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Revealing the basins of convergence in the planar equilateral restricted four-body problem

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Abstract

The planar equilateral restricted four-body problem where two of the primaries have equal masses is used in order to determine the Newton-Raphson basins of convergence associated with the equilibrium points. The parametric variation of the position of the libration points is monitored when the value of the mass parameter \(m_{3}\) varies in predefined intervals. The regions on the configuration \((x,y)\) plane occupied by the basins of attraction are revealed using the multivariate version of the Newton-Raphson iterative scheme. The correlations between the attracting domains of the equilibrium points and the corresponding number of iterations needed for obtaining the desired accuracy are also illustrated. We perform a thorough and systematic numerical investigation by demonstrating how the dynamical parameter \(m_{3}\) influences the shape, the geometry and the degree of fractality of the converging regions. Our numerical outcomes strongly indicate that the mass parameter is indeed one of the most influential factors in this dynamical system.

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Notes

  1. It should be clarified and clearly emphasized that the Newton-Raphson basins of convergence should not be mistaken, by no means, with the classical basins of attraction which exist in dissipative systems. The difference between the Newton-Raphson basins of convergence and the basins of attraction in dissipative systems is huge. This is true because the attraction in the first case is just a numerical artifact of the Newton-Raphson iterative method, while in dissipative systems the attraction is a real dynamical phenomenon, observed through the numerical integration of the initial conditions.

  2. Needless to say the initial conditions corresponding to the three centers \((P_{1}, P_{2}, P_{3})\) of the primaries are excluded from the grid because for these values the distances \(r_{i}\), \(i = 1,2,3\) to the primaries are zero and therefore several terms of the formulae (11) become singular.

  3. When we state that a domain displays fractal structure we simply mean that it has a fractal-like geometry however, without conducting, at least for now, any specific calculations for computing the fractal dimensions as in Aguirre et al. (2009).

  4. By the term “tails” of the distributions we refer to the right-hand side of the histograms, that is, for \(N > N^{*}\).

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Acknowledgements

I would like to express my warmest thanks to the anonymous referee for the careful reading of the manuscript and for all the apt suggestions and comments which allowed us to improve both the quality and the clarity of the paper.

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  1. Department of Physics, School of Science, Aristotle University of Thessaloniki, 541 24, Thessaloniki, Greece

    Euaggelos E. Zotos

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Zotos, E.E. Revealing the basins of convergence in the planar equilateral restricted four-body problem. Astrophys Space Sci 362, 2 (2017). https://doi.org/10.1007/s10509-016-2973-z

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