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Dr. Vidarte-Olivera, Jhon
Research Outputs
Symplectic Reeb atlas and determination of periodic solutions in perturbed isotropic n-oscillators
2025, Crespo, Francisco, Dr. Vidarte-Olivera, Jhon, Villafañe, Jersson
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.
Periodic solutions in a 2D-symmetric Hamiltonian system through reduction and averaging method
2024, Dr. Uribe-Santibañez, Marco, Dr. Vidarte-Olivera, Jhon, Carrasco, D.
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.