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# Rousseau, Christiane

Adjunct and Emeritus and Honorary professor

Faculty of Arts and Science - Department of Mathematics and Statistics

514 343-7729

### Affiliations

• Membre Centre de recherches mathématiques
• Membre CRM  Centre de recherches mathématiques

### Research area

My research interests are around dynamical systems in small dimension, either ODE or difference equations.

In the case of ODE, I am interested in the qualitative theory of ODE and the development of methods allowing understanding the geometric organization of the solutions, often summarized in the phase portrait. I am especially interested to parameter dependent ODE and bifurcation analysis: bifurcations correspond to qualitative changes on the phase portraits occurring for particular values of the parameters. I am interested in applications to Hilbert 19s 16th problem on one side and, occasionally, to some predator-prey models in mathematical biology.

The main part of my recent research deals with the study of equilibrium positions of analytic dynamical systems depending on parameters, more precisely with the problem
of analytic classification of singularities of families of dynamical systems depending on parameters: when are two analytic families of dynamical systems equivalent modulo an analytic change of parameters and possibly a reparameterization of time? There are many obstructions to such equivalences and I am interesting in understanding their geometric meaning.

I am also very involved in popularization of mathematics and the training of future high school teachers.
I was the instigator and international coordinator of the international year Mathematics of Planet Earth 2013 (MPE2013).

### Student supervision Expand allCollapse all

Classification analytique des points fixes paraboliques de germes antiholomorphes et de leurs déploiements Theses and supervised dissertations / 2020-12
Godin, Jonathan
Abstract
We are interested in the dynamics in a neighbourhood of a fixed point of an antiholomorphic function of one variable. First, we want to describe and understand the space of orbits in a neighbourhood of a multiple fixed point, called a parabolic point, and to explore the geometric properties preserved by changes of coordinate. In particular, we solve the problem of analytical classification of parabolic fixed points. To solve this problem, we define a complete modulus of classification that allows to determine whether two germs of antiholomorphic diffeomorphisms are analytically conjugate in a neighbourhood of their parabolic fixed point. We also consider the applications of the modulus to different problems: i) extraction of an n-th antiholomorphic root, ii) existence of an invariant real analytical curve under the dynamics of a parabolic antiholomorphic germ, and iii) centraliser of a parabolic antiholomorphic germ. In the second part, we study generic unfoldings of a double fixed point, i.e. a parabolic point of codimension 1. The questions are similar in nature, namely to understand the space of orbits and the geometric properties of unfoldings. In order to classify generic unfoldings, the modulus of classification of the parabolic point is unfolded, thus providing the necessary and sufficient conditions to determine when two generic unfoldings are equivalent.

Configurations centrales en toile d'araignée Theses and supervised dissertations / 2018-10
Hénot, Olivier
Abstract
This thesis is dedicated to the study of a specific class of solutions for the N-body problem called central configurations. These configurations, especially in the planar case, are closely related to homographic solutions: at any time, the position of the bodies can be obtained by a rotation and/or a rescaling of the initial position. Our aim is to prove the existence of spiderweb central configurations given by n × ℓ masses located at the intersection of n concentric circles with ℓ concurrent half-lines, under the hypothesis that the ℓ masses on the i-th circle are equal to a positive constant mᵢ ; we also discuss the case where we add a central mass m₀ located at the intersection of the ℓ halflines. A first analytical method leads to the existence of these central configurations when n = 2,3,4 and ℓ arbitrary for any strictly positive values of m₁, . . . ,mₙ. A second analytical method yields the existence and uniqueness of such central configurations when we restrict ℓ to be equal to 2, . . . ,9 and n arbitrary for any strictly positive values of m₁, . . . , mₙ. In addition, we extend the result for ℓ = 10, . . . ,18 by requiring m₁ ≥ · · · ≥ mₙ and bounding the value of n in each case. Furthermore, for these two analytical methods, we demonstrate that the results hold for spiderweb configurations with N = n × ℓ + 1 bodies, that is when we add a strictly positive mass at the center of mass. Finally, we give an algorithm providing a rigorous proof of the existence and local uniqueness of such a central configuration with an arbitrary choice of n, ℓ and m₁, . . . , mₙ. The algorithm is applied to all n ≤ 100 and all even values ℓ ≤ 200 when m₁ = . . . = mₙ = 1/ℓ. This is enough to show the existence of spiderweb central configurations for all n ≤ 100, ℓ ≤ 200 even and such that m₁ = . . . = mₙ for any value strictly positive.

