Double canard cycles in singularly perturbed planar systems with two canard points

TL;DR

This paper investigates the existence and properties of double canard cycles in singularly perturbed planar systems with two canard points, extending previous work on systems with a single canard point.

Contribution

The study introduces a detailed analysis of double canard cycles involving two canard points using normal form, blow-up, and Melnikov theories, and applies results to cubic Lienard equations.

Findings

Identification of conditions for double canard cycles

Extension of canard theory to systems with two canard points

Application to cubic Lienard equations with quadratic damping

Abstract

We consider double canard cycles including two canards in singularly perturbed planar systems with two canard points. Previous work studied the complex oscillations including relaxation oscillations and canard cycles in singularly perturbed planar systems with one-parameter layer equations, which have precisely one canard point, two jump points or one canard point and one jump point. Based on the normal form theory, blow-up technique and Melnikov theory, we investigate double canard cycles induced by two Hopf breaking mechanisms at two non-degenerate canard points. Finally, we apply the obtained results to a class of cubic Lienard equations with quadratic damping.