Analytical and experimental study of two delay-coupled excitable units.

Area of Science:

• Dynamical Systems and Control
• Nonlinear Dynamics
• Theoretical Physics

Background:

• Excitable systems, though nonoscillatory, can exhibit complex dynamics when coupled.
• Time-periodic oscillations, specifically relaxation oscillations, arise from the interplay of slow and fast processes.
• Time delays in coupled systems are crucial for understanding emergent behaviors.

Purpose of the Study:

• To investigate the onset of time-periodic oscillations in a system of two identical delay-coupled excitable units.
• To analyze the role of time delay in the dynamics of these oscillations.
• To compare theoretical predictions with experimental results.

Main Methods:

• Asymptotic methods were employed for theoretical analysis of the delay differential equations.
• Experimental investigation using two coupled electronic circuits that model the mathematical system.
• Bifurcation analysis to compare analytical and experimental findings.

Main Results:

• The study describes the emergence of relaxation oscillations characterized by distinct slow and fast phases.
• Asymptotic analysis revealed the critical role of time delay in the fast transition layers.
• Experimental results showed quantitative agreement with the analytical bifurcation diagrams.

Conclusions:

• Time delay is a key parameter in initiating and shaping oscillations in coupled excitable systems.
• The mathematical model accurately captures the experimentally observed dynamics.
• This work validates theoretical approaches for analyzing complex behaviors in delay-coupled systems.