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

Tilemachos J. Kalvouridis, Maria Ch. Gousidou-Koutita

Department of Mathematics, Faculty of Sciences, Aristotle University of Thessaloniki, Thessaloniki, Greece.
Department of Mechanics, Faculty of Applied Sciences, National Technical University of Athens, Athens, Greece.

DOI: 10.4236/am.2012.36082                

Abstract

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.

Keywords

Copenhagen Problem with Magnetic Dipoles; Equilibrium Locations; Basins of Attraction; Solution of Non-Linear Algebraic Systems; Newton and Quasi-Newton Methods; Comparison of Numerical Methods

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	T. J. Kalvouridis and A. G. Mavraganis, “Symmetric Motions in the Equatorial Magnetic-Binary Problem,” Celestial Mechanics, Vol. 40, No. 2, 1987, pp. 177-196. doi:10.1007/BF01230259
[2]	T. J. Kalvouridis, “Three-Dimensional Equilibria and Their Stability in the Magnetic-Binary Problem,” Astrophysics and Space Science, Vol. 159, No. 1, 1989, pp. 91-97. doi:10.1007/BF00640490
[3]	M. Croustalloudi and T. J. Kalvouridis, “Structure and Parametric Evolution of the Basins of Attraction in the Restricted Three-Body Problem,” Proceedings of the 7th National Congress on Mechanics, Chania Crete Vol. 2, 24-26 June 2004, pp. 144-150.
[4]	M. Croustalloudi and T. J. Kalvouridis, “Attracting Domains in Ring-Type N-Body Formations,” Planetary and Space Science, Vol. 55, No. 1, 2006, pp. 53-69. doi:10.1016/j.pss.2006.04.008
[5]	M. Goussidou-Koutita and T. J. Kalvouridis, “A Comparative Study of the Attracting Regions in the Ring Problem of (N+1) Bodies,” International Conference of Computational Methods in Science and Engineering, Chania Crete, 8 April 2006.
[6]	M. Gousidou-Koutita and T. J. Kalvouridis, “Application of Newton and Broyden Methods for the Investigation of the Attracting Regions in the Ring Problem of (N+1) Bodies: A Comparative Study,” Abstracts of the Conference Gene around the World, Tripolis, 29 February-1 March 2008, p. 2.
[7]	M. Gousidou-Koutita and T. J. Kalvouridis, “Numerical Study of the Attracting Domains in a Non-Linear Problem of Celestial Mechanics,” Recent Approaches to Numerical Analysis: Theory, Methods and Applications, Conference in Numerical Analysis, Kalamata, 1-5 September 1998, pp. 84-87.
[8]	M. Gousidou-Koutita and T. J. Kalvouridis, “On the Efficiency of Newton and Broyden Numerical Methods in the Investigation of the Regular Polygon Problem of (N+1) Bodies,” Applied Mathematics and Computing, Vol. 212, No. 1, 2009, pp. 100-112. doi:10.1016/j.amc.2009.02.015
[9]	Ch. Douskos, “Collinear Equilibrium Points of Hill’s Problem with Radiation and Oblateness and Their Fractal Basins of Attraction,” Astrophysics and Space Science, Vol. 326, No. 2, 2010, pp. 263-271. doi:10.1007/s10509-009-0213-5
[10]	M. N. Vrahatis and K. I. Iordanidis, “A Rapid Generalized Method of Bisection for Solving Systems of Non-Linear Equations,” Numerische Mathematik, Vol. 49, No. 2-3, 1986, pp. 123-138. doi:10.1007/BF01389620
[11]	V. Drakopoulos and A. Bohm, “Basins of Attraction and Julia Sets of Shr?der Iteration Functions,” In: A. Bountis and S. Pnevmatikos, Eds., Proceedings of the 7th and 8th Summer Schools on Non-Linear Dynamical Systems, Vol. 4, 1998, pp. 157-163.
[12]	D. J. Faires and R. L. Burden, “Numerical Methods,” PWS-KENT Publ. Co., Boston, 1993.a