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On the Critical Coupling for Kuramoto Oscillators
Florian Dorfler, Francesco Bullo
TL;DR
The paper addresses critical-coupling estimates for synchronization in Kuramoto models and develops analysis for finite-dimensional and multi-rate settings. It introduces phase cohesiveness, derives explicit coupling conditions, and shows that multi-rate synchronization conditions do not depend on inertia.
Problem
Kuramoto applications require synchronization conditions and critical-coupling estimates across finite-dimensional, first-order, second-order, and multi-rate models.
Method
The paper reviews existing bounds, introduces phase cohesiveness, analyzes finite-dimensional dynamics, and relates multi-rate models to first-order Kuramoto models with scaled natural frequencies.
Results
K > Kcritical = ωmax −ωmin gives exponential synchronization for arbitrary natural-frequency distributions, while multi-rate synchronization conditions are independent of inertial terms.
Takeaways & Limitations
The coupling bound provides phase-cohesiveness and order-parameter guarantees, while multi-rate synchronization can be characterized without inertia in the conditions.
Takeaways & Limitations
For a particular frequency distribution, the explicit bound is only sufficient and may be up to twice the necessary bound.
Abstract
from arXiv · showhide
The Kuramoto model captures various synchronization phenomena in biological and man-made 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 for finite and infinite-dimensional, and for first and second-order Kuramoto models. Third, we present the first explicit necessary and sufficient condition on the critical coupling to achieve synchronization in the finite-dimensional Kuramoto model for an arbitrary distribution of the natural frequencies. The multiplicative gap in the synchronization condition yields a practical stability result determining the admissible initial and the guaranteed ultimate phase cohesiveness as well as the guaranteed asymptotic magnitude of the order parameter. Fourth and finally, we extend our analysis to multi-rate Kuramoto models consisting of second-order Kuramoto oscillators with inertia and viscous damping together with first-order Kuramoto oscillators with multiple time constants. We prove that the multi-rate Kuramoto model is locally topologically conjugate to a first-order Kuramoto model with scaled natural frequencies, and we present necessary and sufficient conditions for almost global phase synchronization and local frequency synchronization. Interestingly, these conditions do not depend on the inertiae which contradicts prior observations on the role of inertiae in synchronization of second-order Kuramoto models.
1. Introduction.
The paper introduces phase cohesiveness, reviews critical-coupling estimates, derives explicit synchronization conditions, and extends the analysis to multi-rate Kuramoto models.
- 1. Introduction.: The Kuramoto model describes coupled oscillators whose synchronization depends on the coupling strength K and natural frequencies ωi.It has applications in biological systems, motion coordination, Josephson junctions, power networks, and deep brain stimulation.
- 1. Introduction.: The multi-rate model combines inertial second-order oscillators with first-order oscillators having multiple time constants.Its synchronization conditions and asymptotic synchronization frequency are independent of inertial coefficients Mi.
- 1. Introduction.: The paper distinguishes synchronization concepts and introduces phase cohesiveness as bounded containment of oscillator angles within an arc of fixed length.It is presented both as an analysis tool and as a performance index analogous to the order parameter.
- 1. Introduction.: The authors unify necessary, sufficient, implicit, and explicit critical-coupling estimates across finite- and infinite-dimensional, first- and second-order models.The review also compares analysis techniques and surveys conflicting findings about inertia.
- 1. Introduction.: For arbitrary natural-frequency distributions, the paper gives an explicit necessary and sufficient condition for exponential synchronization in the finite-dimensional model.The multiplicative gap Kcritical/K determines admissible initial conditions, ultimate phase cohesiveness, order-parameter magnitude, and synchronization-rate estimates.
2. Phase Synchronization, Phase Cohesiveness, and Frequency En-
This section distinguishes phase, frequency, and phase-cohesive synchronization, then connects phase containment geometrically to the order parameter and illustrates critical coupling with two oscillators.