Étude des conditions d'extinction d'un système prédateur-proie généralisé avec récolte contrôlée Theses and supervised dissertations / 2016-09
Courtois, Julien
Abstract
In this master thesis, we study a generalized Gause predator-prey system with controlled prey harvest and a generalized Holling response function of type III. We introduce a controlled prey harvesting function taking into account the number of preys with a harvesting threshold. This makes the system realistic, it optimizes the harvesting, and it prevents the possibility of species' extinction which exists in the system with constant harvest for all parameters. This type of harvesting function a priori implies handling a discontinuous system : therefore we study smoothing techniques of such discontinuities by regularization. We first return on systems without and with constant harvest by drawing the exact bifurcation diagrams and phase portraits of those systems. Then, we study the discontinuous system and the regularization methods in order to choose the optimal one. Finally, we put together everything by studying the regularized prey harvesting system through a complete study of the prey stocking system, and we highlight the different effects on the phase portraits under the initial conditions.

Caractère intrinsèque des matrices de Stokes Theses and supervised dissertations / 2015-08
Gagnon, Jean-François
Abstract
It is well known that a linear differential equation, x^(k+1)Y' = A(x)Y, near a non-resonant irregular singular point is uniquely determined (up to analytic isomorphism) by : (1) its formal normal form, (2) the collection of its Stokes matrices. By definition, the Stokes matrices depend on an order defined on the real parts of the eigenvalues of the system which can be perturbed by a rotation in the x coordinate. In this paper, we have established the intrinsic character of the dependency : we have described how the new Stokes collection is obtained from the first collection after a rotation in x which changes the order on the real parts of the eigenvalues. The first chapter contains preliminaries concerning the normal form of an ordinary differential equation and a chapter on the Stokes phenomenon for linear differential equations. The third chapter contains our results.

Unfolded singularities of analytic differential equations Theses and supervised dissertations / 2014-06
Klimes, Martin
Abstract
The thesis is composed of a chapter of preliminaries and two articles on the theme of unfolding of singularities of analytic differential equations in a complex domain. They are both related to the problem of local analytic classification of parametric families of linear systems: When two parametric families of linear systems are equivalent by means of an analytic change of coordinates in a neighborhood of the singularity? The article Analytic classification of families of linear differential systems unfolding a resonant irregular singularity deals with the question of analytic equivalence of parametric families of systems of linear differential equations in dimension 2 unfolding a generic resonant singularity of Poincaré rank 1 whose leading matrix is a Jordan bloc. The problem is completely solved and the moduli space of analytic equivalence classes is described in terms of a set of formal invariants and a single analytic invariant obtained from the trace of the monodromy. Universal unfoldings are provided for all such singularities. The article Confluence of singularities of non-linear differential equations via Borel-Laplace transformations investigates bounded solutions of systems of differential equations describing a 1-dimensional center manifold of an unfolded saddle-node singularity in a family of complex vector fields. Generally, a system of analytic ODE at a double singular point possesses a unique formal solution in terms of a divergent power series. The classical Borel summation method associates to it true solutions that are asymptotic to the series on certain sectors in the complex plane. The article shows how to unfold the Borel and Laplace integral transformations of the summation procedure. A new kind of solutions of parameter dependent systems of ODE with two simple (regular) singular points unfolding a double (irregular) singularity are constructed, which are bounded on certain “spiraling” domains attached to both singular points, and which at the limit converge uniformly to a pair of the classical sectorial solutions. The method provides a unified treatment for all values of parameter.

Problème centre-foyer et application Theses and supervised dissertations / 2011-04
Laurin, Sophie
Abstract
In this thesis, we study the center-focus problem in a polynomial system. We describe two mechanisms to conclude that a monodromic singular point in this polynomial system is a center. The first one is the method of Darboux. In this method, one uses invariant algebraic curves to build a first integral. The second method is the algebraic (and analytic) reversibility. A monodromic singularity, which is algebraically or analytically reversible at the singular point, is necessarily a center. As an application, in the last chapter, we consider the generalized Gause model with prey harvesting and a generalized Holling response function of type III.

