Synchronization rates in classes of relaxation oscillators.

Area of Science:

• Complex Systems
• Nonlinear Dynamics
• Computational Physics

Background:

• Relaxation oscillators are fundamental in diverse scientific fields, including physics, electronics, mathematics, and biology.
• Their mathematical models allow for diverse oscillatory behaviors based on parameter selection.
• Understanding synchronization in coupled oscillator networks is crucial for modeling complex phenomena.

Purpose of the Study:

• To numerically analyze the synchronization rates of four distinct classes of relaxation oscillators.
• To investigate the impact of different interaction functions (Heaviside, sigmoid) on synchronization dynamics.
• To compare synchronization behavior in one-dimensional chains and two-dimensional networks.

Main Methods:

• Numerical simulation of one-dimensional chains and two-dimensional networks of relaxation oscillators.
• Analysis of synchronization rates using a Heaviside step function interaction.
• Exploration of different oscillator regimes: sinusoidal, relaxation, singular limit, and spike-like oscillations.
• Inclusion of sigmoid interaction to assess coupling effects.

Main Results:

• Oscillators in sinusoidal and relaxation regimes synchronized as ≈ n (chain length).
• Singular limit oscillators showed ≈ n^p (p < 0.5).
• Spike-like oscillators synchronized as ≈ log(n) in chains and ≈ log(L) in 2D networks.
• Sigmoid interaction led to ≈ n^2 for sinusoidal/relaxation regimes.

Conclusions:

• The type of relaxation oscillator significantly influences network synchronization speed.
• The coupling function is a critical factor determining synchronization rates.
• Different network topologies (1D vs. 2D) affect synchronization for spike-like oscillators.