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# Schlomiuk, Dana

Faculty of Arts and Science - Department of Mathematics and Statistics

### Affiliations

• Membre Centre de recherches mathématiques

### Student supervision Expand allCollapse all

Action de groupe, formes normales et systèmes quadratiques à foyer faible d'ordre trois Theses and supervised dissertations / 2015-09
Demers, Myriam
Abstract
We are interessed here in the action of the group of affine transformations and time homotheties on quadratic differential systems which have a weak focus of third order. These systems are important for Hilbert sixteenth problem. The bifurcation diagram was produced using Li's normal form in the articles of Andronova [2], and Artès and Llibre [4], without using the projective plane as parameter space, and without using global methods. In [7], Llibre and Schlomiuk used the projective plane as parameter space and global geometric methods (affine and topological invariants). This diagram contains 18 phase portraits and some of these portraits are repeated in distinct parts of the diagram. This led us to ask the following question : do there exist distinct differential systems, corresponding to distinct values of the parameter, which are on the same orbit of the group action? In this master's thesis, we prove an original result : the action of the group is not trivial on Li's normal form (theorem 3.1), neither is it trivial on Bautin's normal form (theorem 4.1). Using the second result, we construct the quotient topological space of these systems with a weak focus of third order, with respect to the group action.

### Selected publications Expand allCollapse all

#### On the limit cycles bifurcating from an ellipse of a quadratic center

Llibre, Jaume et Schlomiuk, Dana, On the limit cycles bifurcating from an ellipse of a quadratic center 35, 1091--1102 (2015), , Discrete Contin. Dyn. Syst.

#### Global configurations of singularities for quadratic differential systems with total finite multiplicity three and at most two real singularities

Art?s, Joan C., Llibre, Jaume, Schlomiuk, Dana et Vulpe, Nicolae, Global configurations of singularities for quadratic differential systems with total finite multiplicity three and at most two real singularities 13, 305--351 (2014), , Qual. Theory Dyn. Syst.

#### Geometric configurations of singularities for quadratic differential systems with total finite multiplicity $m_f=2$

Art?s, Joan C., Llibre, Jaume, Schlomiuk, Dana et Vulpe, Nicolae, Geometric configurations of singularities for quadratic differential systems with total finite multiplicity $m_f=2$ , No. 159, 79 (2014), , Electron. J. Differential Equations

#### Topological and polynomial invariants, moduli spaces, in classification problems of polynomial vector fields

Schlomiuk, Dana, Topological and polynomial invariants, moduli spaces, in classification problems of polynomial vector fields 58, 461--496 (2014), , Publ. Mat.

#### Geometric configurations of singularities for quadratic differential systems with three distinct real simple finite singularities

Art?s, Joan C., Llibre, Jaume, Schlomiuk, Dana et Vulpe, Nicolae, Geometric configurations of singularities for quadratic differential systems with three distinct real simple finite singularities 14, 555--618 (2013), , J. Fixed Point Theory Appl.

#### Mathematical destinies

Schlomiuk, Dana, Mathematical destinies 33, 9--15 (2013), , Lib. Math. (N.S.)

#### Geometric configurations of singularities for quadratic differential systems with total finite multiplicity lower than 2

Art?s, J. C., Llibre, J., Schlomiuk, D. et Vulpe, N., Geometric configurations of singularities for quadratic differential systems with total finite multiplicity lower than 2 , 72--124 (2013), , Bul. Acad. Stiin e Repub. Mold. Mat.

#### New developments based on the mathematical legacy of C. S. Sibirschi

Schlomiuk, Dana, New developments based on the mathematical legacy of C. S. Sibirschi , 3--10 (2013), , Bul. Acad. Stiin e Repub. Mold. Mat.

#### Geometry of quadratic differential systems with a weak focus of second order

J. C. Artés, J. Llibre et D. Schlomiuk, Geometry of quadratic differential systems with a weak focus of second order , 76 (2012), , International J. of Bifurcations and Chaos,

#### Global topological classification of Lotka-Volterra quadratic differential systems

Schlomiuk, Dana et Vulpe, Nicolae, Global topological classification of Lotka-Volterra quadratic differential systems Schlomiuk, Dana and Vulpe, Nicolae, No. 64, 69 (2012), , Electron. J. Differential Equations

#### The geometry of quadratic polynomial differential systems with a weak focus and an invariant straight line

Art?s, Joan C., Llibre, Jaume et Schlomiuk, Dana, The geometry of quadratic polynomial differential systems with a weak focus and an invariant straight line 20, 3627--3662 (2010), , Internat. J. Bifur. Chaos Appl. Sci. Engrg.

#### Bifurcation diagrams and moduli spaces of planar quadratic vector fields with invariant lines of total multiplicity four and having exactly three real singularities at infinity

Schlomiuk, Dana et Vulpe, Nicolae, Bifurcation diagrams and moduli spaces of planar quadratic vector fields with invariant lines of total multiplicity four and having exactly three real singularities at infinity 9, 251--300 (2010), , Qual. Theory Dyn. Syst.

