Hildeberto Cabral
Universidade Federal de Pernambuco, Recife, Pernambuco
Publications:
Araujo G. C., Cabral H. E.
Parametric Stability of a Charged Pendulum with an Oscillating Suspension Point
2021, vol. 26, no. 1, pp. 3960
Abstract
We consider a planar pendulum with an oscillating suspension point and with the bob carrying an electric charge $q$. The pendulum oscillates above a fixed point with a charge $Q.$ The dynamics is studied as a system in the small parameter $\epsilon$ given by the amplitude of the suspension point. The system depends on two other parameters, $\alpha$ and $\beta,$ the first related to the frequency of the oscillation of the suspension point and the second being the ratio of charges. We study the parametric stability of the linearly stable equilibria and use the DepritHori method to construct the boundary surfaces of the stability/instability regions.

de Menezes Neto J. L., Cabral H. E.
Parametric Stability of a Pendulum with Variable Length in an Elliptic Orbit
2020, vol. 25, no. 4, pp. 323329
Abstract
We study the dynamics of a simple pendulum attached to the center of mass of a
satellite in an elliptic orbit. We consider the case where the pendulum lies in the orbital plane
of the satellite. We find two linearly stable equilibrium positions for the Hamiltonian system
describing the problem and study their parametric stability by constructing the boundary curves
of the stability/instability regions.

Cabral H. E., Amorim T. A.
Subharmonic Solutions of a Pendulum Under Vertical Anharmonic Oscillations of the Point of Suspension
2017, vol. 22, no. 7, pp. 782791
Abstract
We prove the existence of subharmonic solutions in the dynamics of a pendulum whose point of suspension executes a vertical anharmonic oscillation of small amplitude.

Carvalho A. C., Cabral H. E.
Lyapunov Orbits in the $n$Vortex Problem on the Sphere
2015, vol. 20, no. 3, pp. 234246
Abstract
In the phase space reduced by rotation, we prove the existence of periodic orbits of the $(n + 1)$vortex problem emanating from a relative equilibrium formed by $n$ unit vortices at the vertices of a regular polygon at a fixed latitude and an additional vortex of intensity κ at the north pole when the ideal fluid moves on the surface of a sphere.

Carvalho A. C., Cabral H. E.
Lyapunov Orbits in the $n$Vortex Problem
2014, vol. 19, no. 3, pp. 348362
Abstract
In the reduced phase space by rotation, we prove the existence of periodic orbits of the $n$vortex problem emanating from a relative equilibrium formed by $n$ unit vortices at the vertices of a regular polygon, both in the plane and at a fixed latitude when the ideal fluid moves on the surface of a sphere. In the case of a plane we also prove the existence of such periodic orbits in the $(n+1)$vortex problem, where an additional central vortex of intensity κ is added to the ring of the polygonal configuration.

Cabral H. E., Meyer K. R., Schmidt D. S.
Stability and bifurcations for the $N + 1$ vortex problem on the sphere
2003, vol. 8, no. 3, pp. 259282
Abstract
The equations of motion for $N$ vortices on a sphere were derived by V.A.Bogomolov in 1977. References to related work can be found in the book by P.K.Newton. We use the equations of motion found there to discuss the stability of a ring of $N$ vortices of unit strength at the latitude $z$ together with a vortex of strength $\kappa$ at the north pole. The regions of stability are bounded by curves $\kappa = \kappa(z)$. These curves are computed explicitly for all values of $N$.
When the stability of a configuration changes, for example by varying the strength of the vortex at the north pole, bifurcations to new configurations are possible. We compute the bifurcation equations explicitly for $N = 2, 3$ and $4$. For larger values of $N$ the complexity of the formal computations becomes too great and we use a numerical value for the latitude instead. We thus derive the bifurcation equations in a seminumerical form. As expected the new configurations look very similar to those which had been found previously for the planar case. 