Characterization of the unfolding of a weak focus and modulus of analytic classification Theses and supervised dissertations / 2010-06
Abstract
The thesis gives a geometric description for the germ of the singular holomorphic foliation associated with the complexification of a germ of generic analytic family unfolding a real analytic vector field with a weak focus at the origin. We show that two such germs of families are orbitally analytically equivalent if and only if the germs of families of diffeomorphisms unfolding the complexified Poincaré map of the singularities are conjugate by a real analytic conjugacy. The Z2-equivariance of the family of real vector fields in R^4 is called the “real character” of the system. It is expressed by the invariance of the real plane under the flow of the system which, in turn, carries the real asymptotic expansion of the Poincaré map when the parameter is real. After blowing up the singularity, the pullback of the real plane by the standard monoidal map intersects the foliation in a real Möbius strip. The blow up technique allows to “realize” a germ of generic family unfolding a germ of diffeomorphism of codimension one and multiplier −1 at the origin as the semi-monodromy of a generic family unfolding an order one weak focus. In order to study the orbit space of the Poincaré map, we perform a trade-off between geometry and dynamics under the Glutsyuk point of view (where the dynamics is linearizable near the singular points): in the resulting “unwrapping coordinate” the dynamics becomes much simpler, but the price we pay is that the local geometry of the ambient complex plane turns into a much more involved Riemann surface. Over the latter, two notions of translations are defined. After taking the quotient by the lifted dynamics we get the orbit space, which turns out to be the union of three complex tori and the singular points (this space is non- Hausdorff). The Glutsyuk invariant is then defined over annular-like regions on the tori. The translations, the real character and the fact that the Poincaré map is the square of the semi-monodromy map, relate the different components of the Glutsyuk modulus. That property yields only one independent component of the Glutsyuk invariant.

Classification analytique de systèmes différentiels linéaires déployant une singularité irrégulière de rang de Poincaré 1 Theses and supervised dissertations / 2010-04
Lambert, Caroline
Abstract
This thesis deals with the analytic classification of unfoldings of linear differential systems with an irregular singularity. It contains two papers related to this subject: the first paper presents results concerning the confluence of the hypergeometric equation and may be viewed as a particular case of the second one; the second paper contains the main theorems and results. In both papers, we study the confluence of two regular singular points into an irregular one and we give consequences of the divergence of solutions at the irregular singular point for the unfolded system. For this study, a full neighborhood of the origin is covered (in a ramified way) in the space of the unfolding parameter $\epsilon$. Monodromy of a well chosen basis of solutions around the regular singular points is directly linked to the unfolded Stokes matrices. These matrices give a complete geometric interpretation to the well-known Stokes matrices: this includes the link (existing at least for the generic cases) between the divergence of the solutions at $\epsilon=0$ and the presence of logarithmic terms in the solutions for resonant values of $\epsilon$. Monodromy of first integrals of related Riccati systems are also interpreted in terms of the elements of the unfolded Stokes matrices. The second paper goes further into the subject, giving the complete system of analytic invariants for the unfoldings of nonresonant linear differential systems $x^2y'=A(x)y$ with an irregular singularity of Poincaré rank $1$ at the origin over a fixed neighborhood $\mathbb{D}_r$ in the space of the variable $x$. It consists of a formal part, given by polynomials, and an analytic part, given by an equivalence class of unfolded Stokes matrices. For each parameter value $\epsilon$ taken in a sector pointed at the origin of opening larger than $2\pi$, we cover the space of the variable, $\mathbb{D}_r$, with two sectors and, over each of them, we construct a well chosen basis of solutions of the unfolded differential system. This basis is used to define the unfolded Stokes matrices. Finally, we give a realization theorem for the invariants satisfying a necessary and sufficient condition, thus identifying the set of modules.

Étude d'un modèle de Gause généralisé avec récolte de proies et fonction de Holling type III généralisée Theses and supervised dissertations / 2008
Etoua, Remy Magloire Dieudonné
Abstract
Thèse numérisée par la Division de la gestion de documents et des archives de l'Université de Montréal.

Étude d'un système prédateur-proie avec fonction de réponse Holling de type III généralisée Theses and supervised dissertations / 2006
Lamontagne, Yann
Abstract
Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Étude du diagramme de bifurcation d'un système prédateur-proie Theses and supervised dissertations / 2003
Coutu, Caroline
Abstract
Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Cyclicité finie des boucles homoclines dans R3 non dégénérées avec valeurs propres principales réelles en résonance 1:1 Theses and supervised dissertations / 1999
Guimond, Louis-Sébastien
Abstract
Thèse diffusée initialement dans le cadre d'un projet pilote des Presses de l'Université de Montréal/Centre d'édition numérique UdeM (1997-2008) avec l'autorisation de l'auteur.

