Equilibrium points and basins of convergence in the linear restricted four-body problem with angular velocity

The Newton–Raphson basins of convergence, related to the equilibrium points, in the Sitnikov four-body problem with nonspherical primaries are numerically investigated. We monitor the parametric evolution of the positions of the roots, as a function of the oblateness coefficient. The classical Newton–Raphson optimal method is used for revealing the basins of convergence, by classifying dense grids of initial conditions in several types of two-dimensional planes. We perform a systematic and thorough analysis in an attempt to understand how the oblateness coefficient affects the geometry as well as the basin entropy of the convergence regions. The convergence areas are related with the required number of iterations and also with the corresponding probability distributions.

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Exploring the Location and Linear Stability of the Equilibrium Points in the Equilateral Restricted Four-Body Problem

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127420501552 ◽

2020 ◽

Vol 30 (10) ◽

pp. 2050155

Author(s):

Euaggelos E. Zotos

Keyword(s):

Linear Stability ◽

Body Problem ◽

Equilibrium Points ◽

Libration Points ◽

The Other ◽

Classification Method ◽

Four Body Problem ◽

The Masses ◽

Stable Points ◽

Planar Version

The planar version of the equilateral restricted four-body problem, with three unequal masses, is numerically investigated. By adopting the grid classification method we locate the coordinates, on the plane [Formula: see text], of the points of equilibrium, for all possible values of the masses of the primaries. The linear stability of the libration points is also determined, as a function of the masses. Our analysis indicates that linearly stable points of equilibrium exist only when one of the primaries has a considerably larger mass, with respect to the other two primary bodies, when the triangular configuration of the primaries is also dynamically stable.

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Nonlinear Stability of Equilibrium Points in the Planar Equilateral Restricted Mass-Unequal Four-Body Problem

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127421300317 ◽

2021 ◽

Vol 31 (11) ◽

pp. 2130031

Author(s):

José Alejandro Zepeda Ramírez ◽

Martha Alvarez-Ramírez ◽

Antonio García

Keyword(s):

Nonlinear Stability ◽

Mass Formula ◽

Body Problem ◽

Equilibrium Points ◽

Constant Angular Velocity ◽

Gravitational Attraction ◽

Stability Of Equilibrium ◽

Four Body Problem ◽

Elliptic Equilibrium ◽

The Stability

In this paper, we investigate the stability of equilibrium points for the planar restricted equilateral four-body problem in the case that one particle of negligible mass is moving under the Newtonian gravitational attraction of three positive masses [Formula: see text], [Formula: see text] and [Formula: see text] (called primaries). These always lie at the vertices of an equilateral triangle (Lagrangian configuration) and move with constant angular velocity in circular orbits around their center of masses. We consider the case where all the primaries have unequal masses, and investigate the nonlinear stability (in the sense of Lyapunov) of the elliptic equilibrium for the specific values of the mass [Formula: see text] and [Formula: see text] of the primary, fixed on the horizontal axis. Moreover, the [Formula: see text][Formula: see text]:[Formula: see text][Formula: see text] four-order resonant cases are determined and the stability is investigated. In this study, Markeev’s theorem and Arnold’s theorem become key ingredients.

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EQUILIBRIUM POINTS AND THEIR STABILITY IN THE RESTRICTED FOUR-BODY PROBLEM

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127411029707 ◽

2011 ◽

Vol 21 (08) ◽

pp. 2179-2193 ◽

Cited By ~ 73

Author(s):

A. N. BALTAGIANNIS ◽

K. E. PAPADAKIS

Keyword(s):

Body Problem ◽

Equilibrium Points ◽

Relative Equilibrium ◽

Center Of Mass ◽

Basins Of Attraction ◽

Equilibrium Solutions ◽

Velocity Surface ◽

Four Body Problem ◽

The Stability ◽

Zero Velocity Surface

We study numerically the problem of four bodies, three of which are finite, moving in circles around their center of mass fixed at the origin of the coordinate system, according to the solution of Lagrange where they are always at the vertices of an equilateral triangle, while the fourth is infinitesimal. The fourth body does not affect the motion of the three bodies (primaries). The allowed regions of motion as determined by the zero-velocity surface and corresponding equipotential curves as well as the positions of the equilibrium points are given. The existence and the number of collinear and noncollinear equilibrium points of the problem depend on the mass parameters of the primaries. For three unequal masses, collinear equilibrium solutions do not exist. Critical masses associated with the existence and the number of equilibrium points, are given. The stability of the relative equilibrium solutions in all cases is also studied. The regions of the basins of attraction for the equilibrium points of the present dynamical model for some values of the mass parameters are illustrated.

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