Kuramoto model


The Kuramoto model, first proposed by Yoshiki Kuramoto, is a mathematical model used to describe synchronization. More specifically, it is a model for the behavior of a large set of coupled oscillators. Its formulation was motivated by the behavior of systems of chemical and biological oscillators, and it has found widespread applications such as in neuroscience and oscillating flame dynamics. Kuramoto was quite surprised when the behavior of some physical systems, namely coupled arrays of Josephson junctions, followed his model.
The model makes several assumptions, including that there is weak coupling, that the oscillators are identical or nearly identical, and that interactions depend sinusoidally on the phase difference between each pair of objects.

Definition

In the most popular version of the Kuramoto model, each of the oscillators is considered to have its own intrinsic natural frequency, and each is coupled equally to all other oscillators. Surprisingly, this fully nonlinear model can be solved exactly in the limit of infinite oscillators, N→ ∞ ; alternatively, using self-consistency arguments one may obtain steady-state solutions of the order parameter.
The most popular form of the model has the following governing equations:
where the system is composed of N limit-cycle oscillators, with phases and coupling constant K.

Noise can be added to the system. In that case, the original equation is altered to:
where is the fluctuation and a function of time. If we consider the noise to be white noise, then:
with denoting the strength of noise.

Transformation

The transformation that allows this model to be solved exactly is as follows:
Define the "order" parameters r and ψ as
Here r represents the phase-coherence of the population of oscillators and ψ indicates the average phase. Multiplying this equation with and only considering the imaginary part gives:
Thus the oscillators' equations are no longer explicitly coupled; instead the order parameters govern the behavior. A further transformation is usually done, to a rotating frame in which the statistical average of phases over all oscillators is zero. Finally, the governing equation becomes:

Large ''N'' limit

Now consider the case as N tends to infinity. Take the distribution of intrinsic natural frequencies as g. Then assume that the density of oscillators at a given phase θ, with given natural frequency ω, at time t is. Normalization requires that
The continuity equation for oscillator density will be
where v is the drift velocity of the oscillators given by taking the infinite-N limit in the transformed governing equation, such that
Finally, we must rewrite the definition of the order parameters for the continuum limit. must be replaced by its ensemble average and the sum must be replaced by an integral, to give

Solutions

The incoherent state with all oscillators drifting randomly corresponds to the solution. In that case, and there is no coherence among the oscillators. They are uniformly distributed across all possible phases, and the population is in a statistical steady-state.
When coupling K is sufficiently strong, a fully synchronized solution is possible. In the fully synchronized state, all the oscillators share a common frequency, although their phases can be different.
A solution for the case of partial synchronization yields a state in which only some oscillators synchronize; other oscillators drift incoherently. Mathematically, the state has
for locked oscillators, and
for drifting oscillators. The cutoff occurs when.

Connection to Hamiltonian systems

The dissipative Kuramoto model is contained in certain conservative Hamiltonian systems with Hamiltonian of the form:
After a canonical transformation to action-angle variables with actions and angles , exact Kuramoto dynamics emerges on invariant manifolds of constant. With the transformed Hamiltonian:
Hamilton's equation of motion become:
and
So the manifold with is invariant because and the phase dynamics becomes the dynamics of the Kuramoto model. The class of Hamiltonian systems characterizes certain quantum-classical systems including Bose–Einstein condensates.

Variations of the models

There are a number of types of variations that can be applied to the original model presented above. Some models change to topological structure, others allow for heterogeneous weights, and other changes are more related to models that are inspired by the Kuramoto model but do not have the same functional form.

Variations of network topology

Beside the original model, which has an all-to-all topology, a sufficiently dense complex network-like topology is amenable to the mean-field treatment used in the solution of the original model. Network topologies such as rings and coupled populations support chimera states. One also may ask for the behavior of models in which there are intrinsically local, like one-dimensional topologies which the chain and the ring are prototypical examples. In such topologies, in which the coupling is not scalable according to 1/N, it is not possible to apply the canonical mean-field approach, so one must rely upon case-by-case analysis, making use of symmetries whenever it is possible, which may give basis for abstraction of general principles of solutions.
Uniform synchrony, waves and spirals can readily be observed in two-dimensional Kuramoto networks with diffusive local coupling. The stability of waves in these models can be determined analytically using the methods of Turing stability analysis. Uniform synchrony tends to be stable when the local coupling is everywhere positive whereas waves arise when the long-range connections are negative. Waves and synchrony are connected by a topologically distinct branch of solutions known as ripple. These are low-amplitude spatially-periodic deviations that emerge from the uniform state via a Hopf bifurcation. The existence of ripple solutions was predicted by Wiley, Strogatz and Girvan, who called them multi-twisted q-states.
The topology on which the Kuramoto model is studied can be made adaptive by use of fitness model showing enhancement of synchronization and percolation in a self-organised way.

Variations of network topology and network weights: from vehicle coordination to brain synchronization

Some works in the control community have focused on the Kuramoto model on networks and with heterogeneous weights. The dynamics of this model reads as follows:
where is a nonzero positive real number if oscillator is connected to oscillator.
Such model allows for a more realistic study of, e.g., flocking, schooling, and vehicle coordination. In the work from Dörfler and colleagues, several theorems provide rigorous conditions for phase and frequency synchronization of this model. Further studies, motivated by experimental observations in neuroscience, focus on deriving analytical conditions for cluster synchronization of heterogeneous Kuramoto oscillators on arbitrary network topologies. Since the Kuramoto model seems to play a key role in assessing synchronization phenomena in the brain, theoretical conditions that support empirical findings may pave the way for a deeper understanding of neuronal synchronization phenomena.

Variations of the phase interaction function

Kuramoto approximated the phase interaction between any two oscillators by its first Fourier component, namely, where.
Better approximations can be obtained by including higher-order Fourier components,
where parameters and must be estimated.
For example, synchronization among a network of weakly-coupled Hodgkin-Huxley neurons can be replicated using coupled oscillators that retain the first four Fourier components of the interaction function. The introduction of higher-order phase interaction terms can also induce interesting dynamical phenomena such as partially synchronized states, heteroclinic cycles, and chaotic dynamics.

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