Dr. Uribe-Santibañez, Marco

We consider a completely integrable system of differential equations in arbitrary dimensions whose phase space contains an open set foliated by periodic orbits. This research analyzes the persistence and stability of the periodic orbits under a nonlinear periodic perturbation. For this purpose, we use the Melnikov method and Floquet theory to establish conditions for the existence and stability of periodic orbits. Our approach considers periods of the unperturbed orbits depending on the integrals and constant periods. In the applications, we deal with both cases. Precisely, we study the existence of periodic orbits in a perturbed generalized Euler system. In the degenerate case, we analyze the existence and stability of periodic orbits for a perturbed harmonic oscillator.

In this work, we show the existence of zero-Hopf periodic orbits in a 10-parametric differential equation of third order x′′′ + (a1x′ + b1x + c1)x′′ + (a2x′ + b2x + c2)x′ + (a3x′ + b3x + c3)x + k = 0, where ai, bi, ci, k ∈ R for i = 1, 2, 3. This family is based on a generalization of the equation associated to the Hiemenz flow, when the boundary conditions are neglected, and it will be named as generalized Hiemenz equation. Our approach relies in the use of averaging method. Moreover, the kind of stability of the periodic orbits is determined according to the parameters.