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2013
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# Claudio Vidal

Casilla 5-C, Concepción, VIII-Región, Chile
 Andrade J., Vidal C., Sierpe C. Stability of the Relative Equilibria in the Two-body Problem on the Sphere 2021, vol. 26, no. 4, pp.  402-438 Abstract 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. Keywords: two-body-problem on the sphere, Hamiltonian formulation, normal form, resonance, nonlinear stability Citation: Andrade J., Vidal C., Sierpe C.,  Stability of the Relative Equilibria in the Two-body Problem on the Sphere, Regular and Chaotic Dynamics, 2021, vol. 26, no. 4, pp. 402-438 DOI:10.1134/S1560354721040067
 Cárcamo-Díaz D., Palacián J. F., Vidal C., Yanguas P. On the Nonlinear Stability of the Triangular Points in the Circular Spatial Restricted Three-body Problem 2020, vol. 25, no. 2, pp.  131-148 Abstract 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. Keywords: restricted three-body problem, $L_4$ and $L_5$, elliptic equilibria, resonances, formal and Lie stability, exponential estimates Citation: Cárcamo-Díaz D., Palacián J. F., Vidal C., Yanguas P.,  On the Nonlinear Stability of the Triangular Points in the Circular Spatial Restricted Three-body Problem, Regular and Chaotic Dynamics, 2020, vol. 25, no. 2, pp. 131-148 DOI:10.1134/S156035472002001X
 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