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This paper deals with singular perturbation problems for vector fields on 2-dimensional manifolds. " Canard solutions " are solutions that, starting near an attracting normally hyperbolic branch of the singular curve, cross a " turning point " and follow for a while a normally repelling branch of the singular curve. Following the geometric ideas developed… (More)

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 Liénard equations. The obtained limit cycles are relaxation oscillations. Both classical Liénard equations and generalized Liénard equations are treated.

- Renato HuzakB, Peter De Maesschalck, Alberto Cabada
- 2014

Using techniques from singular perturbations we show that for any n ≥ 6 and m ≥ 2 there are Liénard equations { ˙ x = y − F(x), ˙ y = G(x)}, with F a polynomial of degree n and G a polynomial of degree m, having at least 2[ n−2 2 ] + [ m 2 ] hyperbolic limit cycles, where [·] denotes " the greatest integer equal or below " .

Continuation techniques have been known to successfully describe bifurcation diagrams appearing in slow-fast systems with more than one slow variable (see eg. [12]). 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,… (More)

In ordinary differential equations of singular perturbation type, the dynamics of solutions near saddle-node bifurcations of equilib-ria are rich. Canard solutions can arise, which, after spending time near an attracting equilibrium, stay near a repelling branch of equilibria for long intervals of time before finally returning to a neighborhood of the… (More)

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