Kick synchronization versus diffusive synchronization

TL;DR

This paper contrasts two foundational oscillator synchronization models: diffusive synchronization, studied via continuous-time, incremental stability, dissipativity, and contraction analyses; and kick synchronization, analyzed through hybrid dynamics and firing maps. It connects the models through phase reduction, showing that in the weak-coupling limit both reduce to a Kuramoto-type phase model with coupling functions tied to the phase response curve. The authors provide concrete results for two-oscillator and large-network scenarios, including conditions like $2K>\mu$ for diffusive synchronization and monotone PRCs yielding phase-locked or finite-time synchronized states in the impulsive case, as well as infinite-population continuum results. The work highlights both the complementary nature and the distinct contraction mechanisms of the two approaches, and argues for broader exploration of hybrid kick dynamics in synchronization theory with practical relevance for natural and engineered systems.

Abstract

The paper provides an introductory discussion about two fundamental models of oscillator synchronization: the (continuous-time) diffusive model, that dominates the mathematical literature on synchronization, and the (hybrid) kick model, that accounts for most popular examples of synchronization, but for which only few theoretical results exist. The paper stresses fundamental differences between the two models, such as the different contraction measures underlying the analysis, as well as important analogies that can be drawn in the limit of weak coupling.

Figures (11)

• Figure 1: The paper is organized according to the coupling models. Sections \ref{['sec:diffusive']} and \ref{['sec:kick']} focus on (possibly strong) diffusive and kick synchronization, respectively. Section \ref{['sec:weak']} deals with the phase models encountered in the limit of weak coupling.
• Figure 2: The asymptotic phase map $\Theta:\mathcal{B}(\gamma)\rightarrow\mathbb{S}^1$ assigns to each point $\mathbf{q}$ in the basin $\mathcal{B}(\gamma)$ a single scalar phase $\theta$ on the unit circle $\mathbb{S}^1$, such that $\lim_{t\rightarrow+\infty}\left\| \mathbf{\Phi}(t,\mathbf{q},0) - \mathbf{\Phi}(t,\mathbf{p},0) \right\|_2=0$ where $\mathbf{p}=\mathbf{x}^\gamma(\theta/\omega)$. The set of all points $\mathbf{q}$ characterized by the same phase $\theta$ is the isochron $\mathcal{I}_{\theta}$.
• Figure 3: The van der Pol oscillator exhibits two different oscillation regimes: the quasi-harmonic ($\mu \ll 1$) and the relaxation ($\mu \gg 1$) oscillation regimes. Quasi-harmonic and relaxation regimes are displayed in $(x,\dot{x})$ and $(x,z)$ state-spaces, respectively (with the transformation $z = x - x^3/3 - \dot{x}/\mu$).
• Figure 4: The shape of the infinitesimal phase response curve (for the van der Pol oscillator) is very different in both regimes. Typically, it is harmonic in the weakly nonlinear oscillation regime and monotone (and hence discontinuous) in the relaxation regime.
• Figure 5: For integrate-and-fire oscillators, the finite PRC $Z_\epsilon$ is directly derived from the iPRC $Z$.

Theorems & Definitions (9)

• Definition 1
• Definition 2
• Remark 1
• Remark 2: Integrate-and-fire oscillators
• Remark 3
• Theorem 1
• Theorem 2
• Remark 4: van der Pol oscillator
• Theorem 3