Dr. Vidarte-Olivera, Jhon

We construct a symplectic atlas adapted to the flow action of an uncoupled isotropic n-oscillator, referred to as the Reeb atlas. In the context of Reeb's Theorem for Hamiltonian systems with symmetry, these variables are very useful for finding periodic orbits and determining their stability in perturbed harmonic oscillators. These variables separate orbits, meaning they are in bijective correspondence with the set of orbits. Hence, they are especially suited for determining the exact number of periodic solutions via reduction and averaging methods. Moreover, for an arbitrary polynomial perturbation, we provide lower and upper bounds for the number of periodic orbits according to the degree of the perturbation.

We investigate the existence of periodic orbits in a perturbed Hamiltonian system of stellar type in 1:1 resonance. The perturbation consists of a potential of degree four with two real parameters. We determine six families of periodic orbits using reduction and averaging theories. Also, we characterize the stability of these orbits and their bifurcation curves in terms of the parameters. Finally, we show a complete picture of the choreographies of critical points originating the periodic orbits.

We study a type of perturbed polynomial Hamiltonian system in 1:1 resonance. The perturbation consists of a homogeneous quartic potential invariant by rotations of 𝜋/2 radians. The existence of periodic solutions is established using reduction and averaging theories. The different types of periodic solutions, linear stability, and bifurcation curves are characterized in terms of the parameters. Finally, some choreography of bifurcations are obtained, showing in detail the evolution of the phase flow.

We investigate analytically the existence of several families of periodic solutions for the planar anisotropic Schwarzschild-type problem. We use reduction and averaging theory, as well as the technique of continuation of Poincaré, for the study of symmetric periodic solutions. Moreover, the determination of KAM 2-tori encasing some of the linearly stable periodic solutions is proved. Finally, we analyze the existence of periodic Hamiltonian pitchfork bifurcation of the periodic solutions.