Isochronous dynamics in pulse coupled oscillator networks with delay.

Area of Science:

• Complex Systems
• Nonlinear Dynamics
• Network Science

Background:

• Pulse-coupled oscillators are fundamental models in neuroscience and physics.
• Understanding synchronized behavior in networks with delays is crucial.
• Discontinuous dynamics can lead to complex emergent phenomena.

Purpose of the Study:

• To investigate the emergence of isochronous regions in pulse-coupled oscillator networks with delay.
• To analyze the properties and existence of these regions.
• To characterize the dynamical behavior within these isochronous subsets of the phase space.

Main Methods:

• Analytical investigation of network dynamics.
• Numerical simulations of oscillator networks.
• Phase space analysis using surfaces of section.

Main Results:

• Demonstrated the existence of isochronous regions induced by discontinuous dynamics.
• Identified subsets of initial conditions leading to periodic orbits with identical periods.
• Characterized the properties and non-zero dimensions of these isochronous regions.
• Provided analytical and numerical proof for the observed phenomena.

Conclusions:

• Isochronous regions represent a significant simplification in the phase space of delayed pulse-coupled oscillator networks.
• The discontinuous nature of the dynamics is key to their formation.
• These findings offer insights into the collective behavior and synchronization patterns in complex oscillatory systems.