### Selected publications Expand allCollapse all

Christiane Rousseau, Yvan Saint-aubin, Traduction brésilienne : matemática e Atualidade 2, 374 (2016), , Sociedade Brasileira de Mateméatica

#### Construire une image médicale

ROUSSEAU, CHRISTIANE, Construire une image médicale 10, (2015), , Accromath

Christiane Rousseau, Yvan Saint-aubin, Traduction brésilienne : matemática e Atualidade 1, 326 (2015), , Sociedade Brasileira de Mateméatica

#### Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank $k$

Hurtubise, Jacques, Lambert, Caroline et Rousseau, Christiane, Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank $k$ 14, 309--338, 427 (2014), , Mosc. Math. J.

#### The moduli space of germs of generic families of analytic diffeomorphisms unfolding a parabolic fixed point

Christopher, Colin et Rousseau, Christiane, The moduli space of germs of generic families of analytic diffeomorphisms unfolding a parabolic fixed point Christopher, Colin and Rousseau, Christiane, 2494--2558 (2014), , Int. Math. Res. Not. IMRN

#### Cristaux

ROUSSEAU, CHRISTIANE, Cristaux 9, (2014), , Accromath

#### Passera, passera pas?

ROUSSEAU, CHRISTIANE, Passera, passera pas? 9, (2014), , Accromath

#### Moduli space of unfolded differential linear systems with an irregular singularity of Poincaré rank 1

Lambert, Caroline et Rousseau, Christiane, Moduli space of unfolded differential linear systems with an irregular singularity of Poincaré rank 1 13, 529--550, 553--554 (2013), , Mosc. Math. J.

#### How Inge Lehmann discovered the inner core of the Earth

Rousseau, Christiane, How Inge Lehmann discovered the inner core of the Earth 44, 399--408 (2013), , College Math. J.

#### Mathématiques de la planète Terre 2013 et chimie

ROUSSEAU, CHRISTIANE, Mathématiques de la planète Terre 2013 et chimie 380, 8-10 (2013), , L'actualité chimique

#### Comment Inge Lehmann a découvert le noyau interne de la Terre

ROUSSEAU, CHRISTIANE, Comment Inge Lehmann a découvert le noyau interne de la Terre 8, (2013), , Accromath

#### L'équation du temps

ROUSSEAU, CHRISTIANE, L'équation du temps 8, (2013), , Accromath

#### Des coquillages aux pelages

ROUSSEAU, CHRISTIANE, Des coquillages aux pelages 7, (2012), , Accromath

#### Que signifie dimension?

ROUSSEAU, CHRISTIANE, Que signifie dimension? 7, (2012), , Accromath

#### Voyager aux confins du système solaire en économisant l'énergie

ROUSSEAU, CHRISTIANE, Voyager aux confins du système solaire en économisant l'énergie 7, (2012), , Accromath

#### The modulus of cusps in conformal geometry

ROUSSEAU, C., The modulus of cusps in conformal geometry 252, 1562-1588 (2012), , Journal of Differential Equations

#### The equation of time

ROUSSEAU, CHRISTIANE, The equation of time 16, (2012), , Pi in the Sky

#### Traduction allemande : Mathematik und technologie

Christiane Rousseau, Yvan saint-Aubin, Traduction allemande : Mathematik und technologie Mathematik und technologie, 609 (2012), , Springer Spektrum, Berlin

#### The modulus of unfoldings of cusps in conformal geometry

Rousseau, C., The modulus of unfoldings of cusps in conformal geometry 252, 1562--1588 (2012), , J. Differential Equations

#### Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank 1

LAMBERT, C. et ROUSSEAU, C., Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank 1 12, 77-138 (2012), , Moscow Mathematical Journal

#### L'effet papillon

ROUSSEAU, CHRISTIANE, L'effet papillon 6, (2011), , Accromath

#### Les sphères de Dandelin

ROUSSEAU, CHRISTIANE, SAINT-AUBIN, YVAN, Les sphères de Dandelin 6, (2011), , Accromath

#### Teorema do ponto fixo de Banach e aplicações

ROUSSEAU CHRISTIANE, Teorema do ponto fixo de Banach e aplicações 164, 32-39 (2011), , Gazeta de Matemática

#### Au delà de l'effet papillon

ROUSSEAU CHRISTIANE, Au delà de l'effet papillon 6, (2011), , Accromath

#### The center and cyclicity problems

Rousseau, Christiane, The center and cyclicity problems 53, 402--405 (2011), , SIAM Rev.

