Papers by Robert Roussarie
Des équations différentielles aux systèmes dynamiques : Tome 1, Théorie élémentaire des équations... more Des équations différentielles aux systèmes dynamiques : Tome 1, Théorie élémentaire des équations différentielles avec éléments de topologie différentielle Click here if your download doesn"t start automatically Des équations différentielles aux systèmes dynamiques : Tome 1, Théorie élémentaire des équations différentielles avec éléments de topologie différentielle Robert Roussarie, Jean Roux Des équations différentielles aux systèmes dynamiques : Tome 1, Théorie élémentaire des équations différentielles avec éléments de topologie différentielle Robert Roussarie, Jean Roux Télécharger Des équations différentielles aux systèmes dyna ...pdf Lire en ligne Des équations différentielles aux systèmes dy ...pdf
Bulletin of the Belgian Mathematical Society - Simon Stevin
This paper is part of the DRR-program of [4] to prove the finiteness part of Hilbert's 16th probl... more This paper is part of the DRR-program of [4] to prove the finiteness part of Hilbert's 16th problem for quadratic vector fields by showing the finite cyclicity of 121 graphics. In this paper we prove the finite cyclicity of 4 graphics passing through a triple nilpotent point of elliptic type surrounding a center, namely the graphics (H 1 7), (F 1 7a), (H 3 11) and (I 1 6a). These four graphics are of pp-type, in the sense that they join two parabolic sectors of the nilpotent point. The exact cyclicity is 2 for (H 1 7) and (H 3 11). The graphics (F 1 7a) and (I 1 6a) occur in continuous families. Their exact cyclicity is 2 except for a discrete subset of such graphics. The method can be applied to most other graphics of the DRR-program [4] through a triple nilpotent point and surrounding a center.
Journal of Differential Geometry
Lecture Notes in Mathematics, 1991
Nonlinearity, 1989
The 16th Hilbert problem for polynomial planar vector fields is a consequence of the following co... more The 16th Hilbert problem for polynomial planar vector fields is a consequence of the following conjecture: any analytic deformation of the planar vector field germ along a limit set has a finite cyclicity (a finite bound for the number of limit cycles near the limit periodic set). The property of finite cyclicity is established for the simplest singular limit sets loops made by homoclinic connection at a hyperbolic saddle point and cuspidal singular points.
Comptes Rendus Mathematique, Nov 15, 2005
A 2-dimensional predator-prey model with five parameters is investigated, adapted from the Volter... more A 2-dimensional predator-prey model with five parameters is investigated, adapted from the Volterra–Lotka system by a non-monotonic response function. A description of the various domains of structural stability and their bifurcations is given. The bifurcation structure is reduced to four organising centres of codimension 3. Research is initiated on time-periodic perturbations by several examples of strange attractors. To cite this
J Differential Equations, 2001
Bull Braz Math Soc, 1995
We provide a characterization of the limit periodic sets for analytic families of vector fields u... more We provide a characterization of the limit periodic sets for analytic families of vector fields under the hypothesis that the first jet is non-vanishing at any singular point. Also, applying the family desingularization method, we reduce the complexity of some of these sets.
Data Revues 1631073x Unassign S1631073x13002094, 2013
ABSTRACT On considère des cycles canard transitoires comportant un mécanisme de cassure générique... more ABSTRACT On considère des cycles canard transitoires comportant un mécanisme de cassure générique, de type Hopf ou bien de saut, en combinaison avec un passage de type rapide–rapide par un point de saut. De tels cycles séparent deux types de cycles canard de formes différentes. On obtient des bornes supérieures sur le nombre dʼorbites périodiques qui peuvent apparaître près du cycle canard, sous certaines conditions très générales.
Discrete Contin Dyn Syst, 2006
Qualitative Theory of Dynamical Systems, 2014
In the original publication of the article, the first author's name was incorrect. The correct na... more In the original publication of the article, the first author's name was incorrect. The correct name is Lilia Mahmoudi and not Lylia Mamouhdi.
Qualitative Theory of Dynamical Systems, May 31, 2014
In the original publication of the article, the first author's name was incorrect. The correct na... more In the original publication of the article, the first author's name was incorrect. The correct name is Lilia Mahmoudi and not Lylia Mamouhdi.
Transactions of the Moscow Mathematical Society, 2015
In this paper we introduce new methods to prove the finite cyclicity of some graphics through a t... more In this paper we introduce new methods to prove the finite cyclicity of some graphics through a triple nilpotent point of saddle or elliptic type surrounding a center. After applying a blow-up of the family, yielding a singular 3-dimensional foliation, this amounts to proving the finite cyclicity of a family of limit periodic sets of the foliation. The boundary limit periodic sets of these families were the most challenging, but the new methods are quite general for treating such graphics. We apply these techniques to prove the finite cyclicity of the graphic (I 1 14), which is part of the program started in 1994 by Dumortier, Roussarie and Rousseau (and called DRR program) to show that there exists a uniform upper bound for the number of limit cycles of a planar quadratic vector field. We also prove the finite cyclicity of the boundary limit periodic sets in all graphics but one through a triple nilpotent point at infinity of saddle, elliptic or degenerate type (with a line of zeros) and surrounding a center, namely the graphics (I 1 6b), (H 3 13), and (DI 2b).
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Papers by Robert Roussarie