Invariant Algebraic Curves Research Articles

Characterization of the Riccati and Abel Polynomial Differential Systems Having Invariant Algebraic Curves

The Riccati polynomial differential systems are differential systems of the form [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] for [Formula: see text] are polynomial functions. We characterize all the Riccati polynomial differential systems having an invariant algebraic curve. We show that the coefficients of the first four highest degree terms of the polynomial in the variable [Formula: see text] defining the invariant algebraic curve determine completely the Riccati differential system. A similar result is obtained for any Abel polynomial differential system.

Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves

Abstract The classification of the phase portraits is one of the classical and difficult problems in the qualitative theory of polynomial differential systems in R 2 {{\mathbb{R}}}^{2} , particularly for quadratic systems. Even with the hundreds of studies on the topology of real planar quadratic vector fields, fully characterizing their phase portraits is still a difficult problem. This paper is devoted to classifying the phase portraits of two polynomial vector fields with two usual invariant algebraic curves, by investigating the geometric solutions within the Poincaré disc. One can notice that these systems yield 26 topologically different phase portraits.

Number of Limit Cycles for Planar Systems with Invariant Algebraic Curves

AbstractFor planar polynomials systems the existence of an invariant algebraic curve limits the number of limit cycles not contained in this curve. We present a general approach to prove non-existence of periodic orbits not contained in this given algebraic curve. When the method is applied to parametric families of polynomial systems that have limit cycles for some values of the parameters, our result leads to effective algebraic conditions on the parameters that force non-existence of the periodic orbits. As applications we consider several families of quadratic systems: the ones having some quadratic invariant algebraic curve, the known ones having an algebraic limit cycle, a family having a cubic invariant algebraic curve and other ones. For any quadratic system with two invariant algebraic curves we prove a finiteness result for its number of limit cycles that only depends on the degrees of these curves. We also consider some families of cubic systems having either a quadratic or a cubic invariant algebraic curve and a family of Liénard systems. We also give a new and simple proof of the known fact that quadratic systems with an invariant parabola have at most one limit cycle. In fact, what we show is that this result is a consequence of the similar result for quadratic systems with an invariant straight line.

Invariant algebraic curves for certain generalized Liénard differential system

In this work we solve the problem of finding the invariant algebraic curves of a generalized Liénard differential system $ \dot{x} = y $, $ \dot{y} = -f(x)y-g(x) $, where $ \deg f = m $ and $ \deg g = n $ and with $ n = m+1 $, generalizing the known previous examples. In particular it is studied the case $ m = 3 $ and $ n = 4 $. The difficulties in applying the Puiseux method are shown even when the degrees of the invariant curves are bounded.

Characterization of the kukles polynomial differential systems having an invariant algebraic curve

Let f(x) and g(x) be complex polynomials. We characterize all Kukles polynomial differential systems of the formx′=y,y′=−y2−f(x)y−g(x) having an invariant algebraic curve. We show that expanding an invariant algebraic curve of these differential systems as a polynomial in the variable y, the first four higher coefficients of the polynomial defining the invariant algebraic curve determine completely these Kukles systems. In particular if the second and third higher coefficients of the polynomial defining the invariant algebraic curve satisfy a simple relation between them the invariant algebraic curve is of the form (y+p(x))n=0 for some polynomial p(x) and y+p(x)=0 is an invariant algebraic curve of the Kukles system for any complex polynomial f(x).

Global Phase Portraits of Quadratic Polynomial Differential Systems Having as Solution Some Classical Planar Algebraic Curves of Degree 6

Abstract The main goal of this paper is to classify the global phase portraits of seven quadratic polynomial differential systems, exhibiting as invariant algebraic curves seven well-known algebraic curves of degree six. We prove that these systems have five topologically different phase portraits in the Poincarédisc.

Invariant hypercomplex structures and algebraic curves

AbstractWe show that ‐invariant hypercomplex structures on (open subsets) of regular semisimple adjoint orbits in correspond to algebraic curves C of genus , equipped with a flat projection of degree k, and an antiholomorphic involution covering the antipodal map on .

