# Nicolae VulpeInstitute of Mathematics and Computer Science, Moldova,Chisinau · Laboratory of Differential Equations

Nicolae Vulpe

Dr. habil., Full Professor, Corresponding Member of the Academy of Sciences of Moldova

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Introduction

**Skills and Expertise**

## Publications

Publications (108)

According to Proposition 6.1 for the quadratic systems having the finite singularities of total multiplicity two, the conditions μ0 = μ1 = 0 and μ2 ≠ 0 must be satisfied. So by Theorem 6.4 the following lemma is valid.

In this chapter we sum up results on finite singularities of quadratic differential systems previously obtained by us and that were published prior to the publication of this book (see [41, 29, 338, 301, 26, 32]). Roughly speaking these results give us global information about the possibilities for the number and multiplicity of finite singularitie...

We will give lots of examples of quadratic systems. Some sections will contain only examples from one specific normal form (having some parameters), and we will give the values of those parameters.

According to Proposition 5.1, for a quadratic system to have finite singularities of total multiplicity three (i.e. mf = 3), the conditions μ0 = 0 and μ1 ≠ 0 must be satisfied. Then by Theorem 6.4 the following lemma is valid.

In general in a classification problem one starts with a set of objects X and an equivalence relation ~ on X. One way of stating the classification problem of the elements of X with respect to ~ is to ask for a complete list of representatives of all equivalence classes of X, which is minimal in the sense that any two representatives in the list ar...

A study of configurations of singularities for the whole family of quadratic systems cannot be considered complete unless we also add the degenerate cases.

Since we are going to talk about finite and infinite singularities, we must first describe the compactified space in which we are going to work.

Consider real the quadratic systems (8.1). According to Proposition 5.1 for a quadratic system (8.1) to have finite singularities of total multiplicity four (i.e. mf = 4), the condition μ0 ≠ 0 must be satisfied. Therefore according to Theorem 6.4 the following lemma is valid.

One of the goals of this book is to enumerate all possible geometrical configurations of singularities, finite and infinite, of real quadratic differential systems. The foliations with singularities of the complexified systems can be compactified over the complex projective space.

Quadratic differential systems occur often in many areas of applied mathematics, in population dynamics [145], nonlinear mechanics [236, 237, 69], chemistry, electrical circuits, neural networks, laser physics, hydrodynamics [347, 328, 183, 191], astrophysics [80] and others [280, 154, 102].

The roots of the invariant theory of polynomial vector fields lie in the classical invariant theory. The idea to adapt to polynomial vector fields the concepts of classical invariant theory is due to C. S. Sibirschi, the founder of the Chişinău school of qualitative theory of differential equations.

Consider the class QS of all non-degenerate planar quadratic systems and its subclass QSE of all its systems possessing an invariant ellipse. This is an interesting family because on one side it is defined by an algebraic geometric property and on the other, it is a family where limit cycles occur. Note that each quadratic differential system can b...

This book addresses the global study of finite and infinite singularities of planar polynomial differential systems, with special emphasis on quadratic systems. While results covering the degenerate cases of singularities of quadratic systems have been published elsewhere, the proofs for the remaining harder cases were lengthier. This book covers a...

The goal of this article is to give invariant necessary and sufficient conditions for a quadratic system, presented in whatever normal form, to have anyone of 17 out of the 20 phase portraits of the family of quadratic systems with two complex conjugate invariant lines intersecting at a finite real point. The systems in this family have a maximum o...

In Artés et al. (Geometric configurations of singularities of planar polynomial differential systems. A global classification in the quadratic case. Birkhäuser, Basel, 2019) the authors proved that there are 1765 different global geometrical configurations of singularities of quadratic differential systems in the plane. There are other 8 configurat...

Let QSH be the whole class of non-degenerate planar quadratic differential systems possessing at least one invariant hyperbola. We classify this family of systems, modulo the action of the group of real affine transformations and time rescaling, according to their geometric properties encoded in the configurations of invariant hyperbolas and invari...

In this work we consider the problem of classifying all configurations of invariant lines of total multiplicity eight (including the line at infinity) of real planar cubic differential systems. The classification was initiated in Bujac and Vulpe (J Math Anal Appl 423:1025–1080, 2015) where the cubic systems which possess 4 distinct infinite singula...

In the article Llibre and Vulpe (Rocky Mt J Math 38:1301–1373, 2006) the family of cubic polynomial differential systems possessing invariant straight lines of total multiplicity 9 was considered and 23 such classes of systems were detected. We recall that 9 invariant straight lines taking into account their multiplicities is the maximum number of...

In this article we consider the class QS of all non-degenerate qua-dratic systems. A quadratic polynomial differential system can be identified with a single point of R 12 through its coefficients. In this paper using the algebraic invariant theory we provided necessary and sufficient conditions for a system in QS to have at least one invariant hyp...

In this article, we consider the class QSLu+vc+wc,∞ 4 of all real quadratic differential systems dx dt = p(x, y), dy dt = q(x, y) with gcd(p, q) = 1, having invariant lines of total multiplicity four and two complex and one real infinite singularities. We first construct compactified canonical forms for the class QSLu+vc+wc,∞ 4 so as to include lim...

