On the “blue sky catastrophe” termination in the restricted four-body problem

Burgos-Garcia, Jaime; Delgado, Joaquín

doi:10.1007/s10569-013-9498-3

Cited by 26 publications

(11 citation statements)

References 31 publications

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“…They also computed some families of symmetric periodic orbits. Similar results were obtained in [17] but for the case of two equal masses, while the existence of blue sky catastrophe around a specific collinear equilibrium point was presented in [16]. In a recent paper [12] a large number of families of non-symmetric periodic orbits around Jupiter and the Trojan asteroids was found.…”

supporting

confidence: 81%

Escape and collision dynamics in the planar equilateral restricted four-body problem

Zotos

2016

International Journal of Non-Linear Mechanics

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We consider the planar circular equilateral restricted four body-problem where a test particle of infinitesimal mass is moving under the gravitational attraction of three primary bodies which move on circular orbits around their common center of gravity, such that their configuration is always an equilateral triangle. The case where all three primaries have equal masses is numerically investigated. A thorough numerical analysis takes place in the configuration (x, y) as well as in the (x, C) space in which we classify initial conditions of orbits into four main categories: (i) bounded regular orbits, (ii) trapped chaotic orbits, (iii) escaping orbits and (iv) collision orbits. Interpreting the collision motion as leaking in the phase space we related our results to both chaotic scattering and the theory of leaking Hamiltonian systems. We successfully located the escape and the collision basins and we managed to correlate them with the corresponding escape and collision times of orbits. We hope our contribution to be useful for a further understanding of the escape and collision properties of motion in this interesting dynamical system.

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supporting

confidence: 81%

Escape and collision dynamics in the planar equilateral restricted four-body problem

Zotos

2016

International Journal of Non-Linear Mechanics

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“…Actually, there has been a considerable progress in understanding the basic but important aspects of this problem. We refer to the interested reader to the works [6], [15], [7], [5], [16], [17] and references therein for a deeper discussion in the dynamics of this fascinating system.…”

Section: The Hill Approximation In the Four Body Problemmentioning

confidence: 99%

“…The analysis performed in [32] shows that, when z > 0 in the Hamiltonian (15), there exists a polar orbit z(t) for which the particle moves between z = 0 and z = d(h), where d(h) is a solution of the equation z 2 2 − 1 z = h, with h a constant energy. Since this equation has solution for any value of h (equivalently for any C) the polar orbit exists for all values of h and its amplitude depends on the energy level as it is shown in Figure 2.…”

Section: 3mentioning

confidence: 99%

The spatial Hill four-body problem I -- An exploration of basic invariant sets

Burgos-Garcia,

Bengochea,

Franco-Perez

2021

Preprint

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In this work we perform a first study of basic invariant sets of the spatial Hill's four-body problem, where we have used both analytical and numerical approaches. This system depends on a mass parameter µ in such a way that the classical Hill's problem is recovered when µ = 0. Regarding the numerical work, we perform a numerical continuation, for the Jacobi constant C and several values of the mass parameter µ by applying a classical predictor-corrector method, together with a high-order Taylor method considering variable step and order and automatic differentiation techniques, to specific boundary value problems related with the reversing symmetries of the system. The solution of these boundary value problems defines initial conditions of symmetric periodic orbits. Some of the results were obtained departing from periodic orbits within Hill's three-body problem. The numerical explorations reveal that a second distant disturbing body has a relevant effect on the stability of the orbits and bifurcations among these families. We have also found some new families of periodic orbits that do not exist in the classical Hill's three-body problem; these families have some desirable properties from a practical point of view.

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“…Another example is the bifurcation at the Lagrangian points in the three-body problem at 27(m 1 m 2 + m 1 m 2 + m 1 m 2 )/(m 1 + m 2 + m 3 ) 2 = 1, where m i are the masses of the bodies; see [10] and references therein. A more recent example was found at some collinear equilibrium points in the restricted four-body problem; see [2] and references within.…”

Section: Introductionmentioning

confidence: 97%

A Note on the Relative Equilibria Bifurcations in the $$(2N+1)$$ ( 2 N + 1 ) -Body Problem

2014

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Consider the planar Newtonian (2N + 1)-body problem, N ≥ 1, with 2N bodies of unit mass and one body of mass m. Using the discrete symmetry due to the equal masses and reducing by the rotational symmetry, we show that solutions with the 2N unit mass points at the vertices of two concentric regular N -gons and m at the centre at all times form invariant manifold. We study the regular 2N -gon with central mass m relative equilibria within the dynamics on the invariant manifold described above. As m varies, we identify the bifurcations, relate our results to previous work and provide the spectral picture of the linearization at the relative equilibria.

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On the “blue sky catastrophe” termination in the restricted four-body problem

Cited by 26 publications

References 31 publications

Escape and collision dynamics in the planar equilateral restricted four-body problem

Escape and collision dynamics in the planar equilateral restricted four-body problem

The spatial Hill four-body problem I -- An exploration of basic invariant sets

A Note on the Relative Equilibria Bifurcations in the $$(2N+1)$$ ( 2 N + 1 ) -Body Problem

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