Foliations on $$\mathbb {P}^2$$ with only one singular point

In this paper we study holomorphic foliations on $${\mathbb {P}}^2$$ with only one singular point. If the singularity has algebraic multiplicity one, we prove that the foliation has no invariant algebraic curve. We also present several examples of such foliations in degree three.

LIMIT CYCLES AND PHASE PORTRAIT FOR A CLASS OF DIFFERENTIAL SYSTEMS

We introduce an explicit expression of invariant algebraic curves for a class of polynomial differential systems and an explicit expression for its first integral. Moreover, we determine sufficient conditions for these systems to possess a limit cycle, which can be expressed by an explicit formula. Concrete examples exhibiting the applicability of our results are introduced.

A characterization of the generalized Liénard polynomial differential systems having invariant algebraic curves

The generalized Liénard polynomial differential systems are the differential systems of the form x′ = y, y′ = − f(x)y − g(x), where f and g are polynomials.We characterize all the generalized Liénard polynomial differential systems having an invariant algebraic curve. We show that the first four higher coefficients of the polynomial in the variable y, defining the invariant algebraic curve, determine completely the generalized Liénard polynomial differential system. This fact does not hold for arbitrary polynomial differential systems. 2010 mathematics subject classificationPrimary 34A05. Secondary 34C05, 37C10.

Phase Portraits of a Class of Cubic Systems with an Ellipse and a Straight Line as Invariant Algebraic Curves

Phase Portraits of a Class of Cubic Systems with an Ellipse and a Straight Line as Invariant Algebraic Curves

Invariant Algebraic Curves of Generalized Liénard Polynomial Differential Systems

In this study, we focus on invariant algebraic curves of generalized Liénard polynomial differential systems x′=y, y′=−fm(x)y−gn(x), where the degrees of the polynomials f and g are m and n, respectively, and we correct some results previously stated.

EXISTENCE AND COMPUTATION OF INVARIANT ALGEBRAIC CURVES FOR PLANAR QUADRATIC DIFFERENTIAL SYSTEMS

<abstract> Some necessary conditions are given for the existence of invariant algebraic curves for planar quadratic differential systems in a special canonical form. An efficient algorithm is then designed for computations of invariant algebraic curves. From the algorithm, a quadratic differential system is found with two Hopf bifurcations as the parameter varies, each leading to an invariant algebraic limit cycle of degree 5. A family of degree 6 invariant algebraic limit cycles is also produced. To further demonstrate the capability of the algorithm, we provide a quadratic system with a family of degree 7 invariant algebraic curves enclosing one or two centers, and a system possessing a degree 16 irreducible invariant algebraic curve with a singular point of multiplicity 8 on the curve. </abstract>

Toward Effective Liouvillian Integration

We prove that foliations on the projective plane admitting a Liouvillian first integral but not admitting a rational first integral always have invariant algebraic curves of degree bounded by a function of the degree of the foliation. We establish, for the same class of foliations, the existence of a bound for the degree of the simplest integrating factor depending only on the degree of the foliation and on the nature of its singularities. We also prove the existence of invariant algebraic curves of small degree for foliations with rational first integral and intermediate Kodaira dimension.

Quadratic systems with an invariant algebraic curve of degree 3 and a Darboux invariantq

Let \(QS\) be the class of non-degenerate planar quadratic differential ystems and \(QS_3\) its subclass formed by the systems possessing an invariant cubic \(f(x,y)=0\). In this article, using the action of the group of real affine transformations and time rescaling on \(QS\), we obtain all the possible normal forms for the quadratic systems in \(QS_3\). Working with these normal forms we complete the characterization of the phase portraits in \(QS_3\) having a Darboux invariant of the form \(f(x, y)e^{st}\), with \(s \in \mathbb R\).&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2021/69/abstr.html

On the dynamic of the isotropic star

In this paper, we investigate the dynamical behavior of the planar polynomial system which describes the evolution of the isotropic star. It is shown that the system has no global [Formula: see text] and no global analytic first integrals. It has an invariant algebraic curve with algebraic multiplicity [Formula: see text] and an exponential factor that comes from the multiplicity of the infinite invariant straight line. It is proved that the system can be changed into the Liénard system. By using a dominant balance analysis, we prove the system has a general solution which eventuates to a finite-time singularity. Finally, we prove the trajectories of the vector field associated with the planar system of the isotropic star do not create a trajectory manifold which means there is no pseudo-Riemannian metric [Formula: see text] in the sense of trajectory metric such that the trajectories of the isotropic star system be geodesic.

