Contribuciones al estudio de órbitas periódicas y variedades invariantes en sistemas dinámicos
- Autores: Clara Cufí Cabré
- Directores de la Tesis: Jaume Llibre (dir. tes.), Ernest Fontich Julià (codir. tes.)
- Lectura: En la Universitat Autònoma de Barcelona ( España ) en 2023
- Idioma: español
- Tribunal Calificador de la Tesis: Xavier Jarque Ribera (presid.), Lluís Alsedà i Soler (secret.), Inmaculada Beldoná Barraca (voc.)
- Programa de doctorado: Programa de Doctorado en Matemáticas por la Universidad Autónoma de Barcelona
- Materias:
- Enlaces
- Tesis en acceso abierto en: TDX
- Resumen
This thesis concerns the study of invariant manifolds and periodic orbits of discrete and continuous dynamical systems. The memoir is divided into two parts that can be read independently. The first part (Chapters 1-6) is dedicated to the study of invariant manifolds associated with parabolic points and parabolic invariant tori. The second part (Chapters 7-9) concerns the study of periodic orbits of dynamical systems on manifolds.
In Chapters 2 and 3 we study the existence and regularity of invariant manifolds of planar maps having a parabolic fixed point with nilpotent part using the parameterization method. The study is done for analytic maps and for finitely differentiable maps. In the analytic case, we prove the existence of an analytic one-dimensional invariant manifold under suitable conditions on the coefficients of the nonlinear terms of the map. In the differentiable case, we prove that if the regularity of the map is bigger than some value, then there exists an invariant manifold of the same regularity, away from the fixed point.
In Chapter 4 we consider an analogous problem as in Chapters 2 and 3, but for planar vector fields. We present the results of existence of invariant curves of such vector fields using the results from the previous chapters and the fact that, under suitable conditions, the invariant manifolds of a vector field are the same ones as the invariant manifolds of its time-t flow.
In Chapters 5 and 6 we consider maps and vector fields having a d-dimensional parabolic invariant torus with nilpotent part. In this context, we give conditions on the coefficients of the nonlinear terms of the map (resp. vector field) under which the invariant torus possesses stable and unstable invariant manifolds. We also consider the same problem for non-autonomous vector fields that depend quasiperiodically on time, and we present some applications of our results.
All the results of existence of invariant manifolds are stated in two steps. In the first step we present an algorithm to compute an approximation of a parameterization of the invariant manifold. In the second step, we present an «a posteriori» result, which ensures that there exists a true invariant manifold close to that approximation. Combining the two results we obtain the existence of an invariant manifold which is well approximated by the parameterization provided in the first step.
In Chapter 8 we use the Lefschetz numbers and the Lefschetz zeta function to obtain information on the set of periods of certain diffeomorphisms on compact manifolds. We consider the class of Morse-Smale diffeomorphisms defined on the n-dimensional sphere, on products of two spheres of arbitrary dimension, on the n-dimensional complex projective space, and on the n-dimensional quaternion projective space. Then, we describe the minimal sets of Lefschetz periods for such Morse-Smale diffeomorphisms, which is a subset of the set of periods that are preserved under homotopy equivalence.
Finally, in Chapter 9 we study the existence of limit cycles of linear vector fields on manifolds. It is well known that linear vector fields in R^n can not have limit cycles, because either they do not have periodic orbits or their periodic orbits form a continuum. In that chapter, we show that linear vector fields defined in some manifolds different from R^n can have limit cycles and we consider the question of how many limit cycles can they have at most.