#### Preface [Classical problems on planar polynomial vector fields]

Llibre, Jaume et Schlomiuk, Dana, Preface [Classical problems on planar polynomial vector fields] 9, 1--3 (2010), , Qual. Theory Dyn. Syst.

#### Global classification of the planar Lotka-Volterra differential systems according to their configurations of invariant straight lines

Schlomiuk, Dana et Vulpe, Nicolae, Global classification of the planar Lotka-Volterra differential systems according to their configurations of invariant straight lines 8, 177--245 (2010), , J. Fixed Point Theory Appl.

#### Integrals and phase portraits of planar quadratic differential systems with invariant lines of at least five total multiplicity

Schlomiuk, Dana et Vulpe, Nicolae, Integrals and phase portraits of planar quadratic differential systems with invariant lines of at least five total multiplicity 38, 2015--2075 (2008), , Rocky Mountain J. Math.

#### The full study of planar quadratic differential systems possessing a line of singularities at infinity

Schlomiuk, Dana et Vulpe, Nicolae, The full study of planar quadratic differential systems possessing a line of singularities at infinity 20, 737--775 (2008), , J. Dynam. Differential Equations

#### On general algebraic mechanisms for producing centers in polynomial differential systems

Christopher, Colin et Schlomiuk, Dana, On general algebraic mechanisms for producing centers in polynomial differential systems 3, 331--351 (2008), , J. Fixed Point Theory Appl.

#### Integrals and phase portraits of planar quadratic differential systems with invariant lines of total multiplicity four

Schlomiuk, Dana et Vulpe, Nicolae, Integrals and phase portraits of planar quadratic differential systems with invariant lines of total multiplicity four , 27--83 (2008), , Bul. Acad. Stiin e Repub. Mold. Mat.

#### Planar quadratic differential systems with invariant straight lines of total multiplicity four

Schlomiuk, Dana et Vulpe, Nicolae, Planar quadratic differential systems with invariant straight lines of total multiplicity four 68, 681--715 (2008), , Nonlinear Anal.

#### The geometry of quadratic differential systems with a weak focus of second order

Art?s, Joan C., Llibre, Jaume et Schlomiuk, Dana, The geometry of quadratic differential systems with a weak focus of second order 16, 3127--3194 (2006), , Internat. J. Bifur. Chaos Appl. Sci. Engrg.

#### On finiteness in differential equations and Diophantine geometry

Schlomiuk, Dana, Bolibrukh, Andrei, Yakovenko, Sergei, Kaloshin, Vadim et Buium, Alexandru, On finiteness in differential equations and Diophantine geometry , x+182 (2005), , American Mathematical Society, Providence, RI

#### Finiteness problems in differential equations and diophantine geometry

D. Schlomiuk, Finiteness problems in differential equations and diophantine geometry American Mathematical Society,, 9 (2005), , Differential Equations and Diophantine Geometry, CRM Monograph Series,

#### Geometry of quadratic differential systems in the neighborhood of infinity

D. Schlomiuk and N. Vulpe, Geometry of quadratic differential systems in the neighborhood of infinity 215,, 357-400 (2005), , J. of Differential Equations

#### Planar quadratic vector fields with invariant lines of total multiplicity at least five

D. Schlomiuk and N. Vulpe, Planar quadratic vector fields with invariant lines of total multiplicity at least five 5,, 135-194 (2004), , Qualitative Theory of Dynamical Systems

#### Aspects of planar polynomial vector fields: Global versus local, real versus complex, analytic versus algebraic and geometric

D. Schlomiuk, Aspects of planar polynomial vector fields: Global versus local, real versus complex, analytic versus algebraic and geometric NATO Science Series, Series II: Mathematics, vol. 137,, 471-509 (2004), , Normal forms, Bifurcations and Finiteness Problems in Differential Equations, Yu.,

#### The Geometry of Quadratic Differential systems with a weak focus of third order

J. Llibre and D. Schlomiuk, The Geometry of Quadratic Differential systems with a weak focus of third order 56,, 310-343 (2004), , Canadian Journal of Math.

#### On the geometric structure of the class of planar quadratic differential systems

R. Roussarie and D. Schlomiuk, On the geometric structure of the class of planar quadratic differential systems 3,, 93-122 (2003), , Qualitative Theory of Dynamical Systems

#### The mathematical legacy of C. S. Sibirsky, basis for future work

D. Schlomiuk, The mathematical legacy of C. S. Sibirsky, basis for future work 1,, 3-6 (2003), , Bulletin of the Academy of Sciences of Moldova

#### The geometry in the neighborhood of infinity of quadratic differential systems with a weak focus

D. Schlomiuk and J. Pal, The geometry in the neighborhood of infinity of quadratic differential systems with a weak focus 2,, 1-43 (2001), , Qualitative Theory of Dynamical Systems

#### On the geometry in the neighborhood of infinity of quadratic differential systems with a weak focus

Schlomiuk, Dana et Pal, Janos, On the geometry in the neighborhood of infinity of quadratic differential systems with a weak focus 2, 1--43 (2001), , Qual. Theory Dyn. Syst.

