
DE MAESSCHALCK, Peter; RebolloPerdomo, S.; Torregrosa, J.
This paper concerns the study of smallamplitude limit cycles that appear in the phase portrait near an unfolded fake saddle singularity. This degenerate singularity is also known as an impassable grain. The canonical form of the unperturbed vector field is like a degenerate flow box. Near the singularity, the phase portrait consists of parallel fibers, all but one of which have no singular points, and at the singular fiber, there is one node. We demonstrate different techniques in order to show that the cyclicity is bigger than or equal to two when the canonical form is quadratic. (C) 2014 Elsevier...

Huzak, Renato; De Maesschalck, Peter
Using techniques from singular perturbations we show that for any n???6 and m???2 there are Li??nard 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".

HUZAK, Renato; DE MAESSCHALCK, Peter; DUMORTIER, Freddy
In this paper we continue the study of ``large" smallamplitude limit cycles in slowfast codimension 3 elliptic bifurcations which is initiated in [8]. Our treatment is based on blowup and good normal forms.

DE MAESSCHALCK, Peter; HUZAK, Renato
We significantly improve lower bounds for the number of limit cycles for polynomial classical Li??nard equations, aiming at stating lower bounds that are reasonable enough to be optimal. The techniques used are the notion of slow divergence integral from the geometric theory of planar slowfast systems.

DE MAESSCHALCK, Peter; DUMORTIER, Freddy; Roussarie, Robert
We introduce transitory canard cycles for slow???fast vector fields in the plane. Such cycles separate ???canards without head??? and ???canardswithhead???,like for example in the Van der Pol equation. We obtain optimal upper bounds on the number of periodic orbits that can appear near the cycle underwhatever condition on the related slow divergence integralI,including the challenging caseI=0.

Bonckaert, Patrick
Given a 1:1 resonant saddle singularity of a planar analytic vector field, we provide a linearization procedure using a series expansion in compensators of Mourtadatype, and show that this series has Gevrey1 asymptotics. In case of an analytic Poincar\'eDulac normal form we show that this transformation is analytic as a function of the compensators

DE MAESSCHALCK, Peter; Popovic, Nikola; KUTAFINA, Ekaterina
We consider an extended threedimensional Bonhoeffervan der Pol oscillator which generalises the planar FitzHughNagumo model from mathematical neuroscience, and which was recently studied by Sekikawa et al. and by Freire and Gallas. Focussing on a parameter regime
which has hitherto been neglected, and in which the governing equations evolve on three distinct timescales, we propose a reduction to a model problem that was formulated by Krupa et al. as a canonical form for such systems. Based on previously obtained results in, we characterise completely the mixedmode dynamics of the resulting three timescale extended Bonhoeffervan der Pol oscillator from the point of...

KUTAFINA, Ekaterina
Following the paper of Shimizu et al. (Phys Lett A 375:1566, 2011), we consider the Bonhoeffervan der Pol oscillator with nonautonomous periodic perturbation. We show that the presence of mixed mode oscillations reported in that paper can be explained using the geometric singular perturbation theory. The considered model can be rewritten as a fourdimensional (locally threedimensional) autonomous system, which under certain conditions has a folded saddlenode singularity and additionally can be treated as a three time scale one.

Coll, B.; DUMORTIER, Freddy; Prohens, R.
We show that every finite configuration of disjoint simple closed curves in the plane is topologically realizable as the set of limit cycles of a polynomial Liénard equation.The related vector field X is Morse–Smale. Moreover it has the minimum numberof singularities required fo rrealizing the configuration in a Liénardequation. We provide an explicit upper bound on the degree of X, which is lower than the results obtained before, obtained in the context of general polynomial vector fields.
©2013ElsevierInc. All rights reserved.

Dumortier, Freddy; Ibanez, Santiago; Kokubu, Hiroshi; Simo, Carles
We study arbitrary generic unfoldings of a Hopfzero singularity of codimension two. They can be written in the following normal form: {x' = y + mu x  axz + A(x,y,z,lambda,mu) y' = x + mu y  ayz + B(x,y,z,lambda,mu) z' = z(2) vertical bar lambda vertical bar b(x(2) vertical bar y(2)) vertical bar C(x,y,z,lambda,mu), with a > 0, b > 0 and where A, B, C are Cinfinity or Comega functions of order O(vertical bar vertical bar(x,y,z,lambda,mu)vertical bar vertical bar(3)). Despite that the existence of Shilnikov homoclinic orbits in unfoldings of Hopfzero singularities has been discussed previously in...

