Mixed-mode oscillations from a constrained extended Bonhoeffer-van der Pol oscillator with a diode

Area of Science:

• Nonlinear Dynamics
• Circuit Theory
• Chaos Theory

Background:

• The two-variable Bonhoeffer-van der Pol (BVP) oscillator is known for exhibiting canard explosions.
• Extended BVP oscillators, naturally forming three-variable systems, are capable of generating mixed-mode oscillations (MMOs).

Purpose of the Study:

• To investigate the dynamics of an extended BVP oscillator with a nonlinear conductor featuring an idealized diode.
• To explain the generation of MMOs and related bifurcations using one-dimensional (1D) return maps.

Main Methods:

• Modeling the extended BVP oscillator with an idealized diode, leading to a constrained equation.
• Constructing 1D Poincaré return maps for the system dynamics.
• Analyzing phenomena such as simple MMOs, MMO-incrementing bifurcations, and asymmetric Farey trees.
• Experimental verification of theoretical findings.

Main Results:

• The idealized diode case leads to a degenerate system where dynamics are defined only forward in time.
• 1D return maps successfully explain simple MMOs and MMO-incrementing bifurcations.
• The oscillator exhibits both small-amplitude oscillations (supercritical Hopf bifurcation) and large-amplitude relaxation oscillations (canard explosion).
• Asymmetric Farey trees characterize the MMO bifurcations between small and large amplitude oscillations.

Conclusions:

• The study provides a theoretical framework, supported by experiments, for understanding MMOs in extended BVP oscillators with idealized diodes.
• The use of 1D return maps offers a simplified yet effective method for analyzing complex oscillatory behaviors in such systems.