Slow-fast normal forms arising from piecewise smooth vector fields

2022/05/04 by Otavio Henrique Perez, Perez, Otavio Henrique, Gabriel Rondón +3 · 1 citation
Mathematics · #34A09 #34C45 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:34A09 #msc:34C45

paper · pdf · doi:10.48550/arxiv.2205.02263

arxiv created 2022/05/04 · arxiv updated 2022/05/06

Abstract

We studied piecewise smooth differential systems of the form z = Z(z) = \dfrac1 + sgn(F)2X(z) + \dfrac1 - sgn(F)2Y(z), where F: ℝⁿ→ ℝ is a smooth map having 0 as a regular value. We consider linear regularizations of the vector field Z given by z= Z_ε(z) = \dfrac1 + φ(F/ε)2X(z) +\dfrac1 - φ(F /ε)2Y(z),where φ 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.

Cited by

• Canard cycles of non-linearly regularized piecewise smooth vector fields

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