- Phase Synchronization, Phase Cohesiveness, and Frequency En-: Phase cohesiveness means that all oscillator angles remain within an arc of fixed length, including when the arc rotates.It addresses non-identical frequencies, where phase differences may converge to nonzero constants.
- Phase Synchronization, Phase Cohesiveness, and Frequency En-: Frequency synchronization means that oscillator frequencies converge, while phase locking corresponds to constant phase differences or a common rotating frequency.The literature uses related terminology inconsistently.
- Phase Synchronization, Phase Cohesiveness, and Frequency En-: For angles contained in an arc of length γ, the order-parameter magnitude satisfies r ∈ [cos(γ/2),1].Conversely, within a semicircle, an order parameter of magnitude r implies an arc of length at least 2 arccos(r).
- Phase Synchronization, Phase Cohesiveness, and Frequency En-: Relative and grounded coordinates remove rotational invariance and support analysis of phase-locked trajectories and exponential synchronization.The paper links this coordinate-based analysis to bounded angular distances and frequency synchronization.
- Phase Synchronization, Phase Cohesiveness, and Frequency En-: In the two-oscillator case, synchronization occurs if and only if K > Kcritical = ω2−ω1.The ratio Kcritical/K determines ultimate phase cohesiveness and admissible initial conditions, with loss of synchronization at a saddle-node bifurcation.
3. A Review of Bounds for the Critical Coupling Strength.
The review organizes necessary, sufficient, implicit, and explicit critical-coupling estimates across finite- and infinite-dimensional Kuramoto models, including first- and second-order settings. It highlights how assumptions, analysis methods, and model formulations affect synchronization bounds and their interpretation.
- Infinite-dimensional models: Continuum-limit studies analyze first-order continuity equations or second-order Fokker–Planck equations and derive distribution-dependent synchronization thresholds.For bounded symmetric distributions, some interval-based bounds are tight for bipolar and uniform distributions.
- Finite-dimensional bounds: Finite-dimensional analyses assume natural frequencies lie in a compact interval, since unbounded frequencies make the critical coupling infinite.The interval width provides a basic scale for necessary conditions.
- Finite-dimensional bounds: Necessary finite-dimensional conditions include bounds based on frequency differences, interval width, and variance.For bipolar distributions and symmetric topologies, related explicit conditions can also be derived.
- Finite-dimensional bounds: Sufficient synchronization bounds commonly use incremental-stability arguments, norms of frequency non-uniformity, and phase-difference constraints.The matrix V can represent either deviations from the average frequency or pairwise frequency differences.
- Finite-dimensional bounds: Two-norm bounds arise from quadratic Lyapunov, contraction, and contraction-analysis methods, while infinity-norm bounds can scale independently of oscillator number.The scale-free form f(γ) = 1/sin(γ) is obtained in some contraction and analyticity arguments.
- Implicit and exact bounds: Implicit critical-coupling equations characterize phase-locked solutions and, in related work, provide local stability information through a saddle-node bifurcation.The reviewed conclusion distinguishes explicit bounds from implicit formulas that compute a threshold for local stability.
4. Necessary and Sufficient Conditions on the Critical Coupling.
Theorem 4.1 gives an explicit necessary-and-sufficient coupling condition for synchronization under arbitrary natural-frequency distributions, together with phase-cohesiveness, convergence, and order-parameter guarantees. The condition is K > Kcritical = ωmax −ωmin, while the ratio Kcritical/K determines guaranteed phase-cohesiveness bounds and the attraction region.
- Explicit synchronization condition: K > Kcritical = ωmax −ωmin is the explicit necessary-and-sufficient synchronization condition for arbitrary natural-frequency distributions.The bound is equivalently K > (ωmax −ωmin)/sin(γ) in the phase-cohesiveness formulation.
- Explicit synchronization condition: Theorem 4.1 establishes equivalent conditions for exponential synchronization, local exponential stability, and a coupling threshold valid for all distributions supported on [ωmin, ωmax].The theorem uses γmax ∈]π/2, π] for global initial-phase guarantees and γmin ∈[0, π/2[ for local stability.
