Generic torus canards

摘要: Abstract Torus canards are special solutions of fast/slow systems that alternate between attracting and repelling manifolds limit cycles the fast subsystem. A relatively new dynamic phenomenon, torus have been found in neural applications to mediate transition from tonic spiking bursting via amplitude-modulated spiking. In R 3 , degenerate: they require one-parameter families 2-fast/1-slow order be observed even then, only occur on exponentially thin parameter intervals. The addition a second slow variable unfolds canard making it generic robust. That is, with (at least) two variables open sets. So far, studied numerically, their behaviour has inferred based averaging theory. This approach, however, not rigorously justified since method breaks down near fold periodics, which is exactly where originate. this work, we combine techniques Floquet theory, geometric singular perturbation theory show average folded singularity canard. so doing, devise an analytic scheme for identification topological classification k variables, any positive integer . We demonstrate predictive power our results model intracellular calcium dynamics, explain mechanisms underlying novel class elliptic rhythms, called bursting, by constructing analogues mixed-mode oscillations. also make explicit connection here prior studies explosion discuss how methods can extended arbitrary (finite) dimension.