Symmetry-breaking rhythms in coupled, identical fast-slow oscillators

Area of Science:

• Dynamical Systems Theory
• Nonlinear Dynamics
• Mathematical Modeling

Background:

• Coupled, identical, fast-slow systems exhibit complex dynamics due to symmetry-breaking.
• These dynamics include significant differences in oscillation amplitudes and frequencies, indicating distinct functional states.

Purpose of the Study:

• To introduce a novel analytical method for studying symmetry-breaking in coupled fast-slow systems.
• To identify the geometric structures underlying these symmetry-breaking phenomena.
• To demonstrate the robustness of various symmetry-breaking rhythms.

Main Methods:

• Development of a new analytical technique to probe symmetry-breaking.
• Geometric analysis to identify key structures driving the dynamics.
• Application to prototypical systems like the van der Pol and Lengyel-Epstein models.

Main Results:

• The method successfully identifies geometric structures responsible for symmetry-breaking.
• Diverse symmetry-breaking rhythms are shown to arise robustly.
• Observed behaviors include phase-shifted oscillations, large-amplitude oscillations, mixed-mode oscillations, and limit cycle canard explosions.

Conclusions:

• The novel analytical method provides deep insights into symmetry-breaking in fast-slow systems.
• Symmetry-breaking leads to a rich variety of robust dynamical behaviors and distinct oscillatory patterns.
• The findings are applicable to electrical circuits and chemical oscillators.