On the efficiency of Newton and Broyden numerical methods in the investigation of the regular polygon problem of (N+1) bodies

Gousidou-Koutita, M.; Kalvouridis, T. J.

doi:10.1016/j.amc.2009.02.015

Cited by 14 publications

(13 citation statements)

References 23 publications

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“…For obtaining the basins of attraction one should use an iterative scheme (e.g., the Newton-Raphson method) and scan a set of initial conditions in order to reveal their final states (attractors). Over the past years a large number of studies have been devoted on determining the Newton-Raphson basins of convergence in many types of dynamical systems, such as the Hill's problem [Douskos, 2010], the Sitnikov problem [Douskos et al, 2012], the restricted three-body problem with oblateness and radiation pressure [Zotos, 2016], the electromagnetic Copenhagen problem [Kalvouridis & Gousidou-Koutita, 2012;Zotos, 2017b], the photogravitational Copenhagen problem [Kalvouridis, 2008], the four-body problem [Baltagiannis & Papadakis, 2011;Kumari & Kushvah, 2014;Zotos, 2017a], the photogravitational four-body problem [Asique et al, 2016], the ring problem of N + 1 bodies [Croustalloudi & Kalvouridis, 2007;Gousidou-Koutita & Kalvouridis, 2009], or even the restricted 2+2 body problem [Croustalloudi & Kalvouridis, 2013].…”

Section: Introductionmentioning

confidence: 99%

Basins of Convergence of Equilibrium Points in the Generalized Hill Problem

Zotos

2017

Int. J. Bifurcation Chaos

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The Newton-Raphson basins of attraction, associated with the libration points (attractors), are revealed in the generalized Hill problem. The parametric variation of the position and the linear stability of the equilibrium points is determined, when the value of the perturbation parameter varies. The multivariate Newton-Raphson iterative scheme is used to determine the attracting domains on several types of two-dimensional planes. A systematic and thorough numerical investigation is performed in order to demonstrate the influence of the perturbation parameter on the geometry as well as of the basin entropy of the basins of convergence. The correlations between the basins of attraction and the corresponding required number of iterations are also illustrated and discussed. Our numerical analysis strongly indicates that the evolution of the attracting regions in this dynamical system is an extremely complicated yet very interesting issue.

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Section: Introductionmentioning

confidence: 99%

Basins of Convergence of Equilibrium Points in the Generalized Hill Problem

Zotos

2017

Int. J. Bifurcation Chaos

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“…In [7] the Newton-Raphson iterative method was used in order to explore the basins of attraction in the Hill's problem with oblateness and radiation pressure. In the same vein, the multivariate version of the same iterative scheme has been used to unveil the basins of convergence in the restricted three-body problem (e.g., [20]), the electromagnetic Copenhagen problem (e.g., [14]), the photogravitational Copenhagen problem (e.g., [13]), the four-body problem (e.g., [4,15]), the ring problem of N + 1 bodies (e.g., [5,11]), or even even the restricted 2+2 body problem (e.g., [6]) have been studied.…”

Section: Introductionmentioning

confidence: 99%

Investigating the Newton–Raphson basins of attraction in the restricted three-body problem with modified Newtonian gravity

Zotos¹

2016

J. Appl. Math. Comput.

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The planar circular restricted three-body problem with modified Newtonian gravity is used in order to determine the Newton-Raphson basins of attraction associated with the equilibrium points. The evolution of the position of the five Lagrange points is monitored when the value of the power p of the gravitational potential of the second primary 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 basins of convergence of the equilibrium points and the corresponding number of iterations needed for obtaining the desired accuracy are also illustrated. We conduct a thorough and systematic numerical investigation by demonstrating how the dynamical quantity p influences the shape as well as the geometry of the basins of attractions. Our results strongly suggest that the power p is indeed a very influential parameter in both cases of weaker or stronger Newtonian gravity.Keywords Three-body problem · Non-linear algebraic equations · Basins of attraction · Fractal basin boundaries

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“…We call the set of the initial points that lead to the discrete equilibrium points or the dynamically equivalent points, "basins of attraction" (or "basins of convergence", or "attracting domains"). These regions have been studied in the past in some problems of Celestial Dynamics, such as the restricted three-body problem [3], the ring problem of (N + 1)-bodies [4][5][6][7][8], and the Hill's problem with radiation and oblateness [9]. The technique to find these regions is based on a double scanning process of the Oxy plane.…”

Section: Basins Of Attractionmentioning

confidence: 99%

Basins of Attraction in the Copenhagen Problem Where the Primaries Are Magnetic Dipoles

Kalvouridis¹,

Gousidou-Koutita²

2012

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We deal with the Copenhagen problem where the two big bodies of equal masses are also magnetic dipoles and we study some aspects of the dynamics of a charged particle which moves in the electromagnetic field produced by the primaries. We investigate the equilibrium positions of the particle and their parametric variations, as well as the basins of attraction for various numerical methods and various values of the parameter λ.

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On the efficiency of Newton and Broyden numerical methods in the investigation of the regular polygon problem of (N+1) bodies

Cited by 14 publications

References 23 publications

Basins of Convergence of Equilibrium Points in the Generalized Hill Problem

Basins of Convergence of Equilibrium Points in the Generalized Hill Problem

Investigating the Newton–Raphson basins of attraction in the restricted three-body problem with modified Newtonian gravity

Basins of Attraction in the Copenhagen Problem Where the Primaries Are Magnetic Dipoles

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