Claudio Vidal

We consider the planar charged restricted elliptic three-body problem (CHRETBP). In this work we consider the parametric stability of the isosceles triangle equilibrium solution denoted by $L_4^{iso}$. We construct the boundary surfaces of the stability/instability regions in the space of the parameters $\mu$, $\beta$ and $e$, which are parameters of the mass, charges associated to the primaries and the eccentricity of the elliptic orbit, respectively.

We consider the 2-body 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 center-saddle bifurcations.

The well-known problem of the nonlinear stability of $L_4$ and $L_5$ in the circular spatial restricted three-body 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.

In this paper we consider a symmetric restricted circular three-body 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 $a=\dfrac{\sqrt{3}}{2}$.

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].