Synchronization of stochastic hybrid oscillators driven by a common switching environment

Area of Science:

• Complex systems dynamics
• Nonlinear dynamics and chaos
• Stochastic processes

Background:

• Ordinary differential equations (ODEs) model many biological, physical, and chemical systems.
• Piecewise smooth ODEs with Markov jump processes exhibit complex dynamics.
• Fast switching limits of these systems converge to deterministic ODEs.

Purpose of the Study:

• Investigate synchronization in uncoupled oscillators driven by a common Markov jump process.
• Analyze the fast switching limit of piecewise smooth ODEs supporting stable limit cycles.
• Determine the Lyapunov coefficient governing phase difference decay.

Main Methods:

• Analysis of synchronization in the fast switching limit of ODEs with Markov jump processes.
• Calculation of the Lyapunov coefficient using quasi-steady-state approximation and second-order perturbation expansion.
• Numerical simulations using the radial isochron clock model.

Main Results:

• Uncoupled oscillators sharing a common Markov jump process can synchronize.
• The Lyapunov coefficient calculation reveals discrepancies between approximation methods.
• A second-order perturbation expansion provides a more accurate Lyapunov coefficient.

Conclusions:

• Synchronization is achievable in systems governed by common random environments modeled by Markov jump processes.
• The discrete nature of the Markov jump process necessitates careful analysis of synchronization dynamics.
• The second-order perturbation expansion offers a more accurate method for predicting synchronization rates in such systems.