Canard-like phenomena in piecewise-smooth Van der Pol systems

Area of Science:

• Dynamical Systems
• Nonlinear Dynamics
• Mathematical Physics

Background:

• Canard phenomena are typically observed in smooth nonlinear systems.
• Piecewise-linear systems exhibit simplified versions of these phenomena.
• Understanding canards in piecewise-smooth systems is crucial for modeling complex dynamics.

Purpose of the Study:

• To investigate the occurrence and characteristics of canard phenomena in nonlinear, piecewise-smooth planar dynamical systems.
• To analyze the bifurcation mechanisms underlying canard formation and related phenomena.
• To compare the behavior of canards in piecewise-smooth systems with those in smooth and piecewise-linear systems.

Main Methods:

• Analysis of nonlinear, piecewise-smooth planar dynamical systems.
• Identification of bifurcations involving the slow-nullcline and splitting manifold.
• Characterization of canard phenomena, including transitions from small cycles to canards with heads.
• Investigation of the super-explosion phenomenon and subcritical bifurcations.

Main Results:

• Nonlinear, piecewise-smooth systems can exhibit canard phenomena.
• Nonlinearity allows for canards to transition from small cycles to canards with heads, resembling smooth system behavior.
• Bifurcations leading to canards occur when the slow-nullcline coincides with the splitting manifold.
• This bifurcation can result in super-explosion: an instantaneous transition from periodic orbits to relaxation oscillations.
• The bifurcation, whether leading to canards or super-explosion, can be subcritical.

Conclusions:

• Nonlinear, piecewise-smooth systems provide a richer platform for studying canard phenomena than previously thought.
• The identified bifurcation mechanism offers new insights into the transition between different types of oscillatory behaviors.
• The possibility of subcritical bifurcations highlights the complex and potentially abrupt nature of dynamical changes in these systems.