Our objectives are on the following three main subjects: (1) The qualitative study of differential equations, (2) the Hamiltonian systems and the Celestial Mechanics, and (3) the discrete dynamical systems.
(1) The first subject consists in the qualitative study of the differential equations putting special emphasis in the analysis of their periodic solutions, their number, their kind of stability, and their bifurcations. Moreover we also are interested in the study of the integrability of these systems. More precisely:
(1.1) The study of the limit cycles for polynomial differential systems in the plain and the 16th Hilbert problem stated in 1900. Our group has solved this problem for some particular classes of polynomial differential systems. We believe that we are close to its complete solution, or at least of its generic solution. In this project we will extend this problem to trigonometric differential equations on the cylinder.
(1.2) The center-focus problem: to distinguish between a focus and a center of an analytic differential system in the plane. This problem already has more than 100 years. We have encouraging results in this direction, but we want to improve them mainly in the computation of the Liapunov constants for the linear type centers using parallel computation.
(1.3) Development and applications of the averaging theory for studying the periodic solutions of the differential systems in arbitrary finite dimension. We have developed this theory recently to arbitrary order in a small parameter and arbitrary finite dimension for differential systems. Now we want to extend this theory to the continuous and discontinuous piecewise differential systems. This kind of differential systems appear in a natural way in problems of the engineering and in the control theory.
(1.4) The existence of first integrals simplifies the study of the dynamics of a differential system. The Darboux theory of integrability and the study of the analytic first integrals have obtained good results in these last years, part of these results have been obtained by our group.
(2) The second subject mainly consists in studying the families of periodic orbits of the Hamiltonian systems, and in particular of 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 averaging theory to the study of the families of periodic orbits of the Hamiltonian systems, and in special to the ones coming from Celestial Mechanics.
(3) We consider the discrete dynamical systems defined by the iteration of a function or a recurrence. In the study of the dynamics of such discrete system, the periodic orbits play the main role. Our objective is the characterization of the set of periods for some of the relevant discrete dynamical systems. We already have several results in this direction. The Lefschetz zeta function is a good tool for these studies when the variety is compact.