Summary: The research project is part of the area of Dynamic Systems. It focuses on three main lines: (1) The Qualitative Theory of Differential Equations, (2) Hamiltonian Systems and Celestial Mechanics, and (3) Discrete Dynamical Systems. (1) The qualitative theory of differential equations is based on the qualitative study of systems of differential equations in low dimensions. Putting special emphasis on the analysis of their periodic solutions, their number, their stability, and their bifurcations. In addition, we are also interested in studying the integrability of these systems. In particular, the existence and number of limit cycles of polynomial differential equation systems in the plane and in the cylinder are studied. With a special interest in all the variants of the so-called 16th Hilbert problem. Parallel calculation and numerical simulation are tools that we have added to the already traditional ones in this field, such as perturbation theory, the study of bifurcations, and Darboux integrability theory. (2) In Celestial Mechanics, the study of families of periodic orbits of Hamiltonian systems is of special interest, and in particular, the families defined by the central configurations of the n-body problem of Celestial Mechanics. We want to develop and improve the application of the average theory to the study of families of periodic orbits of Hamiltonian systems and especially those that come from Celestial Mechanics. (3) Usually discrete dynamical systems are defined by the action of the iteration of a function or a recurrence. In the study of discrete dynamics, periodic orbits play an important role. The study of periodic behavior, limit sets, and the dynamics restricted to invariant objects as well as the characterization of the set of periods are some of the proposed objectives. Although it is true that the objectives detailed in this project are a continuation or direct evolution of our previous projects, each new objective set opens new expectations and areas of knowledge that allow us to evolve toward a better understanding of the main and current challenges posed by mathematics. current. With the ultimate goal of a possible application. It is important to mention that most of the proposed challenges will be developed in collaboration with other researchers of recognized international prestige.