In this article we prove a classification theorem (Main theorem) of real planar cubic vector fields which possess two distinct infinite singularities (real or complex) and eight invariant straight lines, including the line at infinity and including their multiplicities. This classification, which is taken modulo the action of the group of real affi...

In the topological classification of phase portraits no distinctions are made between a focus and a node and neither are they made between a strong and a weak focus or between foci of different orders. These distinctions are however important in the production of limit cycles close to the foci in perturbations of the systems. The distinction betwee...

In this article we obtain the geometric classification of singularities, finite and infinite, for the three subclasses of quadratic differential systems with mf=4 possessing exactly two finite singularities, namely: (i) systems with two double complex singularities (18 configurations); (ii) systems with two double real singularities (33 configurati...

In this work we consider the problem of classifying all configurations of singularities, finite and infinite, of quadratic differential systems, with respect to the geometric equivalence relation defined in Artés et al. (Rocky Mount J Math, 2014). This relation is deeper than the topological equivalence relation which does not distinguish between a...

In [3] we classified globally the configurations of singularities at infinity of quadratic differential systems, with respect to the geometric equivalence relation. The global classification of configurations of finite singularities was done in [2] modulo the coarser topological equivalence relation for which no distinctions are made between a focu...

In this work we classify, with respect to the geometric equivalence relation, the global configurations of singularities, finite and infinite, of quadratic differential systems possessing exactly three distinct finite simple singularities. This relation is finer than the topological equivalence relation which does not distinguish between a focus an...

The goal of this paper is to present applications of symbolic calculations and polynomial invariants to the problem of classifying planar polynomial systems of differential equations. For these applications, we use some previously defined, and some new polynomial invariants. This is part of a much larger work by the authors together with J.C. Artés...

The Lotka-Volterra planar quadratic differential systems have nu-merous applications but the global study of this class proved to be a challenge difficult to handle. Indeed, the four attempts to classify them (Reyn (1987), Wörz-Buserkros (1993), Georgescu (2007) and Cao and Jiang (2008)) produced results which are not in agreement. The lack of adeq...

This article is about weak singularities of quadratic differential systems, that is, non-degenerate singular points with traces of the corresponding linearized systems at such points equal to zero. These could be foci, centers or saddles. Necessary and sufficient conditions for a real quadratic system to possess a fixed number of weak singularities...

In this paper we classify all cubic polynomial differential systems having a rational first integral of degree two. In other words we characterize all the global phase portraits of the cubic polynomial differential systems having all their orbits contained in conies. We also determine their configurations of invariant straight lines. We show that t...

While quadratic differential systems intervene in many areas of applied mathematics and they also have theoretical importance, the topological classification of this class remains an extremely hard problem. However, in recent years much progress has been achieved due to the use of computer algebra and numerical calculations for obtaining complete c...

A quadratic polynomial differential system can be identified with a single point of R 12 through the coefficients. Using the algebraic invariant theory we classify all the quadratic polynomial differential systems of R 12 having an integrable saddle. We show that there are only 47 topologically different phase portraits in the Poincaré disc associa...

A quadratic polynomial differential systemcan be identified with a single point of ℝ12 through its coefficients. The phase portrait of the quadratic systems having a rational first integral of degree 3 have been
studied using normal forms. Here using the algebraic invariant theory, we characterize all the non-degenerate quadratic polynomial
differe...

In this article we consider the class QSL43s¥{{\bf QSL}_{\bf4}^{\bf3s\boldsymbol\infty}} of all real quadratic differential systems
\fracdxdt=p(x,y),\fracdydt=q(x,y){\frac{{\rm{d}}x}{{\rm{d}}t}=p(x,y),\frac{{\rm{d}}y}{{\rm{d}}t}=q(x,y)} with gcd(p, q) = 1, having invariant lines of total multiplicity four and three real distinct infinite singular...

In this article, we study the Lotka–Volterra planar quadratic differential systems. We denote by LV systems all systems which can be brought to a Lotka–Volterra system by an affine transformation and time homotheties. All
these systems possess invariant straight lines. We classify the family of LV systems according to their geometric properties enc...

In this paper we give a complete study using affine invariant conditions for the quadratic systems having centers. Independently we do the same for quadratic systems being Hamiltonian. There are two improvements of the previous results as [33] in which centers where study up to GL-invariant, and of [1] in which Hamiltonian QS where classified witho...

In this paper we are going to apply the invariant theory to give invariant conditions on the coefficients of any non-degenerate quadratic system in order to determine if it has or not a polynomial first integral without using any normal form. We obtain that the existence of polynomial first integral is directly related with the fact that all the ro...

In this article we consider the action of the group Aff (2, R) of affine transformations and time rescaling on real planar quadratic differential systems. Via affine invariant conditions we give a complete stratificationof this family of systems according to the dimension D of affine orbits proving that 3≤ D ≤ 6. Moreover we give a complete topolog...