The method of Puiseux series and invariant algebraic curves

An explicit expression for the cofactor related to an irreducible invariant algebraic curve of a polynomial dynamical system in the plane is derived. A sufficient condition for a polynomial dynamical system in the plane to have a finite number of irreducible invariant algebraic curves is obtained. All these results are applied to Liénard dynamical systems [Formula: see text], [Formula: see text] with [Formula: see text]. The general structure of their irreducible invariant algebraic curves and cofactors is found. It is shown that Liénard dynamical systems with [Formula: see text] can have at most two distinct irreducible invariant algebraic curves simultaneously and, consequently, are not integrable with a rational first integral.

Necessary and sufficient conditions for the existence of invariant algebraic curves

We present a set of conditions enabling a polynomial system of ordinary differential equations in the plane to have invariant algebraic curves. These conditions are necessary and sufficient. Our main tools include factorizations over the field of Puiseux series near infinity of bivariate polynomials generating invariant algebraic curves. The set of conditions can be algorithmically verified. This fact gives rise to a method, which is able not only to find some irreducible invariant algebraic curves, but also to perform their classification. We study in details the problem of classifying invariant algebraic curves in the most difficult case: we consider differential systems with infinite number of trajectories passing through infinity. As an example, we find necessary and sufficient conditions such that a general polynomial Lienard differential system has invariant algebraic curves. We present a set of all irreducible invariant algebraic curves for quintic Lienard differential systems with a linear damping function. It is supposed in scientific literature that the degrees of their irreducible invariant algebraic curves are bounded by 6. While we derive irreducible invariant algebraic curves of degree 9.

Geometry, integrability and bifurcation diagrams of a family of quadratic differential systems as application of the Darboux theory of integrability

During the last forty years the theory of integrability of Darboux, in terms of algebraic invariant curves of polynomial systems has been very much extended and it is now an active area of research. These developments are covered in numerous papers and several books, not always following the conceptual historical evolution of the subject and its significant connections to Poincare’s problem of the center. Our first goal is to give in a concise way, following the history of the subject, its conceptual development. Our second goal is to display the many aspects of the theory of Darboux we have today, by using it for studying the special family of planar quadratic differential systems possessing an invariant hyperbola, and having either two singular points at infinity or the infinity filled up with singularities. We prove the integrability for systems in 11 of the 13 normal forms of the family and the generic non-integrability for the other 2 normal forms. We construct phase portraits and bifurcation diagrams for 5 of the normal forms of the family, show how they impact the changes in the geometry of the systems expressed in their configurations of their invariant algebraic curves and point out some intriguing questions on the interplay between this geometry and the integrability of the systems. We also solve the problem of Poincare of algebraic integrability for 4 of the normal forms we study.

Geometry and integrability of quadratic systems with invariant hyperbolas

Let QSH be the family of non-degenerate planar quadratic differential systems possessing an invariant hyperbola. We study this class from the viewpoint of integrability. This is a rich family with a variety of integrable systems with either polynomial, rational, Darboux or more general Liouvillian first integrals as well as nonintegrable systems. We are interested in studying the integrable systems in this family from the topological, dynamical and algebraic geometric viewpoints. In this work we perform this study for three of the normal forms of QSH, construct their topological bifurcation diagrams as well as the bifurcation diagrams of their configurations of invariant hyperbolas and lines and point out the relationship between them. We show that all systems in one of the three families have a rational first integral. For another one of the three families, we give a global answer to the problem of Poincare by producing a geometric necessary and sufficient condition for a system in this family to have a rational first integral. Our analysis led us to raise some questions in the last section, relating the geometry of the invariant algebraic curves (lines and hyperbolas) in the systems and the expression of the corresponding integrating factors.