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2013
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# Claudio Vidal

Casilla 5-C, Concepción, VIII-Región, Chile
 Andrade J., Vidal C. Stability of the Polar Equilibria in a Restricted Three-body Problem on the Sphere 2018, vol. 23, no. 1, pp.  80-101 Abstract 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 $a=\dfrac{\sqrt{3}}{2}$. Keywords: circular restricted three-body problem on surfaces of constant curvature, Hamiltonian formulation, normal form, resonance, nonlinear stability Citation: Andrade J., Vidal C.,  Stability of the Polar Equilibria in a Restricted Three-body Problem on the Sphere, Regular and Chaotic Dynamics, 2018, vol. 23, no. 1, pp. 80-101 DOI:10.1134/S1560354718010070
 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.  880-892 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]. Keywords: Hamiltonian system, equilibrium solution, type of stability, normal form, critical cases, Lyapunov’s Theorem, Chetaev’s Theorem Citation: Gutierres R., Vidal C.,  Stability of Equilibrium Points for a Hamiltonian Systems with One Degree of Freedom in One Degenerate Case, Regular and Chaotic Dynamics, 2017, vol. 22, no. 7, pp. 880-892 DOI:10.1134/S1560354717070097