Interplay of the mechanisms of synchronization by common noise and global coupling for a general class of limit-cycle oscillators

Abstract

We construct an analytical theory of interplay between synchronizing effects by common noise and by global coupling for a general class of smooth limit-cycle oscillators. Both the cases of attractive and repulsive coupling are considered. The derivation is performed within the framework of the phase reduction, which fully accounts for the amplitude degrees of freedom. Firstly, we consider the case of identical oscillators subject to intrinsic noise, obtain the synchronization condition, and find that the distribution of phase deviations always possesses lower-law heavy tails. Secondly, we consider the case of nonidentical oscillators. For the average oscillator frequency as a function of the natural frequency mismatch, limiting scaling laws are derived; these laws exhibit the nontrivial phenomenon of frequency repulsion accompanying synchronization under negative coupling. The analytical theory is illustrated with examples of Van der Pol and Van der Pol-Duffing oscillators and the neuron-like FitzHugh-Nagumo system; the results are also underpinned by the direct numerical simulation for ensembles of these oscillators.

Phase Models Beyond Weak Coupling

“…While phase reduction is useful in many applications, its applicability degrades as coupling strength increases, often leading to incorrect predictions about dynamical behavior. Due to this limitation, recent years have seen a flurry of interest in the development and use of nonlinear model reduction strategies to characterize the dynamical behavior of coupled limit cycle oscillators in situations where the weak coupling approximation is not sufficient [9], [10], [11], [12], [13], [14], [15], [16]. In general, this is a difficult task for an n-dimensional model as greater accuracy coupling functions must usually be found by considering the dynamical behavior of n − 1 coordinates transverse to a limit cycle.…”

Controlling oscillator coherence by multiple delay feedback

“…Physically, this approximation corresponds to a diffusion approximation for R(τ = D ; if one approximately employs the same law for a finite τ , then R(τ ) ≈ Dτ /2, which is Eq. (27). This approximation is practically not less accurate than the linear-in-noise approximation and the linear-in-feedback one; additionally, it works well for a strong noise [32].…”

“…(10) from (7)-(8) becomes very subtle in the presence of a white noise, and the consideration of the Stratonovich drift term, where it is not negligible, can be insufficiently rigorous. The derivation should be generally performed via the multiple scale expansion for the Fokker-Planck equation (e.g., see [26,27] or [28]); importantly, for such a derivation, one must strictly use the genuine phase variable, but not a protophase [26]. The Stratonovich drift, if it does not vanish after averaging, merely results in the average frequency bias [29][30][31] and does not make the phase equation more complex [27].…”

Ott-Antonsen ansatz truncation of a circular cumulant series

“…Noticeably, the distribution of phase deviations in high-synchrony regimes is Cauchy even though both the common and intrinsic noises are Gaussian. When the common-noise synchronization mechanism is affected by global coupling, the distribution changes to [42]), where θ is the phase deviation from the cluster center; the distribution half-width σ is proportional to the intrinsic noise strength and σ ∝ (−λ) −1/2 , λ is the Lyapunov exponent of an oscillator without intrinsic noise; µ = [coupling strength]/(−2λ). Perfect synchrony of identical oscillators without intrinsic noise occurs for µ > −1/2.…”