Folded Saddles and Faux Canards

We study the two parameter family of faux canards associated with the folded saddle singularity within the folded singularity normal form. Recently, rotational behavior of folded saddle faux canards has been reported by Vo and Wechselberger in [SIAM J. Math. Anal., 47 (2015), pp. 3235-3283] where they studied examples of systems close to a folded saddle-node type I limit. This is a surprising observation and merits a closer look at faux canards which have been somewhat neglected in the literature. We address this gap in canard knowledge and provide a comprehensive analysis of folded saddle faux canards, both numerically and analytically. We show that for certain values of mu-the eigenvalue ratio of the associated folded singularity within the reduced flow faux canards may possess rotations about the primary faux canard, and that the stable and unstable fast manifolds (i.e., the nonlinear stable and unstable fast fiber bundles) of the primary faux canard form the boundaries of sets of solutions with similar rotational behavior. This is in contrast to the folded node case where the stable and unstable slow manifolds of the primary weak canard organize the family of weak canards. We develop a numerical scheme by which we obtain these fast manifolds and locate and characterize the family of secondary faux canards and their bifurcations. We confirm the bifurcations of secondary faux canards via an extended Melnikov analysis where we find an alternating pattern of bifurcations transcritical and pitchfork similar to the alternating pattern of bifurcations of secondary weak canards in the case of the folded node. In particular, we are able to calculate third order Melnikov integrals to demonstrate pitchfork bifurcations, a result that has only been conjectured in [SIAM J. Appl. Dyn. Syst., 4 (2005), pp. 101-139] for the folded node case.