Claudio Vidal
Publications:
Andrade J., Vidal C., Sierpe C.
Stability of the Relative Equilibria in the Twobody Problem on the Sphere
2021, vol. 26, no. 4, pp. 402438
Abstract
We consider the 2body problem in the sphere $\mathbb{S}^2$. This problem is modeled by a Hamiltonian system with $4$ degrees of freedom and, following the approach given in [4], allows us to reduce the study to a system of $2$ degrees of freedom. In this work we will use theoretical tools such as normal forms and some nonlinear stability results on Hamiltonian systems for demonstrating a series of results that will correspond to the open problems proposed in [4] related to the nonlinear stability of the relative equilibria. Moreover, we study the existence of Hamiltonian pitchfork and centersaddle bifurcations.

CárcamoDíaz D., Palacián J. F., Vidal C., Yanguas P.
On the Nonlinear Stability of the Triangular Points in the Circular Spatial Restricted Threebody Problem
2020, vol. 25, no. 2, pp. 131148
Abstract
The wellknown problem of the nonlinear stability of $L_4$ and $L_5$ in the circular
spatial restricted threebody problem is revisited. Some new results in the light of the concept of
Lie (formal) stability are presented. In particular, we provide stability and asymptotic estimates
for three specific values of the mass ratio that remained uncovered. Moreover, in many cases
we improve the estimates found in the literature.

Andrade J., Vidal C.
Stability of the Polar Equilibria in a Restricted Threebody Problem on the Sphere
2018, vol. 23, no. 1, pp. 80101
Abstract
In this paper we consider a symmetric restricted circular threebody problem on the surface $\mathbb{S}^2$ of constant
Gaussian curvature $\kappa=1$. This problem consists in the description of the dynamics of an infinitesimal mass particle attracted
by two primaries with identical masses, rotating with constant angular velocity in a fixed parallel of radius $a\in (0,1)$.
It is verified that both poles of $\mathbb{S}^2$ are equilibrium points for any value of the parameter $a$. This problem is
modeled through a Hamiltonian system of two degrees of freedom depending on the parameter $a$. Using results concerning nonlinear
stability, the type of Lyapunov stability (nonlinear) is provided for the polar equilibria, according to the resonances.
It is verified that for the north pole there are two values of bifurcation (on the stability) $a=\dfrac{\sqrt{4\sqrt{2}}}{2}$ and $a=\sqrt{\dfrac{2}{3}}$,
while the south pole has one value of bifurcation $a=\dfrac{\sqrt{3}}{2}$.

Gutierres R., Vidal C.
Stability of Equilibrium Points for a Hamiltonian Systems with One Degree of Freedom in One Degenerate Case
2017, vol. 22, no. 7, pp. 880892
Abstract
This paper concerns with the study of the stability of one equilibrium solution of an autonomous analytic Hamiltonian system in a
neighborhood of the equilibrium point with $1$degree of freedom in the degenerate case $H= q^4+ H_5+ H_6+\cdots$. Our main results complete the study initiated by Markeev in [9].