#### Organizing center for the bifurcation analysis of a generalized Gause model with prey harvesting and Holling response function of type III

Laurin, Sophie et Rousseau, Christiane, Organizing center for the bifurcation analysis of a generalized Gause model with prey harvesting and Holling response function of type III 251, 2980--2986 (2011), , J. Differential Equations

#### The modulus of analytic classification for the unfolding of the codimension-one flip and Hopf bifurcations

Arriagada-Silva, Waldo et Rousseau, Christiane, The modulus of analytic classification for the unfolding of the codimension-one flip and Hopf bifurcations 20, 541--580 (2011), , Ann. Fac. Sci. Toulouse Math. (6)

#### Apprendre à frauder ou à détecter les fraudes

ROUSSEAU, CHRISTIANE, Apprendre à frauder ou à détecter les fraudes , (2010), , Accromath

#### Bifurcation analysis of a generalized Gause model with prey harvesting anf a geenralized Holling Response of type III

ETOUA, R.M. & ROUSSEAU, C., Bifurcation analysis of a generalized Gause model with prey harvesting anf a geenralized Holling Response of type III 249, 2316-2356 (2010), , Journal of Differential Equations

#### The moduli space of germs of generic families of analytic diffeomorphisms unfolding a codimension 1 resonant diffeomorphism or resonant saddle

ROUSSEAU, C., The moduli space of germs of generic families of analytic diffeomorphisms unfolding a codimension 1 resonant diffeomorphism or resonant saddle 248, 1794-1825 (2010), , Journal of Differential Equations

#### Study of the cyclicity of some degenerate graphics inside quadratic systems

Dumortier, Freddy et Rousseau, Christiane, Study of the cyclicity of some degenerate graphics inside quadratic systems 8, 1133--1157 (2009), , Commun. Pure Appl. Anal.

#### Traduction anglaise : Mathematics and technology

Christiane Rousseau, Yvan Saint-Aubin, Traduction anglaise : Mathematics and technology Texts in Mathematics and Technology, 580 (2008), , Springer, New-York

#### Mathématiques et technologie

Christiane Rousseau, Yvan Saint-Aubin, Mathématiques et technologie Texts in Mathematics and technology, 594 (2008), , Springer, New-York

#### Finite cyclicity of nilpotent graphics of pp-type surrounding a center

Roussarie, R. et Rousseau, C., Finite cyclicity of nilpotent graphics of pp-type surrounding a center 15, 889--920 (2008), , Bull. Belg. Math. Soc. Simon Stevin

#### Analytical moduli for unfoldings of saddle-node vector fields

Rousseau, Christiane et Teyssier, Loïc, Analytical moduli for unfoldings of saddle-node vector fields 8, 547--614, 616 (2008), , Mosc. Math. J.

#### Bifurcation analysis of a predator-prey system with generalised Holling type III functional response

Lamontagne, Yann, Coutu, Caroline et Rousseau, Christiane, Bifurcation analysis of a predator-prey system with generalised Holling type III functional response 20, 535--571 (2008), , J. Dynam. Differential Equations

#### The Stokes phenomenon in the confluence of the hypergeometric equation using Riccati equation

Lambert, Caroline et Rousseau, Christiane, The Stokes phenomenon in the confluence of the hypergeometric equation using Riccati equation 244, 2641--2664 (2008), , J. Differential Equations

#### Modulus of analytic classification for the generic unfolding of a codimension 1 resonant diffeomorphism or resonant saddle

Rousseau, Christiane et Christopher, Colin, Modulus of analytic classification for the generic unfolding of a codimension 1 resonant diffeomorphism or resonant saddle 57, 301--360 (2007), , Ann. Inst. Fourier (Grenoble)

#### The root extraction problem

Rousseau, C., The root extraction problem 234, 110--141 (2007), , J. Differential Equations

#### The moduli space of germs of generic families and analytic diffeomorphisms unfolding a parabolic fixed point

Christopher, Colin et Rousseau, Christiane, The moduli space of germs of generic families and analytic diffeomorphisms unfolding a parabolic fixed point 345, 695--698 (2007), , C. R. Math. Acad. Sci. Paris

#### Modulus of orbital analytic classification for a family unfolding a saddle-node

Rousseau, Christiane, Modulus of orbital analytic classification for a family unfolding a saddle-node 5, 245--268 (2005), , Mosc. Math. J.

