Professor Rivera’s research focuses on the use of concepts of mathematical analysis and differential equations in order to understand physical, biological and economic problems. He uses mathematical tools like Perturbation theory, Bifurcation theory, Piecewise linear systems and Global Continuation Methods to predict and explain the complex motion of various dynamical systems, most of them coming from nonlinear oscillators such as satellites (In Celestial Mechanics: Restricted N-body problems), Nonlinear circuits with passive elements (e.g. Memristors, which behave as a resistance with non-linear memory by relate electric charge with magnetic flux). Prof. Rivera’s contributions include the analytic proof of the existence of a family of even and periodic solutions bifurcates from the equilibrium solution in a Generalized Sitnikov (N+1)-body problem (A special case of the restricted (N+1) - body problem) that’s includes the Sitnikov problem. The proof of the existence of at least three limit cycles in discontinuous piecewise linear systems with two zones and a straight line of discontinuity of Saddle-Focus type. In driven nonlinear oscillators with parametric external force, he obtain a quantification of the interval where the linear stability of periodic solutions obtains as bifurcations of the trivial one is guarantee.
Currently Funded Research
- Qualitative theory of differential equations.
- Dynamical Systems. Discrete, piecewise lineal and continuous systems.
- Nonlinear oscillations of physical, biological and economic models.