- Phase cohesiveness and performance: sin(γmin) = sin(γmax) = Kcritical/K uniquely relates the guaranteed ultimate and admissible phase arc lengths to the coupling margin.Trajectories starting in ∆(γmax) approach the positively invariant set ¯∆(γmin).
- Convergence guarantees: Frequency synchronization converges exponentially at a rate no worse than λfs = K cos(γ), while identical natural frequencies yield phase synchronization at rate no worse than λps = K sinc(γ).The frequency-synchronization limit is the average natural frequency ωavg.
- Phase cohesiveness and performance: The asymptotic order-parameter magnitude satisfies 1 ≥ r ≥ cos(γmin/2) once trajectories become ultimately phase cohesive in ¯∆(γmin).Phase cohesiveness therefore supplies a direct asymptotic performance guarantee for synchronization.
- Time-varying frequencies: The common-Lyapunov analysis extends the guarantees to switching natural frequencies, preserving asymptotic phase cohesiveness under arbitrary switching sequences.With a uniform dwell time, additional convergence properties follow from the same framework.
5. Synchronization of Multi-Rate Kuramoto Models.
The multi-rate Kuramoto model has synchronization conditions and local equilibrium properties that are independent of inertia, while remaining locally equivalent to scaled first-order dynamics. This equivalence supports synchronization results for mixed first- and second-order oscillators.
- Model equivalence: Inertia does not affect the equilibria, local stability properties, synchronization conditions, or synchronization frequency of the multi-rate model.These properties instead depend on viscous damping and time constants.
- Parameterized dynamics: The parameterized family continuously interpolates between gradient-like and mixed dissipative Hamiltonian/gradient-like dynamics.At λ = 1 it is gradient-like, while λ = 0 combines gradient-like and dissipative Hamiltonian dynamics.
- Parameterized dynamics: All systems in the parameterized family share the same equilibria and local stability properties, independently of λ and positive definite D1, D2, and M.Equilibria satisfy ∇H(x) = F, and local stability is determined by the inertia of −∇^2H(x∗).
- Synchronization equivalence: Locally near synchronization manifolds, the multi-rate, first-order, and scaled Kuramoto models are topologically conjugate when their synchronized trajectories are locally exponentially stable.The corresponding linearized dynamics share the same hyperbolic inertia apart from the common translational center eigenspace.
- Synchronization conditions: K > Kcritical = ω̄max − ω̄min is equivalent to locally exponentially stable, phase-cohesive synchronization for all n ≥ 2, admissible m, positive inertiae, damping, and natural frequencies.The phase-cohesiveness arc γmin satisfies Kcritical/K = sin(γmin).
- Synchronization conditions: Almost globally exponentially stable phase synchronization is equivalent to ωi = Di s̄ for every oscillator, with s̄ equal to the constant synchronization frequency.The asymptotic synchronization phase is then determined by the weighted initial phase and frequency terms.
- Scope and transients: The critical-coupling bound can be conservative for particular natural-frequency sets, although implicit formulae can determine the exact critical coupling.Inertia may still affect convergence rates, separatrices, attraction basins, and qualitative transient dynamics.
6. Conclusions.
The paper synthesizes critical-coupling results, formalizes phase cohesiveness, derives explicit synchronization conditions, and extends the analysis to multi-rate models. It concludes that inertia does not affect synchronization conditions, while tighter bounds remain needed for more general coupling and dynamics.
- Contributions: The paper reviews critical-coupling bounds, introduces phase cohesiveness, and derives an explicit tight synchronization bound that is necessary and sufficient for arbitrary natural-frequency distributions.The multiplicative gap also determines practical stability and synchronization-performance measures.
- Conclusions: The multi-rate extension shows that inertial terms do not affect synchronization conditions.The paper identifies broader open settings requiring tight explicit bounds, including arbitrary topologies, delays, non-gradient-like dynamics, and non-uniform coupling.