Importance of sufficient precision in stable dynamics for the numerical computation of canards in singularly perturbed systems

Abstract

Canards are characteristic phenomena of singularly perturbed systems with multiple time scales. Using the Bonhoeffer-van der Pol equations, we perform asymptotic analysis to obtain the parameter values where a canard solution exists. Although the system is two-dimensional, the numerical computations of the equations show a seemingly chaotic phenomenon at these parameter values. Consequently, high-precision numerical computations are employed to quantitatively evaluate both the convergent and divergent dynamics near a canard solution. The numerically produced fake chaos is attributed to insufficient precision of the stable dynamics.