Synchronization Phenomenon in Three-Time-Scale Systems
Navojit Dhali Pallab
TL;DR
The paper addresses synchronization in networks of heterogeneous three-time-scale oscillators with canard dynamics, proposing a relaxed near-synchrony criterion based on $V_v$ and a tolerance $\epsilon$. It develops a mathematical framework that yields a differential inequality for $V_v$, uses the change of variables $W = \sqrt{V_v}$, and applies Gronwall's lemma to obtain a bound showing how coupling strength $k$ must overcome heterogeneity $M$ and time-scale separation. The main contribution is an explicit sufficient condition on $k$, namely $k > \max\left( \frac{2M}{\sqrt{\epsilon}}, \frac{1}{\delta t_{linger}^{\min}} \ln\left( \frac{2 W_0}{\sqrt{\epsilon}} \right) \right)$, ensuring synchronization within the minimum linger time across the network. This result provides a practical design criterion for achieving stable near-synchrony in complex slow-fast networks, with implications for biological systems such as $\beta$-cell networks where coordinated bursting is essential.
Abstract
This paper investigates synchronization phenomena in networks of coupled oscillators governed by three-time-scale dynamical systems exhibiting canard dynamics. A mathematical framework has been developed to analyze the synchronization of fast variables across heterogeneous systems, deriving a sufficient condition for the synchronization error to fall below a specified threshold within the minimum linger time. This condition accounts for coupling strength, heterogeneity, and time-scale separation, ensuring stable oscillatory behavior in the network. The result, supported by rigorous mathematical analysis, advances the understanding of synchronization in complex multi-time-scale systems.