Slow switching in globally coupled oscillators: robustness and occurrence through delayed coupling.

Area of Science:

• Physics
• Nonlinear Dynamics
• Complex Systems

Background:

• Collective dynamics in globally coupled oscillators are crucial for understanding complex systems.
• Slow switching, a specific collective behavior, was initially observed in phase oscillator models.
• This phenomenon involves transitions between clustered states mediated by heteroclinic loops.

Purpose of the Study:

• To investigate the emergence of slow switching in a broader range of oscillator models, including those with amplitude dynamics.
• To demonstrate that heteroclinic loops arise robustly in homogeneous populations of globally coupled oscillators.
• To explore the role of time delay and perturbations in inducing and triggering slow switching.

Main Methods:

• Analysis of oscillator models with amplitude degrees of freedom.
• Mathematical investigation of heteroclinic loop formation in homogeneous populations.
• Introduction of time delay to induce heteroclinic loops and study bifurcations (transcritical, saddle-node).
• Perturbation analysis using a vector model with weak noise and symmetry-breaking forces.

Main Results:

• Slow switching dynamics are not limited to phase oscillators but can occur in models with amplitude dynamics.
• Heteroclinic loops are shown to form inevitably and persist robustly in globally coupled oscillator populations.
• Time-delayed coupling readily induces heteroclinic loops, characterized by specific bifurcation types.
• Weak perturbations, such as noise, can trigger slow switching in systems exhibiting heteroclinic loops.

Conclusions:

• Heteroclinic loops provide a general mechanism for slow switching in globally coupled oscillators.
• Time delay is a key factor in robustly generating these loops, even in systems tending towards synchrony.
• The study expands the understanding of collective dynamics in coupled oscillator systems and their sensitivity to perturbations.