Regularizations of two-fold bifurcations in planar piecewise smooth systems using blowup

We use blowup to study the regularization of codimension-one two-fold singularities in planar piecewise smooth (PWS) dynamical systems. We focus on singular canards, pseudoequilibria, and limit cycles that can occur in the PWS system. Using the regularization of Sotomayor and Teixeira, we show rigorously how singular canards can persist, how the bifurcation of pseudoequilibria is related to bifurcations of equilibria in the regularized system, and that PWS limit cycles are connected to Hopf bifurcations of the regularization. In addition, we show how regularization can create another type of limit cycle that does not appear to be present in the original PWS system. For both types of limit cycle, we show that the criticality of the Hopf bifurcation that gives rise to periodic orbits is strongly dependent on the precise form of the regularization. Finally, we analyze the limit cycles as locally unique families of periodic orbits of the regularization and connect them, when possible, to limit cycles of the PWS system. We illustrate our analysis with numerical simulations and show how the regularized system can undergo a canard explosion phenomenon.