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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 by… (More)

In ordinary differential equations of singular perturbation type, the dynamics of solutions near saddle-node bifurcations of equilibria 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)

- Peter De Maesschalck, M. Desroches
- SIAM J. Applied Dynamical Systems
- 2013

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)

- P. De Maesschalck, S. Rebollo-Perdomo, J. Torregrosac, Salomón Rebollo-Perdomo, Joan Torregrosa
- 2015

This paper concerns the study of small-amplitude limit cycles that appear in the phase portrait near an unfolded fake saddle singularity. This degenerate singularity is also known as a impassable grain. The normal form of the unperturbed vector field is like a degenerate flow box. Near the singularity,the phase portrait consists of parallel fibers, all of… (More)

- Peter De Maesschalck, Ekaterina Kutafina, Nikola Popovic
- Applied Mathematics and Computation
- 2016

- 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 {ẋ = y− F(x), ẏ = 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”.

In this paper, we prove the presence of limit cycles of given multiplicity, 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.

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