Navojit Dhali Pallab | Searcharxiv

arXiv2025

Synchronization Phenomenon in Three-Time-Scale Systems

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.

math.DS↗

arXiv2025

Pancreatic $β-$Cell Dynamics with Three-Time-Scale Systems

Pancreatic $β-$cells regulate insulin secretion through complex oscillations, which are vital for glucose control and diabetes research. In this paper, an existing mathematical model of $β-$cell dynamics is analyzed using a three-time-scale framework to study interactions among fast, intermediate, and slow variables. Through Geometric Singular Perturbation Theory (GSPT), the influence of ATP on oscillatory dynamics via membrane potential is explored. At the non-hyperbolic point, where standard methods fail, blow-up analysis is applied to investigate canard dynamics shaped by intermediate and slow variables. Numerical simulations with varied parameters reveal the glucose-dependent oscillations linked to slow dynamics near the pseudo-singular points. By leveraging the pseudo-singular point, the linger time is defined, and simulated results for the coupling strength needed for bursting initiation synchronization are presented as a sufficient condition. This study links mathematics and biology, offering insights into diabetic studies.

math.DS↗