Slow-fast normal forms arising from piecewise smooth vector fields

TL;DR

This paper investigates the emergence of slow-fast dynamics from regularized piecewise smooth vector fields, introducing an algorithm to construct transition functions that generate various singularities in the regularized systems.

Contribution

It develops an algorithm for constructing transition functions that produce slow-fast singularities from piecewise smooth vector field normal forms.

Findings

Regularization leads to slow-fast systems with typical singularities.

Constructed transition functions can generate fold, transcritical, and pitchfork singularities.

Application of the method to normal forms demonstrates its effectiveness.

Abstract

We studied piecewise smooth differential systems of the form $$\dot{z} = Z(z) = \dfrac{1 + \operatorname{sgn}(F)}{2}X(z) + \dfrac{1 - \operatorname{sgn}(F)}{2}Y(z),$$ where $F: \mathbb{R}^{n}\rightarrow \mathbb{R}$ is a smooth map having 0 as a regular value. We consider linear regularizations of the vector field $Z$ given by $$\dot{z}= Z_{\varepsilon}(z) = \dfrac{1 + \varphi(F/\varepsilon)}{2}X(z) +\dfrac{1 - \varphi(F /\varepsilon)}{2}Y(z),$$where $\varphi$ is a transition function (not necessarily monotonic) and nonlinear regularizations of the vector field $Z$ whose transition function is monotonic. It is a well-known fact that the regularized system is a slow-fast system. The main contribution of this paper is the study of typical singularities of slow-fast systems that arise from (linear or nonlinear) regularizations. We developed an algorithm to construct suitable transition functions, and we apply these ideas in order to create slow-fast singularities from normal forms of piecewise smooth vector fields. We present examples of transition functions that, after regularization of a PSVF normal form, generate normally hyperbolic, fold, transcritical, and pitchfork singularities.