Canard explosion and relaxation oscillation in planar, piecewise-smooth, continuous systems

Classical canard explosion results in smooth systems require the vector field to be at least $C^3$, since canard cycles are created as the result of a Hopf bifurcation. The work on canards in nonsmooth, planar systems is recent and has thus far been restricted to piecewise-linear or piecewise-smooth Van der Pol systems, where an extremum of the critical manifold arises from the nonsmoothness. In both of these cases, a canard (or canard-like) explosion is created through a nonsmooth bifurcation as the slow nullcline passes through a corner of the critical manifold. Additionally, it is possible for these systems to exhibit a superexplosion bifurcation where the canard explosion is skipped. This paper extends the results to more general piecewise-smooth systems, finding conditions for when a periodic orbit is created through either a smooth or nonsmooth bifurcation. In the case the bifurcation is nonsmooth, conditions determining whether the bifurcation is a super-explosion or canards are created.