On the differentiability of first integrals of two dimensional flows

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 straight lines that a cubic polynomial differential systems can have if this number is finite. Here we complete the classification given in Llibre and Vulpe (Rocky Mt J Math 38:1301–1373, 2006) by adding a new class of such cubic systems and for each one of these 24 such classes we perform the corresponding first integral as well as its phase portrait. Moreover we present necessary and sufficient affine invariant conditions for the realization of each one of the detected classes of cubic systems with maximum number of invariant straight lines when this number is finite.

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 singularities are considered and 17 such distinct configurations were detected. That work was continued in Bujac and Vulpe (Qual Theory Dyn Syst 14(1):109–137, 2015) where the classification was done for the family of cubic systems with three distinct infinite singularities and where the existence of 5 distinct configurations was proved. The cubic systems with 2 infinite singularities are considered in Bujac (Bul Acad Stiinte Repub Mold Mat 1:48–86, 2015) and Bujac and Vulpe (Classification of a subfamily of cubic differential systems with invariant straight lines of total multiplicity eight. Universitat Autonoma de Barselona, 35, 2015) where 25 configurations were constructed. This article deals with the last remaining subfamily of cubic systems. We prove here a classification theorem (Main Theorem) of real planar cubic vector fields which possess exactly one infinite singular point and eight invariant straight lines, including the line at infinity and including their multiplicities. We show that these systems could possess only 4 such configurations. We underline that the classifications of all the above mentioned families of cubic systems are taken modulo the action of the group of real affine transformations and time rescaling and are given in terms of invariant polynomials. The algebraic invariants and comitants allow one to verify for any given real cubic system whether or not it has invariant lines of total multiplicity eight, and to specify its configuration of lines endowed with their corresponding real singularities of this system. The calculations can be implemented on computer and the results can therefore be applied for any family of cubic systems, given in any normal form.

First, doing a combination of analytical and algebraic computations, we determine by first time an explicit normal form depending only on three parameters for all cubic homogeneous polynomial differential systems having a center. After using the averaging method of first order we show that we can obtain at most one limit cycle bifurcating from the periodic orbits of the mentioned centers when they are perturbed inside the class of all cubic polynomial differential systems. Moreover, there are examples with one limit cycles.

In this article we prove a classification theorem (Main Theorem) of real planar cubic vector fields which possess eight invariant straight lines, including the line at infinity and including their multiplicities and in addition they possess three distinct infinite singularities. This classification, which is taken modulo the action of the group of real affine transformations and time rescaling, is given in terms of affine invariant polynomials. The invariant polynomials allow one to verify for any given real cubic system whether or not it has invariant straight lines of total multiplicity eight, and to specify its configuration of straight lines endowed with their corresponding real singularities of this system. The calculations can be implemented on computer and the results can therefore be applied for any family of cubic systems in this class, given in any normal form.

We study the integrability of homogeneous rational linear and homo-geneous rational quadratic differential equations. We prove that these equations can be integrated by using their algebraic solutions which are invariant straight lines.

Dynamic agglomeration patterns in a two-country new economic geography model with four regions

Until now all the examples of polynomial differential systems in R 3 exhibiting an invariant sphere have degree larger than one. In this note we show, for the first time, that the polynomial differential systems in R 3 of degree one can exhibit invariant spheres. Moreover, for such polynomial differential systems we also describe their phase portraits on the invariant sphere.

Using Einstein’s notation [4]), the polynomial differential systems of finite dimension n and of degree at most k with coefficients in a field k of characteristic zero can be written as dx dt = a + ajα1x α1 + ajα1α2x 1x2 + ajα1···αrx α1 · · · xr (1) where j, α1, αr ∈ {1, . . . , n}, 1 ≤ r ≤ k, and for j = 1, . . . , n and for 2 ≤ r ≤ k, the tensor ajα1···αr (1 time contravariant and r times covariant) is symmetric with respect to the lower subscripts and we consider the action of the general linear group GL(n, k) on the phase space k. In this work, starting from a minimal system of generators of the ideal of algebraic invariants of polynomial differential systems (1) we will develop an algorithmic method to study the local stability of a given cubic differential system with the help of Grobner bases and using the GL(n, k)-invariants of this cubic system [2]. Our algorithmic method is motivated by the important role of the invariant theory in the survey of polynomial differential systems and the numerous works on Groebner basis one of the main practical tools for solving algebraic systems and computing algebraic varieties. Keywords Polynomial differential systems,linear transformations, cubic differential systems,singular points, invariant, covariant, Grobner bases.

Polynomial differential systems having a given Darbouxian first integral*1

In this work we consider the polynomial differential system $\dot x = -y + x y^{n-1}$, $\dot y =x + a y x^{n-1}$, where $a \in \mathbb{R}$ and $n \ge 2$ with $n \in \mathbb{N}$. This system is a certain generalization of the classical Liénard system. We study the center problem and consequently the analytic integrability problem for such family around the origin for any value of $n$.

The existence of limit cycles is interesting and very important in applications. It is a key to understand the dynamic of polynomial differential systems. The aim of this paper is to investigate a class of planar differential systems of degree seven. Under some suitable conditions, the existence of three limit cycles two of them are non-algebraic while the third is algebraic is proved. Furthermore, these limit cycles are explicitly given in polar coordinates. Some examples are presented in order to illustrate the applicability of our results.

We give sufficient conditions in terms of the Melnikov functions in order that an analytic or a polynomial differential system in the real plane has a period annulus. We study the first nonzero Melnikov function of the analytic differential systems in the real plane obtained by perturbing a Hamiltonian system having either a nondegenerate center, a heteroclinic cycle, a homoclinic cycle, or three cycles obtained connecting the four separatrices of a saddle. All the singular points of these cycles are hyperbolic saddles. Finally, using the first nonzero Melnikov function we give a new proof of a result of Roussarie on the finite cyclicity of the homoclinic orbit of the integrable system when we perturb it inside the class of analytic differential systems.

One of the classical problems of the differential systems in the plane R 2 is to distinguish between a focus and a centre, and when we have a centre to distinguish if it is global or not. Let ( x 0 , y 0 ) be an equilibrium point of a differential system in R 2 . If there is a neighbourhood V of ( x 0 , y 0 ) such that V ∖ { ( x 0 , y 0 ) } is filled with periodic orbits, then ( x 0 , y 0 ) is a center. When V can be chosen as R 2 , the centre is a global center. We deal with a family of cubic polynomial differential systems in R 2 studied by Devlin, Lloyd and Pearson in J. Devlin, N.G. Lloyd and J.M. Pearson, [Cubic systems and Abel equations, J. Differ. Equ. 147 (1997), pp. 435–454.]. These authors analyse when the equilibrium point at the origin of coordinates can be a centre. We show two mistakes in the paper (J. Devlin, N.G. Lloyd and J.M. Pearson, Cubic systems and Abel equations, J. Differ. Equ. 147 (1997), pp. 435–454.), and additionally we prove that when the origin of coordinates is a centre, it can not be a global centre.

This note presents some advances regarding the Lyapunov constants of some families of planar polynomial differential systems, as a first step toward the resolution of the center and cyclicity problems. First, a parallelization approach is computationally implemented to achieve the 14th Lyapunov constant of the complete cubic family. Second, a technique based on interpolating some specific quantities so as to reconstruct the structure of the Lyapunov constants is used to study a Kukles system, some fifth-degree homogeneous systems, and a quartic system with two invariant lines.

In this paper we prove the Darboux integrability of a cubic differential system with a singular point of a center typer having at least two parallel invariant straight lines.