#### Normalizable, integrable and linearizable saddle points in the Lotka-Volterra system

Christopher, Colin et Rousseau, Christiane, Normalizable, integrable and linearizable saddle points in the Lotka-Volterra system 5, 11--61 (2004), , Qual. Theory Dyn. Syst.

#### Modulus of analytic classification for unfoldings of generic parabolic diffeomorphisms

Marde\v si\'c, P., Roussarie, R. et Rousseau, C., Modulus of analytic classification for unfoldings of generic parabolic diffeomorphisms 4, 455--502, 535 (2004), , Mosc. Math. J.

#### Normalizability, synchronicity, and relative exactness for vector fields in $\Bbb C^2$

Christopher, C., Marde\v si\'c, P. et Rousseau, C., Normalizability, synchronicity, and relative exactness for vector fields in $\Bbb C^2$ 10, 501--525 (2004), , J. Dynam. Control Systems

#### PP-graphics with a nilpotent elliptic singularity in quadratic systems and Hilbert's 16th problem

Rousseau, Christiane et Zhu, Huaiping, PP-graphics with a nilpotent elliptic singularity in quadratic systems and Hilbert's 16th problem 196, 169--208 (2004), , J. Differential Equations

#### Normal forms, bifurcations and finiteness problems in differential equations

ROUSSEAU Christiane - ILYASHENKO Yulij, Normal forms, bifurcations and finiteness problems in differential equations , (2004), , Kluwer editor

#### Normalizable, integrable, and linearizable saddle points for complex quadratic systems in $\Bbb C^2$

Christopher, C., Marde\v si\'c, P. et Rousseau, C., Normalizable, integrable, and linearizable saddle points for complex quadratic systems in $\Bbb C^2$ 9, 311--363 (2003), , J. Dynam. Control Systems

#### Finite cyclicity of graphics with a nilpotent singularity of saddle or elliptic type

ROUSSEAU C. & ZHU H., Finite cyclicity of graphics with a nilpotent singularity of saddle or elliptic type 178, 325-436 (2002), , J. Differential Equations.

#### Finite cyclicity of elementary graphics surrounding a focus or center in quadratic systems

Dumortier, F., Guzmàn, A. et Rousseau, C., Finite cyclicity of elementary graphics surrounding a focus or center in quadratic systems 3, 123--154 (2002), , Qual. Theory Dyn. Syst.

#### Normal forms near a saddle-node and applications to finite cyclicity of graphics

Dumortier, F., Ilyashenko, Y. et Rousseau, C., Normal forms near a saddle-node and applications to finite cyclicity of graphics 22, 783--818 (2002), , Ergodic Theory Dynam. Systems

#### Normal forms, bifurcations and finiteness problems in differential equations?

ROUSSEAU Christiane , Normal forms, bifurcations and finiteness problems in differential equations? , 40 (2002), , Kluwer editor

#### Nondegenerate linearizable centres of complex planar quadratic and symmetric cubic systems in $\Bbb C^2$

Christopher, C. et Rousseau, C., Nondegenerate linearizable centres of complex planar quadratic and symmetric cubic systems in $\Bbb C^2$ 45, 95--123 (2001), , Publ. Mat.

#### Finite cyclicity of finite codimension nondegenerate homoclinic loops with real eigenvalues in $\Bbb R^3$

Guimond, Louis-Sébastien et Rousseau, Christiane, Finite cyclicity of finite codimension nondegenerate homoclinic loops with real eigenvalues in $\Bbb R^3$ 2, 151--204 (2001), , Qual. Theory Dyn. Syst.

#### Genericity conditions for finite cyclicity of elementary graphics

Guzmàn, Ana et Rousseau, Christiane, Genericity conditions for finite cyclicity of elementary graphics 155, 44--72 (1999), , J. Differential Equations

#### Global study of a family of cubic Liénard equations

Khibnik, Alexander I., Krauskopf, Bernd et Rousseau, Christiane, Global study of a family of cubic Liénard equations 11, 1505--1519 (1998), , Nonlinearity

#### Cyclicity of graphics with semi-hyperbolic points inside quadratic systems

Rousseau, C., \'Swirszcz, G. et \.Zolpolhk adek, H., Cyclicity of graphics with semi-hyperbolic points inside quadratic systems 4, 149--189 (1998), , J. Dynam. Control Systems

#### Codimension-three unfoldings of reflectionally symmetric planar vector fields

Krauskopf, Bernd et Rousseau, Christiane, Codimension-three unfoldings of reflectionally symmetric planar vector fields 10, 1115--1150 (1997), , Nonlinearity

#### Local bifurcations of critical periods in the reduced Kukles system

Rousseau, C. et Toni, B., Local bifurcations of critical periods in the reduced Kukles system 49, 338--358 (1997), , Canad. J. Math.

