Polynomial inverse integrating factors of quadratic differential systems and other results

Resumen

This thesis is divided into two different parts. In the first one, we study the quadratic systems (polynomial systems of degree two) having a polynomial inverse integrating factor. In the second one, we study three different problems related to polynomial differential systems. The ?rst part. It is very important, for planar differential systems, the knowledge of a ?rst integral. Its level sets are formed by orbits and they let us draw the phase portrait of the system, which is the main objective of the qualitative theory of planar differential equations. As it is known, there is a bijection between the study of the ?rst integrals and the study of inverse integrating factors. In fact, it is easier to study the inverse integrating factors than the ?rst integrals. A widely studied class of planar differential systems is the quadratic one. There are more than a thousand published articles about this subject of differential systems, but we are far away of knowing which quadratic systems are integrable, that is, if they have a ?rst integral. In this work, we study the quadratic systems having a polynomial inverse integrating factor V = V (x, y), so they also have a ?rst integral, de?ned where V does not vanish. This class of quadratic systems is important for several reasons: 1. The ?rst integral is always Darboux. 2. It contains the class of homogeneous quadratic system, widely studied (Date, Sibirskii, Vulpe,...). 3. It contains the class of quadratic systems having a center, also studied (Dulac, Kapteyn, Bautin,...). 4. It contains the class of Hamiltonian quadratic systems (Artés, Llibre, Vulpe). 5. It contains the class of quadratic systems having a polynomial ?rst integral (Chavarriga, García, Llibre, Pérez de Rio, Rodríguez). 6. It contains the class of quadratic systems having a rational ?rst integral of de gree two (Cairó, Llibre). The classi?cation of the quadratic systems having a polynomial inverse integrating factor is not completely ?nished. There remain near a 5% of the cases to study. We leave their study for an immediate future. The second part. We present the following three articles: 1. A. Ferragut, J. Llibre and A. Mahdi, Polynomial inverse integrating factors for polynomial vector ?elds, to appear in Discrete and Continuous Dynamical Systems. 2. A. Ferragut, J. Llibre and M.A. Teixeira, Periodic orbits for a class of C(1) threedimensional systems, submitted. 3. A. Ferragut, J. Llibre and M.A. Teixeira, Hyperbolic periodic orbits coming from the bifurcation of a 4-dimensional non-linear center, to appear in Int. J. Of Bifurcation and Chaos. In the first article we give three main results. First we prove that a polynomial vector field having a polynomial must have a polynomial inverse integrating factor. The second one is an example of a polynomial vector ?eld having a rational ?rst integral and having neither polynomial ?rst integral nor polynomial inverse integrating factor. It was an open problem to know if there exist polynomial vector ?elds verifying these conditions. The third one is an example of a polynomial vector ?eld having a center and not having a polynomial inverse integrating factor. An example of this type was expected but unknown in the literature. In the second article we study reversible polynomial vector ?elds of degree four in R(3) which have, under certain generic conditions, an arbitrary number of hyperbolic periodic orbits. Without these conditions, they have an arbitrary number of periodic orbits. Finally, in the third article, we study the perturbation of a center in R(4) which comes from a problem of physics. By the ?rst order averaging theory and perturbing inside the polynomial vector ?elds of degree four, the perturbed system may have at most sixteen hyperbolic periodic orbits bifurcating from the periodic orbits of the center.