Periodic orbits and chaos in fast–slow systems with Bogdanov–Takens type fold points

The existence of stable periodic orbits and chaotic invariant sets of singularly perturbed problems of fast–slow type having Bogdanov–Takens bifurcation points in its fast subsystem is proved by means of the geometric singular perturbation method and the blow-up method. In particular, the blow-up method is effectively used for analyzing the flow near the Bogdanov–Takens type fold point in order to show that a slow manifold near the fold point is extended along the Boutroux's tritronquée solution of the first Painlevé equation in the blow-up space.