coupling strength, algebraic connectivity, natural frequencies, and the inertial term to guarantee the occurrence of frequency synchronization. We also make a comparison with the existing conditions proposed for the first-order Kuramoto model and conclude that the inertial effect, if appropriately small, has little influence on frequency synchronization as long as the initial phase configurations are distributed in a half circle.

Effects of assortative mixing in the second-order Kuramoto model.

In this paper we analyze the second-order Kuramoto model in the presence of a positive correlation between the heterogeneity of the connections and the natural frequencies in scale-free networks. We numerically show that discontinuous transitions emerge not just in disassortative but also in strongly assortative networks, in contrast with the first-order model. We also find that the effect of assortativity on network synchronization can be compensated by adjusting the phase damping. Our results show that it is possible to control collective behavior of damped Kuramoto oscillators by tuning the network structure or by adjusting the dissipation related to the phases' movement.

Onset and suffusing transitions towards synchronization in complex networks

Based on the basin of attraction, the basin stability is quantified to measure a system's ability to regain an equilibrium state subjected to even large perturbations. In this letter, we demonstrate a novel phenomenon uncovered by the basin stability that in complex networks of second-order Kuramoto models successively undergoes two first-order transitions: an onset transition from an unstable to a locally stable synchronous state, and a suffusing transition from a locally stable to a globally stable synchronous state; we call this sequence onset-suffusing transitions. We provide an analytical treatment of basin stability by a mean-field analysis. Our findings are in good agreement with simulations and fundamentally deepen the understanding of stability of microscopic mechanisms towards synchronization.

Low-dimensional behavior of Kuramoto model with inertia in complex networks.

Low-dimensional behavior of large systems of globally coupled oscillators has been intensively investigated since the introduction of the Ott-Antonsen ansatz. In this report, we generalize the Ott-Antonsen ansatz to second-order Kuramoto models in complex networks. With an additional inertia term, we find a low-dimensional behavior similar to the first-order Kuramoto model, derive a self-consistent equation and seek the time-dependent derivation of the order parameter. Numerical simulations are also conducted to verify our analytical results.

Cluster Explosive Synchronization in Complex Networks

The emergence of explosive synchronization has been reported as an abrupt transition in complex networks of first-order Kuramoto oscillators. In this Letter we demonstrate that the nodes in a second-order Kuramoto model perform a cascade of transitions toward a synchronous macroscopic state, which is a novel phenomenon that we call cluster explosive synchronization. We provide a rigorous analytical treatment using a mean-field analysis in uncorrelated networks. Our findings are in good agreement with numerical simulations and fundamentally deepen the understanding of microscopic mechanisms toward synchronization.

On the Critical Coupling for Kuramoto Oscillators

The celebrated Kuramoto model captures various synchronization phenomena in biological and man-made dynamical systems of coupled oscillators. It is well known that there exists a critical coupling strength among the oscillators at which a phase transition from incoherency to synchronization occurs. This paper features four contributions. First, we characterize and distinguish the different notions of synchronization used throughout the literature and formally introduce the concept of phase cohesiveness as an analysis tool and performance index for synchronization. Second, we review the vast literature providing necessary, sufficient, implicit, and explicit estimates of the critical coupling strength in the finite- and infinite-dimensional cases and for both first-order and second-order Kuramoto models. Third, we present the first explicit necessary and sufficient condition on the critical coupling strength to achieve synchronization in the finite-dimensional Kuramoto model for an arbitrary distribution of...