DE MAESSCHALCK, Peter; DUMORTIER, Freddy; Roussarie, Robert
We consider transitory canard cycles that consist of a generic breaking mechanism, i.e. a Hopf or a jump breaking mechanism, in combination with a fastfast passage through a jump point. Such cycle separates two types of canard cycles with a different shape. We obtain upper bounds on the number of periodic orbits that can appear near the canard cycle, and this under very general conditions.

BONCKAERT, Patrick; VERSTRINGE, Freek
We explore the convergence/divergence of the normal form for a singularity of a vector field on Cn with nilpotent linear part. We show that a Gevreyalpha vector field X with a nilpotent linear part can be reduced to a normal form of Gevrey1 + alpha type with the use of a Gevrey1 + alpha transformation. We also give a proof of the existence of an optimal order to stop the normal form procedure. If one stops the normal form procedure at this order, the remainder becomes exponentially small.

De Maesschalck, Peter; Dumortier, Freddy; Roussarie, Robert
We introduce transitory canard cycles for slowfast vector elds in the plane. Such cycles separate "canards without head" and "canards with head", like for example in the Van der Pol equation. We obtain optimal upper bounds on the number of periodic orbits that can appear near the cycle under whatever condition on the related slow divergence integral I, including the challenging case I = 0.

De Maesschalck, Peter; Huzak, Renato; Dumortier, Freddy
In this paper we study singular perturbation problems occuring in planar slowfast systems. We distinguish between slowfast elliptic and slowfast saddle bifurcations of codimension 3. These bifurcations are singular limits of wellknown codimension 3 bifurcations. In this paper we focus on the appearance of smallamplitude limit cycles, using tools from geometric singular perturbation theory, including blowup.

DE MAESSCHALCK, Peter
This paper deals with normal forms about contact points (turning points) of nilpotent type that one frequently encounters in the study of planar slowfast systems. In case the contact point of an analytic slowfast vector field is of order two, we prove that the slowfast vector field can locally be written as a slowfast Li??nard equation up to exponentially small error. The proof is based on the use of Gevrey asymptotics. Furthermore, for slowfast jump points, we eliminate the exponentially small remainder.

Luca, Stijn; Karsmakers, Peter; Cuppens, Kris; Croonenborghs, Tom; Van de Vel, Anouk; Ceulemans, Berten; Lagae, Lieven; Van Huffel, Sabine; Vanrumste, Bart
Nocturnal home monitoring of epileptic children is often not feasible due to the cumbersome manner of seizure detection with the standard method of video/EEGmonitoring. We propose a method for hypermotor seizure detection based on accelerators attached to the extremities. Hypermotor seizures often involve violent movements with the arms or legs, which increases the need for an alarm system as the patient can injure himself during the seizure. In the literature, classification models are commonly estimated in a supervised manner. Such models are estimated using annotated examples. This annotation of data requires expert (human) interaction and results therefore in a substantial...

Coll, B.; Dumortier, F.; Prohens, R.
This paper aims at providing an example of a family of polynomial Lienard equations exhibiting an alien limit cycle. This limit cycle is perturbed from a 2saddle cycle in the boundary of an annulus of periodic orbits given by a Hamiltonian vector field. The Hamiltonian represents a truncated pendulum of degree 4. In comparison to a former polynomial example, not only the equations are simpler but a lot of tedious calculations can be avoided, making the example also interesting with respect to simplicity in treatment. (C) 2012 Elsevier Inc. All rights reserved.

DUMORTIER, Freddy
In [1] and [2] upperbounds have been given for the number of large amplitude limit cycles in polynomial Lienard systems of type (m, n) with m < 2n + 1, m and n odd. In the current paper we improve the upperbounds from [1] and [2] by one unity, obtaining sharp results. We therefore introduce the "method of cloning variables" that might be useful in other cyclicity problems.

De Maesschalck, P.; Desroches, M.
Continuation techniques have been known to successfully describe bifurcation diagrams appearing in slowfast systems with more than one slow variable. 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 the numerical tools) on the appearance of multiple limit cycles of relaxationoscillation type and on the existence of multiple critical periods in wellchosen annuli of slowfast periodic orbits in the plane. We then apply the technique to study a notion of...

Kutafina, Ekaterina; Vladimirov, V.A.; Zorychta, B.
We consider a hydrodynamic system of balance equations for mass and momentum. This system is closed by the dynamic equation of state, taking into account the effects of spatiotemporal nonlocalities. Using group theory reduction, we obtain a system of ODEs, describing a set of approximate traveling wave solutions to the source system. The factorized system, containing a small parameter, proves to be Hamiltonian when the parameter is zero. Using Melnikov's method, we show that the factorized system possesses, in general, a oneparameter family of homoclinic loops, corresponding to the approximate solitonlike solutions of the source system.