#### Darboux linearization and isochronous centers with a rational first integral

Marde\v si\'c, P., Moser-Jauslin, L. et Rousseau, C., Darboux linearization and isochronous centers with a rational first integral 134, 216--268 (1997), , J. Differential Equations

#### Almost planar homoclinic loops in ${\bf R}^3$

Roussarie, Robert et Rousseau, Christiane, Almost planar homoclinic loops in ${\bf R}^3$ 126, 1--47 (1996), , J. Differential Equations

#### A stratum of cubic vector fields with an integrable saddle and $Z_2\times Z_2$ symmetry

Guimond, Louis-Sébatien et Rousseau, Christiane, A stratum of cubic vector fields with an integrable saddle and $Z_2\times Z_2$ symmetry 9, 761--785 (1996), , Nonlinearity

#### Hilbert's 16th problem for quadratic systems and cyclicity of elementary graphics

Dumortier, F., El Morsalani, M. et Rousseau, C., Hilbert's 16th problem for quadratic systems and cyclicity of elementary graphics 9, 1209--1261 (1996), , Nonlinearity

#### Cubic vector fields symmetric with respect to a center

Rousseau, C. et Schlomiuk, D., Cubic vector fields symmetric with respect to a center 123, 388--436 (1995), , J. Differential Equations

#### Linearization of isochronous centers

Marde\v si\'c, P., Rousseau, C. et Toni, B., Linearization of isochronous centers 121, 67--108 (1995), , J. Differential Equations

#### The centres in the reduced Kukles system

Rousseau, Christiane, Schlomiuk, Dana et Thibaudeau, Pierre, The centres in the reduced Kukles system 8, 541--569 (1995), , Nonlinearity

#### Hilbert's 16th problem for quadratic vector fields

Dumortier, F., Roussarie, R. et Rousseau, C., Hilbert's 16th problem for quadratic vector fields 110, 86--133 (1994), , J. Differential Equations

#### Elementary graphics of cyclicity $1$ and $2$

Dumortier, F., Roussarie, R. et Rousseau, C., Elementary graphics of cyclicity $1$ and $2$ 7, 1001--1043 (1994), , Nonlinearity

#### Local bifurcation of critical periods in vector fields with homogeneous nonlinearities of the third degree

Rousseau, C. et Toni, B., Local bifurcation of critical periods in vector fields with homogeneous nonlinearities of the third degree 36, 473--484 (1993), , Canad. Math. Bull.

#### Bifurcation at infinity in polynomial vector fields

Blows, T. R. et Rousseau, C., Bifurcation at infinity in polynomial vector fields 104, 215--242 (1993), , J. Differential Equations

#### Bifurcations et orbites périodiques de champs de vecteurs

ROUSSEAU Christiane, Bifurcations et orbites périodiques de champs de vecteurs , 50 (1993), , Kluwer editor

#### Zeroes of complete elliptic integrals for $1:2$ resonance

Rousseau, Christiane et \.Zolpolhk adek, Henryk, Zeroes of complete elliptic integrals for $1:2$ resonance 94, 41--54 (1991), , J. Differential Equations

#### Codimension $2$ symmetric homoclinic bifurcations and application to $1:2$ resonance

Li, Cheng Zhi et Rousseau, Christiane, Codimension $2$ symmetric homoclinic bifurcations and application to $1:2$ resonance 42, 191--212 (1990), , Canad. J. Math.

#### Cubic Liénard equations with linear damping

Dumortier, Freddy et Rousseau, Christiane, Cubic Liénard equations with linear damping 3, 1015--1039 (1990), , Nonlinearity

#### Codimension $1$ and $2$ bifurcations of fixed points of diffeomorphisms and periodic solutions of vector fields

Rousseau, Christiane, Codimension $1$ and $2$ bifurcations of fixed points of diffeomorphisms and periodic solutions of vector fields 13, 55--91 (1990), , Ann. Sci. Math. Québec

#### A simple proof for the unicity of the limit cycle in the Bogdanov-Takens system

Li, Chengzhi, Rousseau, Christiane et Wang, Xian, A simple proof for the unicity of the limit cycle in the Bogdanov-Takens system 33, 84--92 (1990), , Canad. Math. Bull.

