Periodic Orbits, Bifurcations, Integrability in Dynamical Systems and its Applications

The project is framed within the field of Dynamical Systems, focusing on two main areas: Qualitative Theory of Differential Equations and Discrete Dynamical Systems. In the first area, we develop a qualitative theory centered on the study of low-dimensional differential equation systems. Special emphasis is placed on analyzing periodic solutionstheir number, stability, and bifurcationsas well as the integrability of these systems. In particular, we investigate the existence and number of limit cycles in polynomial differential systems defined on the plane and the cylinder, with special interest in all variants of the well-known Hilberts 16th Problem. To achieve this, we incorporate modern tools such as parallel computing and numerical simulation, complementing traditional methods like perturbation theory, bifurcation analysis, and Darboux integrability theory. Furthermore, we integrate the study of periodic orbits in Hamiltonian systems, especially those associated with central configurations in the n-body problem of Celestial Mechanics, using averaging theory as the main approach. The second area focuses on Discrete Dynamical Systems, defined by the iteration of functions or recurrences. In this context, periodic orbits play a fundamental role. Our objectives include studying periodic behaviors, limit sets, dynamics restricted to invariant objects, and characterizing the set of periods. Although these objectives represent a natural evolution of our previous projects, each new goal opens opportunities to advance our understanding of the main challenges posed by contemporary mathematics, with the ultimate aim of potential applications. It is important to highlight that most of the proposed challenges will be addressed in collaboration with internationally recognized researchers, ensuring a solid and high-impact scientific approach. Impacto científico técnico o internacional esperable: The project is expected to generate results of the highest scientific quality, with publications in internationally recognized, high-impact journals. This expectation is supported by the groups consistent track record: for over 15 years, the vast majority of our work has appeared in top-tier journals, primarily those ranked in the first quartile of the JCR. We anticipate maintaining this level of excellence throughout the project. Our dissemination strategy prioritizes visibility and credibility. Results will be published in prestigious journals with rigorous anonymous peer review and presented at leading international conferences. More than half of our team members have already been invited to speak at such events, and this trend is expected to continue, reinforcing the groups international standing. The UAB Dynamic Systems Group has a long-established commitment to transparency and outreach. For over 25 years, our website (http://www.gsd.uab.cat) has provided a comprehensive record of publications, defended theses, visiting professors, and organized scientific activities. All articles are also available through the UAB digital repository (http://ddd.uab.cat/collection/gsd), ensuring open access and global dissemination. Since the 1980s, the group has organized weekly research seminars, offering a platform for presenting results and fostering collaboration with leading international researchers. These seminars provide doctoral students with exceptional opportunities to learn advanced techniques and engage with top experts. In recent years, we have expanded this initiative globally by hosting online sessions via Zoom, attracting significant international participation and enhancing the groups visibility worldwide. In addition to publications and seminars, the project will actively promote knowledge transfer through participation in international and local conferences, research stays at universities and research centers, and the organization of scientific events. Specifically, during the projects duration, we plan to organize three international conferences (one annually), several specialized workshops, and two advanced courses. These activities will consolidate the groups leadership in the field and amplify its international impact. In summary, the project builds on a proven history of excellence and global engagement. Its outcomes will not only advance the state of the art in mathematics but also strengthen international collaboration and visibility, ensuring a significant scientific and societal return on investment.