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 Poincaré 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

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 Poincaré’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 Poincaré of algebraic integrability for 4 of the normal forms we study

Global phase portraits for the Abel quadratic polynomial differential equations of second kind with Z₂-symmetries

We provide normal forms and the global phase portraits in the Poincaré disk for all Abel quadratic polynomial differential equations of the second kind with Z₂-symmetries

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

We describe the origin and evolution of ideas on topological and polynomial invariants and their interaction, in problems of classification of polynomial vector fields. The concept of moduli space is discussed in the last section and we indicate its value in understanding the dynamics of families of such systems. Our interest here is in the concepts and the way they interact in the process of topologically classifying polynomial vector fields. We survey the literature giving an ample list of references and we illustrate the ideas on the testing ground of families of quadratic vector fields. In particular, the role of polynomial invariants is illustrated in the proof of our theorem in the section next to last. These concepts have proven their worth in a number of classification results, among them the most recent work on the geometric classification of the whole class of quadratic vector fields, according to their configurations of infinite singularities. An analog work including both finite and infinite singularities of the whole quadratic class, joint work with J. C. Artés, J. Llibre, and N. Vulpe, is in progress

Detection of special curves via the double resultant

We introduce several applications of the use of the double resultant through some examples of computation of different nature: special level sets of rational first integrals for rational discrete dynamical systems; remarkable values of rational first integrals of polynomial vector fields; bifurcation values in phase portraits of polynomial vector fields; and the different topologies of the offset of curves.The authors are partially supported by MINECO/ FEDER MTM2013-40998-P Grant. Johanna D. García-Saldaña is also partially supported by FONDECyT postdoctoral fellowship 3150131/2015. Armengol Gasull is also partially supported by Generalitat de Catalunya Grant 2014SGR568

First integrals and phase portraits of planar polynomial differential cubic systems with the maximum number of invariant straight lines

In the article LliVul2006 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 LliVul2006 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

A new algorithm for finding rational first integrals of polynomial vector fields

We present a new method to compute rational first integrals of a planar polynomial vector field. The algorithm is in general much faster than the usual methods and also allows to compute the remarkable curves associated to the rational first integral of the system.Preprin

An integrating factor matrix method to find first integrals

In this paper we developed an integrating factor matrix method to derive conditions for the existence of first integrals. We use this novel method to obtain first integrals, along with the conditions for their existence, for two and three dimensional Lotka-Volterra systems with constant terms. The results are compared to previous results obtained by other methods