
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.

Dumortier, Freddy; Gyssens, Marc; Van Gucht, Dirk; Vandeurzen, Luc
Several authors have suggested to use firstorder logic over the real numbers to describe spatial database applications. Geometric objects are then described by polynomial inequalities with integer coefficients involving the coordinates of the objects. Such geometric objects are called semialgebraic sets. Similarly, queries are expressed by polynomial inequal ities. The query language thus obtained is usually referred to as FO + poly.
From a practical point of view, it has been argued that a linear restriction of this socalled polynomial model is more desirable. In the socalled linear model, geometric objects are described by linear inequalities, and are called semilinear sets....

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...

De Maesschalck, Peter; Popovic, Nikola
We consider front propagation in a family of scalar reaction–diffusion equations in the
asymptotic limit where the polynomial degree of the potential function tends to infinity.
We investigate the Gevrey properties of the corresponding critical propagation speed,
proving that the formal series expansion for that speed is Gevrey1 with respect to the
inverse of the degree. Moreover, we discuss the question of optimal truncation. Finally, we
present a reliable numerical algorithm for evaluating the coefficients in the expansion with
arbitrary precision and to any desired order, and we illustrate that algorithm by calculating
explicitly the first ten coefficients. Our analysis builds on results obtained previously in
[F. Dumortier,...

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.

De Maesschalck, Peter; Dumortier, Freddy; Roussarie, Robert
We study the limit cycles of planar slow–fast vector fields, appearing near a given slow–fast cycle, formed by an arbitrary sequence of slow parts and fast parts, and where the slow parts can meet the fast parts in a nilpotent contact point of arbitrary order. Using the notion slow divergence integral, we delimit a large subclass of these slow–fast cycles out of which at most one limit cycle can perturb, and a smaller subclass out of which exactly one limit cycle will perturb. Though the focus lies on common slow–fast cycles, i.e. cycles with only attracting or only repelling slow...

Bonckaert, Patrick; Verstringe, Freek
dynamical systems; normal forms; renormalization; majorant method

Bonckaert, Patrick; Verstringe, Freek
We explore the convergence/divergence of the normal
form for a singularity of a vector eld on Cn with nilpotent linear
part. We prove two main theorems in this article. We show that a
Gevrey vector eld X with a nilpotent linear part can be reduced
to a normal form of Gevrey1+ type with the use of a Gevrey1+
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
In this paper, we prove the presence of limit cycles of given multiplicity, together with a complete unfolding, in families of (singularly perturbed) polynomial Lienard equations. The obtained limit cycles are relaxation oscillations. Both classical Lienard equations and generalized Lienard equations are treated.

DE MAESSCHALCK, Peter; DUMORTIER, Freddy
Based on geometric singular perturbation theory we prove the existence of classical Lienard equations of degree 6 having 4 limit cycles. It implies the existence of classical Lienard equations of degree n >= 6, having at least [n1/2] + 2 limit cycles. This contradicts the conjecture from Lins, de Melo and Pugh formulated in 1976, where an upperbound of [n1/2] limit cycles was predicted. This paper improves the counterexample from Dumortier, Panazzolo and Roussarie (2007) by supplying one additional limit cycle from degree 7 on, and by finding a counterexample of degree 6. We also give a precise system of...

DE MAESSCHALCK, Peter; DUMORTIER, Freddy
In this paper we study perturbations from planar vector fields having a line of zeros and representing a singular limit of BogdanovTakens (BT) bifurcations. We introduce, among other precise definitions, the notion of slowfast BTbifurcation and we provide a complete study of the bifurcation diagram and the related phase portraits. Based on geometric singular perturbation theory, including blowup, we get results that are valid on a uniform neighborhood both in parameter space and in the phase plane. (C) 2010 Elsevier Inc. All rights reserved.

DUMORTIER, Freddy; Popovic, Nikola; Kaper, Tasso J.
'Cutoffs' were introduced to model front propagation in reactiondiffusion systems in which the reaction is effectively deactivated at points where the concentration lies below some threshold. In this article, we investigate the effects of a cutoff on fronts propagating into metastable states in a class of bistable scalar equations. We apply the method of geometric desingularization from dynamical systems theory to calculate explicitly the change in front propagation speed that is induced by the cutoff. We prove that the asymptotics of this correction scales with fractional powers of the cutoff parameter, and we identify the source of these exponents, thus...

DE MAESSCHALCK, Peter; DUMORTIER, Freddy
This paper deals with the study of limit cycles that appear in a class of planar slowfast systems, near a "canard" limit periodic set of FSTStype. Limit periodic sets if FSTStype are closed orbits, composed of a Fast branch, an attracting Slow branch, a Turning point, and a repelling Slow branch. Techniques to bound the number of limit cycles near a FSTSl.p.s. are based on the study of the socalled slow divergence integral, calculated along the slow branches. In this paper, we extend the technique to the case where the slow dynamics has singulariteis of any (finite) order that accumulate...

Caubergh, M.; DUMORTIER, Freddy; LUCA, Stijn
The paper deals with the cyclicity of unbounded semihyperbolic 2saddle cycles in polynomial Lienard systems of type (m, n) with m < 2n+1, m and n odd. We generalize the results in [1] (case m = 1), providing a substantially simpler and more transparant proof than the one used in [1].

DE MAESSCHALCK, Peter; DUMORTIER, Freddy
The paper deals with planar slowfast cycles containing a unique generic turning point. We address the question on how to study canard cycles when the slow dynamics call be singular at the turning point. We more precisely accept a generic saddlenode bifurcation to pass through the turning point. It reveals that in this case the slow divergence integral is no longer the good tool to use, but its derivative with respect to the layer Variable still is. We provide general results as Well as a number of applications. We show how to treat the open problems presented in Artes et...

LUCA, Stijn; DUMORTIER, Freddy

DUMORTIER, Freddy

DE MAESSCHALCK, Peter; DUMORTIER, Freddy; Wechselberger, Martin

DUMORTIER, Freddy; Roussarie, Robert

BONCKAERT, Patrick; Hoveijn, I.; VERSTRINGE, Freek
We study local analytic simplification of families of analytic maps near a hyperbolic fixed point. A particularly important application of the main result concerns families of hyperbolic saddles, where Siegel's theorem is too fragile, at least in the analytic category. By relaxing on the formal normal form we obtain analytic conjugacies. Since we consider families, it is more convenient to state some results for analytic maps on a Banach space; this gives no extra complications. As an example we treat a family passing through a 1 : 1 resonant saddle. (C) 2010 Elsevier Inc. All rights reserved.