Controlling Canard Cycles

TL;DR

This paper develops a control method combining geometric singular perturbation theory, the blow-up technique, and control theory to stabilize canard cycles in planar fast-slow systems, enabling the generation of stable mixed-mode oscillations.

Contribution

It introduces a novel control approach that stabilizes canard cycles in fast-slow systems, which are traditionally difficult to detect and reproduce.

Findings

Successfully stabilizes canard cycles in planar systems

Enables controlled generation of mixed-mode oscillations

Demonstrates effectiveness on van der Pol oscillator

Abstract

Canard cycles are periodic orbits that appear as special solutions of fast-slow systems (or singularly perturbed Ordinary Differential Equations). It is well known that canard cycles are difficult to detect, hard to reproduce numerically, and that they are sensible to exponentially small changes in parameters. In this paper we combine techniques from geometric singular perturbation theory, the blow-up method, and control theory, to design controllers that stabilize canard cycles of planar fast-slow systems with a folded critical manifold. As an application, we propose a controller that produces stable mixed-mode oscillations in the van der Pol oscillator.