Periodic canard trajectories with multiple segments following the unstable part of critical manifold

Discrete and Continuous Dynamical Systems - Series B. 2013. Vol. 18. No. 2. P. 467–482.

Krasnosel'skii Alexander M., O'Grady E., Pokrovskii A., Rachinskii D.

We consider a scalar fast differential equation which is periodically driven by a slowly varying input. Assuming that the equation depends on scalar parameters, we present simple sufficient conditions for the existence of a periodic canard solution, which, within a period, makes n fast transitions between the stable branch and the unstable branch of the folded critical curve. The closed trace of the canard solution on the plane of the slow input variable and the fast phase variable has n portions elongated along the unstable branch of the critical curve. We show that the length of these portions and the length of the time intervals of the slow motion separated by the short time intervals of fast transitions between the branches are controlled by the parameters.

Keywords: stability, Periodic solutions, degree theory, canard trajectory, Периодические траектории, решения-утки, теория степени, устойчивость решений