P. De Maesschalck

BIFURCATIONS OF MULTIPLE RELAXATION OSCILLATIONS IN POLYNOMIAL LIENARD EQUATIONS

In this paper, we prove the presence of limit cycles of given multi- plicity, together with a complete unfolding, in families of (singularly perturbed) polynomial Li enard equations. The obtained limit cycles are relaxation oscil- lations. Both classical Li enard equations and generalized Li enard equations are treated.

Numerical Continuation Techniques for Planar Slow-Fast Systems ∗

Continuation techniques have been known to successfully describe bifurcation diagrams appearing in slow-fast systems with more than one slow variable (see, e.g., [M. Desroches, B. Krauskopf, and H. M. Osinga, Nonlinearity, 23 (2010), pp. 739--765]). In this paper we investigate the usefulness of numerical continuation techniques dealing with some solved and some open problems in the study of planar singular perturbations. More precisely, we first verify known theoretical results (thereby showing the reliability of this numerical tool) on the appearance of multiple limit cycles of relaxation-oscillation type and on the existence of multiple critical periods in well-chosen annuli of slow-fast periodic orbits in the plane. We then apply the technique to study the period function in detail.

Sector-delayed-Hopf-type mixed-mode oscillations in a prototypical three-time-scale model

We consider a three-dimensional three-time-scale system that was first proposed by Krupa et?al. (2008) under the additional assumption that two singular perturbation parameters are present in the equations. While the presence of three scales was shown to give rise to canard-induced periodic mixed-mode oscillations (MMOs) (Desroches et?al., 2012) in the parameter regime studied by Krupa et?al. (2008), we additionally observe mixed-mode patterns that display delayed-Hopf-type behaviour (Ne?shtadt, 1987). We present analytical and numerical evidence for the occurrence of stable periodic dynamics that realises both mechanisms, and we discuss the transition between them. To the best of our knowledge, the resulting mixed sector-delayed-Hopf-type MMO trajectories represent a novel class of mixed-mode dynamics in singularly perturbed systems of ordinary differential equations.