Bursting multistability induced by double-Hopf bifurcation

Area of Science:

• Nonlinear Dynamics
• Computational Neuroscience
• Physical Systems

Background:

• Coupled Bonhöffer-van der Pol oscillators exhibit complex dynamics.
• Slow-fast systems with bifurcations are crucial for understanding phenomena like bursting.
• Multistability in dynamical systems leads to diverse solution behaviors.

Purpose of the Study:

• To analyze the slow-fast dynamics of coupled Bonhöffer-van der Pol oscillators under a slowly varying parameter near a double-Hopf bifurcation.
• To investigate the emergence and coexistence of single-mode and two-mode bursting solutions.
• To elucidate the underlying mechanisms of bursting solution coexistence and their robustness.

Main Methods:

• Numerical simulations of coupled Bonhöffer-van der Pol oscillators.
• Analysis of slow-fast dynamics and bifurcation theory.
• Examination of phase portraits for the fast subsystem.

Main Results:

• Two distinct bursting solutions (single-mode and two-mode) were observed.
• A double-Hopf bifurcation was found to induce a stable coexistence of these two bursting solutions.
• The coexistence mechanism was explained through slowly changing phase portraits, showing robustness.

Conclusions:

• The double-Hopf bifurcation in this slow-fast system facilitates a stable coexistence of different bursting patterns.
• The findings highlight the intricate interplay between bifurcations and multistability in generating complex dynamics.
• The observed bursting coexistence is robust, offering insights into oscillatory systems.