Many-circuit canard trajectories and their applications

We study the case when two distinct curves of slow motion in a two-dimensional relaxation system with cylindrical phase space intersect each other in a generic way. We establish that the so-called canard trajectories can arise in this situation. They differ from ordinary canard trajectories in the following respect. The passage from the stable curve of slow motion to the unstable one is performed via finitely many asymptotically quick rotations of the phase point around the axis of the cylinder. The results obtained are used in the asymptotic analysis of eigenvalues of a boundary-value problem of Sturm–Liouville type for a singularly perturbed linear Schrödinger equation.