Universal phase transitions to synchronization in Kuramoto-like models with heterogeneous coupling

Xu, Can; Boccaletti, Stefano; Zheng, Zhigang; Guan, Shuguang

doi:10.1088/1367-2630/ab4f59

Cited by 28 publications

(5 citation statements)

References 44 publications

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“…To highlight the heterogeneity of coupling, we restrict ourselves to a special setting, namely, K i j = K|ω i | (in-coupling) with K > 0 [37][38][39][40][41][42][43][44]. The effect is to endow the oscillators with heterogeneous coupling establishing the frequency-coupling correlations in the coupled system.…”

Section: Dynamical Model and Numerical Resultsmentioning

confidence: 99%

Exact solutions of the abrupt synchronization transitions and extensive multistability in globally coupled phase oscillator populations

Tang¹,

Lü²,

Xu³

2021

J. Phys. A: Math. Theor.

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The Kuramoto model consisting of large ensembles of globally coupled phase oscillators serves as a paradigm for modelling synchronization and collective behavior in diverse self-sustained systems. As interest in the effects of higher-order interactions, we study an extension of the Kuramoto model with higher-order structure by considering the correlations between frequency and coupling. The resulting model is exactly solvable and several novel dynamical phenomena including clustering, multistability, and abrupt synchronization (desynchronization) transition emerge. We demonstrate that the extensive multiclusters corresponding to different arrangements of oscillator populations on the circle are established in a universal way that are independent of the choices of frequency distributions (heterogeneity of the ensembles). Using the partial dimensionality reduction of the Ott–Antonsen, we present a rigorous analysis of various multi-cluster states by studying their spectrum in the thermodynamic limit. In particular, we prove that a large multiplicity of the synchronous states are asymptotically stable to perturbation in the tangent space, thereby determining an explicit stability condition for their occurrences in the high-dimensional phase space.

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Section: Dynamical Model and Numerical Resultsmentioning

confidence: 99%

Exact solutions of the abrupt synchronization transitions and extensive multistability in globally coupled phase oscillator populations

Tang¹,

Lü²,

Xu³

2021

J. Phys. A: Math. Theor.

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“…In this setting, the randomness is intrinsic to the oscillators themselves rather than to the coupling between them [41][42][43][44][45][46][47][48][49]. Moreover, β ≥ 1 (β < 1) sets up a positive (negative) feedback between the coupling and the coherence of the system [50][51][52][53][54][55].…”

Section: Mathematical Model and Its Stationary Solutionsmentioning

confidence: 99%

Collective dynamics of heterogeneously and nonlinearly coupled phase oscillators

Xu,

Tang,

Lü

et al. 2021

Preprint

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Coupled oscillators have been used to study synchronization in a wide range of social, biological, and physical systems, including pedestrian-induced bridge resonances, coordinated lighting up of firefly swarms, and enhanced output peak intensity in synchronizing laser arrays. Here we advance this subject by studying a variant of the Kuramoto model, where the coupling between the phase oscillators is heterogeneous and nonlinear. In particular, the quenched disorder in the coupling strength and the intrinsic frequencies are correlated, and the coupling is extended to depend on the global rhythm of the population. We show that the interplay of these factors leads to a fascinatingly rich collective dynamics, including explosive synchronization transitions, hybrid transitions with hysteresis absence, abrupt irreversible desynchronization transitions, and tiered phase transitions with or without a vanishing onset. We develop an analytical treatment that enables us to determine the observed equilibrium states of the system, as well as to explore their asymptotic stability at various levels. Our research thus provides theoretical foundations for a number of self-organized phenomena that may be responsible for the emergence of collective rhythms in complex systems.

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“…In this setup, the bifurcation mechanism for phase transition in oscillator system becomes clear. On the one hand, ĝ(0) = 0 implies that the incoherent state loses its stability via Hopf bifurcation at the critical coupling K a = 2 πĝ(Ωc) [30,31], where Ω c is the imaginary part of the eigenvalue of the linear stability analysis about R = W = 0. On the other hand, the characteristic function F (s) indicates that a saddle node bifurcation takes place at K c = F (s c ) −1 , at which a number of oscillators lock their phases forming a macroscopic order characterized by a non-zero W c = s c F (s c ).…”

Section: A Tiered Phase Transitionmentioning

confidence: 99%

Universal scaling and phase transitions of coupled phase oscillator populations

Xu,

Wang,

Skardal

2020

Preprint

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The Kuramoto model, which serves as a paradigm for investigating synchronization phenomenon of oscillatory system, is known to exhibit second-order, i.e., continuous, phase transitions in the macroscopic order parameter. Here, we generalize a number of classical results by presenting a general framework for capturing, analytically, the critical scaling of the order parameter at the onset of synchronization. Using a self-consistent approach and constructing a characteristic function, we identify various phase transitions toward synchrony and establish scaling relations describing the asymptotic dependence of the order parameter on coupling strength near the critical point. We find that the geometric properties of the characteristic function, which depends on the natural frequency distribution, determines the scaling properties of order parameter above the criticality.

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Universal phase transitions to synchronization in Kuramoto-like models with heterogeneous coupling

Cited by 28 publications

References 44 publications

Exact solutions of the abrupt synchronization transitions and extensive multistability in globally coupled phase oscillator populations

Exact solutions of the abrupt synchronization transitions and extensive multistability in globally coupled phase oscillator populations

Collective dynamics of heterogeneously and nonlinearly coupled phase oscillators

Universal scaling and phase transitions of coupled phase oscillator populations

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