#### A system with three limit cycles appearing in a Hopf bifurcation and dying in a homoclinic bifurcation: the cusp of order $4$

Li, Cheng Zhi et Rousseau, Christiane, A system with three limit cycles appearing in a Hopf bifurcation and dying in a homoclinic bifurcation: the cusp of order $4$ 79, 132--167 (1989), , J. Differential Equations

Joyal, Pierre et Rousseau, Christiane, Saddle quantities and applications 78, 374--399 (1989), , J. Differential Equations

#### Elementary characterization of orbits and strata in the classical Lie and Jordan algebras

Rousseau, Christiane, Elementary characterization of orbits and strata in the classical Lie and Jordan algebras 32(80), 75--88 (1988), , Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.)

#### Generalized Hopf bifurcations and applications to planar quadratic systems

Rousseau, C. et Schlomiuk, D., Generalized Hopf bifurcations and applications to planar quadratic systems 49, 1--16 (1988), , Ann. Polon. Math.

#### Example of a quadratic system with two cycles appearing in a homoclinic loop bifurcation

Rousseau, Christiane, Example of a quadratic system with two cycles appearing in a homoclinic loop bifurcation 66, 140--150 (1987), , J. Differential Equations

#### Clebsch-Gordan coefficients for ${\rm SU}(5)$ unification models

del Olmo, M. A., Patera, J., Rodriguez, M. A. et Rousseau, C., Clebsch-Gordan coefficients for ${\rm SU}(5)$ unification models 28, 258--271 (1987), , J. Math. Phys.

#### Spectral decomposition theorem for real symmetric matrices in topoi and applications

Rousseau, Christiane, Spectral decomposition theorem for real symmetric matrices in topoi and applications 38, 91--102 (1985), , J. Pure Appl. Algebra

#### Clebsch-Gordan coefficients for $E_{6}$ and ${\rm SO}(10)$ unification models

Koh, In Guy, Patera, J. et Rousseau, C., Clebsch-Gordan coefficients for $E_{6}$ and ${\rm SO}(10)$ unification models 25, 2863--2872 (1984), , J. Math. Phys.

#### Versal deformations of elements of classical Jordan algebras

Patera, J. et Rousseau, C., Versal deformations of elements of classical Jordan algebras 24, 1375--1380 (1983), , J. Math. Phys.

#### Clebsch-Gordan coefficients for ${\rm SU}(5)\supset{\rm SU}(3)\times{\rm SU}(2)\times{\rm U}(1)$ theories

Koh, In Gyu, Patera, J. et Rousseau, C., Clebsch-Gordan coefficients for ${\rm SU}(5)\supset{\rm SU}(3)\times{\rm SU}(2)\times{\rm U}(1)$ theories 24, 1955--1967 (1983), , J. Math. Phys.

#### Versal deformations of elements of real classical Lie algebras

Patera, J., Rousseau, C. et Schlomiuk, D., Versal deformations of elements of real classical Lie algebras 15, 1063--1086 (1982), , J. Phys. A

#### Complex orthogonal and symplectic matrices depending on parameters

Patera, J. et Rousseau, C., Complex orthogonal and symplectic matrices depending on parameters 23, 705--714 (1982), , J. Math. Phys.

#### Dimensions of orbits and strata in complex and real classical Lie algebras

Patera, J., Rousseau, C. et Schlomiuk, D., Dimensions of orbits and strata in complex and real classical Lie algebras 23, 490--494 (1982), , J. Math. Phys.

#### Formes normales des matrices rectangulaires dans un topos et applications

Rousseau, Christiane, Formes normales des matrices rectangulaires dans un topos et applications 5, 81--85 (1981), , Ann. Sci. Math. Québec

#### Nombres réels et complexes dans les topos spatiaux

Rousseau, Christiane, Nombres réels et complexes dans les topos spatiaux 3, 143--154 (1979), , Ann. Sci. Math. Québec

#### Topos theory and complex analysis

Rousseau, Christiane, Topos theory and complex analysis 10, 299--313 (1978), , J. Pure Appl. Algebra