La ecuación de Ince-Hill

Laburpena

The Ince-Hill's equation is a linear dierential equation of second degree with periodic coecients that depends on four parameters a; b; c and d (1 + a cos 2x) d2y dx2 + b sin 2x dy dx + (c + d cos 2x) y = 0 (1) The interest of (1) is that all the known cases of equations of Hill's kind with analytic coecients and solved problems of coexistence are particular cases of this equation with four parameters. (1) is in general no much studied, but particular cases of this equation appears on variational equations associated to problems of Celestial Mechanics. Frequently the parameter a denotes ¡e, where e is the eccentricity. Precisely, one of the initial motivations of this thesis is the study of the stability of the trivial solution of (1) for the numerous problems of mechanics that is applied. This study is not easy from the analysis viewpoint. Note that the Ince-Hill's equation is a linear dierential equation of second degree with regular singularities, this singularities are namely singularities of Fuchs kind. So the thesis is divided in two parts. The rst, more general, includes the two rst chapters and it is dedicated to local study of linear dierential equations systems with a singularity of Fuchs. In this part, in chapter 2, we make a local study of the foliation dened for a complex linear system in a neighbourhood of a singularity of Fuchs. This general study is one of the more emphasized results of the thesis is due to that although the linear equations with holomorphic coecients usually have been analyze from the analysis viewpoint, its consideration such us dynamical system and the introduction of algebraic methods is more recent. So the foliation for linear systems has been realized for Camacho, C., H. Kuiper, N., Palis, J. Our case is about the study of systems with singularities of Fuchs. In contrast to Camacho, C., H. Kuiper, N., Palis, J. which is realized an study of the foliation with methods independents of the coordinates, the existence of singularities forces to work with these. As a result we get a characterization of the foliation determined from the vectorial eld. Continue with the analogy between Camacho, C., H. Kuiper, N., Palis, J. and the case of a system with singularities, in chapter 3 we study about the generalization of the results in chapter 2 to vectorial elds in CPn. We generalize the work of Zakeri, S. and obtain a relation between the linear vectorial elds in Cn+1 and the vectorial elds in CPn with a singularity of Fuchs. The second part of the thesis, that includes the chapter 4 is dedicated to particular study of the Ince-Hill's equation. One of the results that we obtain is the classication of the equation in function of the kind of the singularities for the dierent values of the parameters, for this we analyze the algebraic form of the equation. Finally, we make an study of the equation in the case that the equation maybe expressed like a hamiltonian system and we study the stability for some concreted cases of the parameters.