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Summary
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This study investigates slow-fast dynamics in coupled Bonhöffer-van der Pol oscillators. A double-Hopf bifurcation leads to the stable coexistence of single-mode and two-mode bursting solutions, demonstrating robust dynamics.

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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.