|One of the most important aspects of the dynamical systems is that they are one of the best tools for understanding qualitative and quantitative the mathematical models of the experimental sciences: physics, chemistry, economics, ... Most of them are formulated using continuous dynamical systems (i.e. differential equations), or discrete dynamical systems (i.e. iteration of a function or application).
The goals of this project are to advance in the knowledge of these systems with emphasis on the study of following three main research themes, with their corresponding sections. The first two topics are on continuous dynamical systems, and the third one on discrete dynamical systems.
(1) Qualitative study of the differential systems, putting special emphasis first in their periodic orbits, their stability and bifurcations, and second in their integrability.
(1.1) Limit cycles of polynomial differential systems plane and the 16th Hilbert problem.
(1.2) Development and applications of the theory of averaging for studying the periodic orbits of the differentiable systems.
(1.3) Homoclinic and heteroclinic orbits: study of the dynamics in their neigborhood and explicit computation of them.
(1.4) Integrability of polynomial differential systems in finite dimension.
(2) Study of the families of periodic orbits of the Hamiltonian systems, with special attention to Celestial Mechanics, and in particular to the central configurations of the n body problem.
(3) Characterization of the dynamics and specially the periodic dynamics of different classes of functions.
(3.1) Study of periodic orbits of the functions via the Lefschetz numbers.
(3.2) Global Dynamics of some rational recurrences.
(3.3) Algebraic entropy of the birational applications of the plane.
The objectives presented are subjects of maximum actuality in the mathematical research.
Since the early eighties our research group has been working on these issues or in closely related topics, and the results obtained have been published in some of the best journals of our speciality, and we must hope that the expected results will also be published in hight quality and impact journals.