Controlling Canard Cycles

Hildeberto Jardón-Kojakhmetov,Christian Kuehn

doi:10.1007/s10883-021-09553-2

Hildeberto Jardón-Kojakhmetov, Christian Kuehn

Open Access

https://doi.org/10.1007/s10883-021-09553-2

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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.

Highlights

IntroductionFast-slow systems ( known as singularly perturbed ordinary differential equations, see more details in Section 2) are often used to model phenomena occurring in two or more time scales
Fast-slow systems are often used to model phenomena occurring in two or more time scales
We have presented a methodology combining the blow-up method with Lyapunov-based control techniques to design a controller that stabilizes canard cycles

Summary

Introduction

Fast-slow systems ( known as singularly perturbed ordinary differential equations, see more details in Section 2) are often used to model phenomena occurring in two or more time scales Examples of these are vast and range from oscillatory patters in biochemistry and neuroscience [6, 18, 25, 26], all the way to stability analysis and control of power networks [10, 14], among many others [41, Chapter 20]. Canards are extremely important in the theory of fast-slow systems, and in applied sciences, and especially in neuroscience, as they have allowed, for example, the detailed description of the very fast onset of large amplitude oscillations due to small changes of a parameter in neuronal models [18, 26] and of other complex oscillatory patterns [9, 12, 47]. Due to their very nature, canard orbits are not robust, meaning that small perturbations may drastically change the shape of the orbit

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