Journal
JOURNAL OF DYNAMICAL AND CONTROL SYSTEMS
Volume -, Issue -, Pages -Publisher
SPRINGER/PLENUM PUBLISHERS
DOI: 10.1007/s10883-023-09657-x
Keywords
Piecewise smooth vector fields; Geometric singular perturbation theory; Regularization of piecewise smooth vector fields; Transition function
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We study planar piecewise smooth differential systems with a regular value of 0. Linear regularizations and nonlinear regularizations are considered, with transition functions that may or may not be monotonic. The paper focuses on the typical singularities of slow-fast systems that arise from regularizations, such as fold, transcritical, and pitchfork singularities. The dependence of the slow-fast system on the graphical properties of the transition function is also investigated.
We study planar piecewise smooth differential systems of the form [GRAPHICS] . where F : R-2 -> R is a smooth map having 0 as a regular value. We consider linear regularizations Z(epsilon)(phi) of Z by replacing sgn(F) by phi(F/epsilon) in the last equation, with epsilon > 0 small and phi being a transition function (not necessarily monotonic). Nonlinear regularizations of the vector field Z whose transition function is monotonic are considered too. It is a wellknown fact that the regularized system is a slow-fast system. In this paper, we study typical singularities of slow-fast systems that arise from (linear or nonlinear) regularizations, namely, fold, transcritical and pitchfork singularities. Furthermore, the dependence of the slow-fast system on the graphical properties of the transition function is investigated.
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