In this article we make a full study of the class of non-degenerate real planar quadratic differential systems having all
points at infinity (in the Poincaré compactification) as singularities. We prove that all such systems have invariant affine
lines of total multiplicity 3, give all their configurations of invariant lines and show that all these...

In this article we prove that all real quadratic differential systems dx/dt = p(x, y), dy/dt = q(x, y), with gcd(p, q) = 1, having invariant lines of total multiplicity at least five and a finite set of singularities at infinity, are Darboux integrable having integrating factors whose inverses are polynomials over R. We also classify these systems...

When one considers a quadratic differential system, one realizes that it depends on 12 parameters of which one can be fixed by means of a time change. One also can notice that fixing 4 finite real singular points plus 3 infinite real ones (all its possible singular points) implies to fix 11 conditions, that is, 11 equations that the parameters must...

In this article we prove a classification theorem of real planar quadratic vector fields which possess four invariant straight lines, including the line at infinity and including their multiplicities. This classification, which is taken modulo the action of the group of real affine transformations and time rescaling, is given in terms of algebraic...

A quadratic polynomial differential system can be identified with a single point of
\mathbbR12 \mathbb{R}^{12} through the coefficients. Using the algebraic invariant theory we classify all the quadratic polynomial differential systems
of
\mathbbR12 \mathbb{R}^{12} having a rational first integral of degree 2. We show that there are only 24 topol...

In this article we give a complete global classification of the class QSess of planar, essentially quadratic differential systems (i.e. defined by relatively prime polynomials and whose points at infinity are not all singular), according to their topological behavior in the vicinity of infinity. This class depends on 12 parameters but due to the ac...

In this article we consider the action of affine group and time rescaling on planar quadratic differential systems. We construct a system of representatives of the orbits of systems with at least five invariant lines, including the line at infinity and including multiplicities. For each orbit we exhibit its configuration. We characterize in terms o...

In this article we consider the behavior in the vicinity of infinity of the class of all planar quadratic differential systems. This family depends on twelve parameters but due to action of the affine group and re-scaling of time the family actually depends on five parameters. We give simple, integer-valued geometric invariants for this group actio...

We classify all cubic systems possessing the maximum number of invariant straight lines (real or complex) taking into account their multiplicities. We prove that there are exactly 23 topological different classes of such systems. For every class we provide the configuration of its invariant straight lines in the Poincare disc. Moreover, every class...

We classify the family of planar quadratic differential systems with a center of symmetry and two invariant straight lines according to the topology of their phase portraits. The case of the existence of simple infinite singular points is only considered. For each of the obtained distinct topological classes we give necessary and sufficient conditi...

The new necessary and sufficient affine invariant conditions for the existence and for determining the number of centers for general quadratic system are pointed out. These conditions correspond to the partition of 12-dimensional coefficient space of indicated system with respect to the number and the multiplicity of its finite critical points.

Boris Alexeevich Shcherbakov, widely recognized mathematician, Professor of the State University of Moldova, was born February 18, 1924 in Belgorod-Dnestrovsk. He is a veteran of the Second World War.
In 1956 Boris Alexeevich graduated from the Kishinev State University (now called SUM). His fruitful scientific activity had begun at the Institute o...

The affine invariant partition of the set of quadratic systems without finite singular points with respect to different topological classes is done.

It is proved that quadratic systems of autonomous differential equations in the plane admit forty-five topologically distinct phase configurations at infinity. The related conditions on the coefficients are given in affine invariant form.

The affine invariant partition of the set of quadratic systems with one finite singular point of the 4th multiplicity with respect to different topological classes is accomplished. The conditions corresponding to this partition are semi-algebraic, i.e. they are expressed as equalities or inequalities between polynomials.

We give an affine-invariant partition of the space of the nondegenerate quadratic systems of differential equations with respect to the number and the multiplicity of their infinite singular points.

The authors give the necessary and sufficient conditions for the quadratic differential systems to possess two centers. Moreover, the condions are expressed with affine invariants.

Necessary and sufficient conditions of existence of a center in the plane for the "general" quadratic differential systems are given. These conditions are expressed with the affine invariants depending on coefficients of the considered system.

Abstract In this article we consider the behavior in the vicinity of infinity of the class,of all planar quadratic dierential,systems. This family depends on twelve parameters but due to the ane,group action, the family actually depends on five parameters. We give simple, integer-valued geometric invariants for this group action which classify this...

In this article the following problem is solved: Give necessary and sucient conditions in order that a real quadratic dierential system possesses a weak focus of order i, where i 2 {1,2,3}, or a center. We note that such singular point does not need to be at the origin. The conditions are stated in terms of ane invariant polynomials in the 12-dimen...

The affine invariant necessary and sufficient conditions for distribution of multiplicities between four infinite real distinct singular points of full cubic differential system are constructed.

The affine invariant necessary and sufficient conditions for the distribution of multiplicities between three infinite real singular points of full cubic differential system are constructed.

Affine invariant coefficient conditions for quadratic systems to possess two conjugate imaginary straight line solutions as well as two couples of such lines as solutions are established. For some classes of quadratic systems with two imaginary invariant straight lines, necessary and sufficient affine invariant conditions for possessing just one li...