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• W35962554 abstract "Aquesta tesi esta dividida en dues parts diferents. En la primera, estudiam els sistemes quadratics (sistemes polinomials de grau dos) que tenen un invers de factor integrant polinomial. En la segona, estudiam tres problemes diferents referits als sistemes diferencials polinomials.La primera partEn lestudi dels sistemes diferencials plans el coneixement duna integral primera es molt important. Els seus conjunts de nivell estan formats per orbites i ens permeten dibuixar el retrat de fase del sistema, objectiu principal de la teoria qualitativa de les equacions diferencials al pla. Com ja se sap, existeix una bijeccio entre lestudi de les integrals primeres i lestudi dels inversos de factor integrant. De fet, es mes senzill lestudi dels inversos de factor integrant que el de les integrals primeres. Una classe es dels sistemes quadratics ampliament estudiada dins els sistemes diferencials al pla es la dels sistemes quadratics. Hi ha mes dun miler darticles publicats sobre aquest tipus de sistemes, pero encara som lluny de coneixer quins daquests sistemes son integrables, es a dir, si tenen una integral primera.En aquest treball, estudiam els sistemes quadratics que tenen un invers de factor integrant polinomial V = V(x, y), i per tant tambe tenen una integral primera, definida alla on no sanul·la. Aquesta classe de sistemes diferencials es important per diferents motius:1. La integral primera es sempre Darboux.2. Conte la classe dels sistemes quadratics homogenis, ampliament estudiada (Date, Sibirskii, Vulpe...).3. Conte la classe dels sistemes quadratics amb un centre, tambe estudiada (Dulac, Kapteyn, Bautin,...).4. Conte la classe dels sistemes quadratics Hamiltonians (Artes, Llibre, Vulpe).5. Conte la classe dels sistemes quadratics amb una integral primera polinomial (Chavarriga, Garcia, Llibre, Perez de Rio, Rodriguez).6. Conte la classe dels sistemes quadratics amb una integral primera racional de grau dos (Cairo, Llibre).La segona partPresentam els seguents tres 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.En el primer article donam tres resultats principals. Primer provam que un camp vectorial polinomial que te una integral primera polinomial te un invers de factor integrant polinomial. El segon resultat es un exemple dun camp vectorial polinomial que te una integral primera racional i no te ni una integral primera polinomial ni un invers de factor integrant polinomial. Era un problema obert el fet de sebre si existien camps vectorials polinomials veri?cant aquestes condicions. El tercer resultat es un exemple dun camp vectorial polinomial que te un centre i no te invers de factor integrant polinomial. Un exemple daquest tipus era esperat pero desconegut en la literatura.En el segon article estudiam camps vectorials polinomials reversibles de grau quatre en R(3) que tenen, sota certes condicions generiques, un nombre arbitrari d`orbitesperi`odiques hiperb`oliques. Sense aquestes condicions, tenen un nombre arbitrari dorbites periodiques hiperboliques. Sense aquestes condicions, tenen un nombre arbitrari dorbites periodiques.Finalment, en el tercer article, estudiam la pertorbacio dun centre de R(4) que prove dun problema de la fisica. Mitjancant la teoria dels termes mitjans de primer ordre dins els camps vectorials polinomials de grau quatre, el sistema pertorbat pot tenir fins a setze orbites periodiques hiperboliques bifurcant de les orbites perodiques del centre._______________________________________________________________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 (Artes, Llibre, Vulpe).5. It contains the class of quadratic systems having a polynomial ?rst integral (Chavarriga, Garcia, Llibre, Perez de Rio, Rodriguez).6. It contains the class of quadratic systems having a rational ?rst integral of de gree two (Cairo, 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." @default.
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• W35962554 title "Polynomial inverse integrating factors of quadratic differential systems and other results" @default.
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