#### On the geometry of planar vector fields

D. Schlomiuk, On the geometry of planar vector fields 21,, 65-86 (1999), , Mathematical Reports of the Royal Society of Canada

#### Basic Algebro-Geometric concepts in the study of planar polynomial vector fields

D. Schlomiuk, Basic Algebro-Geometric concepts in the study of planar polynomial vector fields 41, 269-295 (1997), , Publicacions Mathématiques

#### On the global analysis of the planar quadratic vector fields

D. Schlomiuk, On the global analysis of the planar quadratic vector fields 41,, 269-295 (1997), , Nonlinear Analysis Theory, Methods and Applications

#### Summing up the dynamics of quadratic Hamiltonian systems with a center

J. Pal and D. Schlomiuk, Summing up the dynamics of quadratic Hamiltonian systems with a center 49,, 583-599 (1997), , Canadian Journal of Mathematics

#### The centers in the reduced Kukles system

C. Rousseau, P. Thibaudeau and D. Schlomiuk, The centers in the reduced Kukles system 8,, 388-436 (1995), , Nonlinearity

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

C. Rousseau and D. Schlomiuk, Cubic vector fields symmetric with respect to a center 123,, 388-436 (1995), , J. of Diff. Eq.

#### Elementary first integrals and algebraic invariant curves of differential equations

Schlomiuk, Dana, Elementary first integrals and algebraic invariant curves of differential equations 11, 433--454 (1993), , Exposition. Math.

#### Elementary first integrals of differential equations and invariant algebraic curves

D. Schlomiuk, Elementary first integrals of differential equations and invariant algebraic curves 11,, 433-454 (1993), , Expositiones Mathematicae

#### Algebraic and geometric aspects of the theory of planar polynomial vector fields

D. Schlomiuk, Algebraic and geometric aspects of the theory of planar polynomial vector fields NATO Advanced Study Institutes Series, Series C: Mathematical and Physical Sciences - vol. 408,, 429-467 (1993), , Bifurcations and Periodic Orbits of Vector Fields

#### Algebraic particular integrals, integrability and the problem of the center

D. Schlomiuk, Algebraic particular integrals, integrability and the problem of the center 338,, 799-841 (1993), , Transactions of the A.M.S.

#### The nilpotent part and distinguished form of resonant vector fields or diffeomorphisms

J. Ecalle and D. Schlomiuk, The nilpotent part and distinguished form of resonant vector fields or diffeomorphisms 43,, 1407-1483 (1993), , Annales de l'Institut Fourier

#### Une caract?risation g?om?trique g?n?rique des champs de vecteurs quadratiques avec un centre

Schlomiuk, Dana, Une caract?risation g?om?trique g?n?rique des champs de vecteurs quadratiques avec un centre 310, 723--726 (1990), , C. R. Acad. Sci. Paris S?r. I Math.

#### Integrability of plane quadratic vector fields

Schlomiuk, Dana, Guckenheimer, John et Rand, Richard, Integrability of plane quadratic vector fields 8, 3--25 (1990), , Exposition. Math.

#### Degenerate homoclinic cycles in perturbations of quadratic Hamiltonian systems

Guckenheimer, John, Rand, Richard et Schlomiuk, Dana, Degenerate homoclinic cycles in perturbations of quadratic Hamiltonian systems 2, 405--418 (1989), , Nonlinearity

#### 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.

#### La logica dei topos

D. Schlomiuk, La logica dei topos , 135 (1982), Traduction en italien du livre indiqué ci-dessous , La Goliardica Editrice

#### 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.

#### 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

#### Logique des topos (introduction ? la th?orie des topos ?l?mentaires)

Schlomiuk, Dana I., Logique des topos (introduction ? la th?orie des topos ?l?mentaires) , 132 (1977), , Les Presses de l'Universit? de Montr?al, Montr?al, Que.

#### Logique des topos

D. Schlomiuk, Logique des topos , 132 (1976), , Presses de l'Université de Montréal

#### Topos di Grothendieck e topos di Lawvere e Tierney

Schlomiuk, Dana I., Topos di Grothendieck e topos di Lawvere e Tierney 7, 513--553 (1974), , Rend. Mat. (6)

#### An elementary theory of the category of topological spaces

Schlomiuk, Dana I., An elementary theory of the category of topological spaces 149, 259--278 (1970), , Trans. Amer. Math. Soc.