Kuramoto model
More specifically, it is a model for the behavior of a large set of coupled oscillators.
[9][10] Kuramoto was quite surprised when the behavior of some physical systems, namely coupled arrays of Josephson junctions, followed his model.
In the most popular version of the Kuramoto model, each of the oscillators is considered to have its own intrinsic natural frequency
Surprisingly, this fully nonlinear model can be solved exactly in the limit of infinite oscillators, N→ ∞;[5] alternatively, using self-consistency arguments one may obtain steady-state solutions of the order parameter.
[3] The most popular form of the model has the following governing equations: where the system is composed of N limit-cycle oscillators, with phases
A further transformation is usually done, to a rotating frame in which the statistical average of phases over all oscillators is zero (i.e.
Take the distribution of intrinsic natural frequencies as g(ω) (assumed normalized).
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, the definition of the order parameters must be rewritten for the continuum (infinite N) limit.
) and the sum must be replaced by an integral, to give The incoherent state with all oscillators drifting randomly corresponds to the solution
In the fully synchronized state, all the oscillators share a common frequency, although their phases can be different.
, the angle falls into a stable attractor (that is, the two oscillators lock in phase).
Similarly, the state space of the N=3 case is a 2-dimensional torus, and so the system evolves as a flow on the 2-torus, which cannot be chaotic.
, exact Kuramoto dynamics emerges on invariant manifolds of constant
There are a number of types of variations that can be applied to the original model presented above.
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[15] (see Transformation and Large N limit above for more info).
Network topologies such as rings and coupled populations support chimera states.
[16] 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.
Waves and synchrony are connected by a topologically distinct branch of solutions known as ripple.
[19] The existence of ripple solutions was predicted (but not observed) by Wiley, Strogatz and Girvan,[20] who called them multi-twisted q-states.
goes to infinity will be connected and it has been conjectured[24] that this value is too the number at which these random graphs undergo synchronization which a 2022 preprint claims to have proved.
[25][26] Some works in the control community have focused on the Kuramoto model on networks and with heterogeneous weights (i.e. the interconnection strength between any two oscillators can be arbitrary).
Such model allows for a more realistic study of, e.g., power grids,[28] flocking, schooling, and vehicle coordination.
[29] 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.
[30] Since the Kuramoto model seems to play a key role in assessing synchronization phenomena in the brain,[31] theoretical conditions that support empirical findings may pave the way for a deeper understanding of neuronal synchronization phenomena.
Kuramoto approximated the phase interaction between any two oscillators by its first Fourier component, namely
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.
