Chaotic Dynamics of the Kepler Problem with Oscillating Singularity

Alessandro Margheri 1  and Pedro J. Torres 2
  • 1 Faculdade de Ciências da Universidade de Lisboa e CMAF-CIO, Campo Grande, Edifício C6, piso 2, P-1749-016 Lisboa, Portugal
  • 2 Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
Alessandro Margheri and Pedro J. Torres

Abstract

We prove the presence of chaotic dynamics for the classical two-body Kepler problem with a time-periodic gravitational coefficient oscillating between two fixed values. The set of chaotic solutions we detect is coded by the number of revolutions in each period. The chaotic dynamics is obtained for large period T as well as for small angular momentum μ. In particular, we provide an explicit lower bound on T and explicit upper bound on μ which guarantee the existence of complex dynamics. We get our results by applying a simple and well-known topological method, the stretching along the path technique. Our results are robust with respect to small perturbations of the gravitational coefficient and to the addition of a small friction term.

  • [1]

    Arnold V. I., Kozlov V. V. and Neishtadt A. I., Mathematical Aspects of Classical and Celestial Mechanics. Dynamical Systems III, Encyclopaedia Math. Sci. 3, Springer, Berlin, 1993.

  • [2]

    Aulbach B. and Kieninger B., On three definitions of chaos, Nonlinear Dyn. Syst. Theory 1 (2001), 23–37.

  • [3]

    Bekov A. A., Periodic solutions of the Gylden–Merscherskii problem, Astron. Rep. 37 (1993), 651–654.

  • [4]

    Burns K. and Weiss H., A geometric criterion for positive topological entropy, Comm. Math. Phys. 172 (1995), 95–118.

  • [5]

    Burton R. and Easton R. W., Ergodicity of linked twist maps, Global Theory of Dynamical Systems (Evanston 1979), Lecture Notes in Math. 819, Springer, Berlin (1980), 35–49.

  • [6]

    Chu J., Torres P. J. and Wang F., Radial stability of periodic solutions of the Gylden-Meshcherskii-type problem, Discrete Contin. Dyn. Syst. A 35 (2015), no. 5, 1921–1932.

  • [7]

    Deprit A., The secular accelerations in Gylden’s problem, Celestial Mech. 31 (1983), 1–22.

  • [8]

    Devaney R. L., Subshifts of finite type in linked twist mappings, Proc. Amer. Math. Soc. 71 (1978), 334–338.

  • [9]

    Diacu F. and Selaru D., Chaos in the Gýlden problem, J. Math. Phys. 39 (1998), 6537–6546.

  • [10]

    Hadjidemetriou J., Two-body problem with variable mass: A new approach, Icarus 2 (1963), 440–451.

  • [11]

    Hadjidemetriou J., Analytic solutions of the two-body problem with variable mass, Icarus 5 (1966), 34–46.

  • [12]

    Kennedy J. and Yorke J. A., Topological horseshoes, Trans. Amer. Math. Soc. 353 (2001), 2513–2530.

  • [13]

    Margheri A., Rebelo C. and Zanolin F., Chaos in periodically perturbed planar Hamiltonian systems using linked twist maps, J. Differential Equations 249 (2010), 3233–3257.

  • [14]

    Medio A., Pireddu M. and Zanolin F., Chaotic dynamics for maps in one and two dimensions: A geometrical method and applications to economics, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 19 (2009), 3283–3309.

  • [15]

    Mischaikow K. and Mrozek M., Isolating neighborhoods and chaos, Japan J. Indust. Appl. Math. 12 (1995), 205–236.

  • [16]

    Moser J., Stable and Random Motions in Dynamical Systems, Ann. of Math. Stud. 77, Princeton University Press, Princeton, 1973.

  • [17]

    Omarov T. B., The restricted problem of perturbed motion of two bodies with variable mass, Soviet Astronomy 8 (1964), 127–131.

  • [18]

    Omarov T. B., Two-body motion with corpuscular radiation, Soviet Astronomy 7 (1964), 707–719.

  • [19]

    Pal A., Selaru D., Mioc V. and Cucu-Dumitrescu C., The Gyldén-type problem revisited: More refined analytical solutions, Astron. Nachr. 327 (2006), 304–308.

  • [20]

    Papini D. and Zanolin F., On the periodic boundary value problem and chaotic-like dynamics for nonlinear Hill’s equations, Adv. Nonlinear Stud. 4 (2004), 71–91.

  • [21]

    Pascoletti A., Pireddu M. and Zanolin F., Multiple periodic solutions and complex dynamics for second order ODEs via linked twist maps, Electron. J. Qual. Theory Differ. Equ. 14 (2008), 1–32.

  • [22]

    Pascoletti A. and Zanolin F., Example of a suspension bridge ODE model exhibiting chaotic dynamics: A topological approach, J. Math. Anal. Appl. 339 (2008), 1179–1198.

  • [23]

    Przytycki F., Ergodicity of toral linked twist mappings, Ann. Sci. École Norm. Sup. 16 (1983), 345–354.

  • [24]

    Przytycki F., Periodic points of linked twist mappings, Studia Math. Sup. 83 (1986), 1–18.

  • [25]

    Saslaw W. C., Motion around a source whose luminosity changes, Astrophys. J. 226 (1978), 240–252.

  • [26]

    Selaru D., Cucu-Dumitrescu C. and Mioc V., On a two-body problem with periodically changing equivalent gravitational parameter, Astron. Nachr. 313 (1993), 257–263.

  • [27]

    Selaru D. and Mioc V., Le probleme de Gyldén du point de vue de la théorie KAM, C. R. Acad. Sci. Paris Sér. II b 325 (1997), 487–490.

  • [28]

    Selaru D., Mioc V. and Cucu-Dumitrescu C., The periodic Gyldén-type problem in astrophysics, AIP Conf. Proc. 895 (2007), 163–170.

  • [29]

    Srzednicki R., A generalization of the Lefschetz fixed point theorem and detection of chaos, Proc. Amer. Math. Soc. 128 (2000), 1231–1239.

  • [30]

    Sturman R., Ottino J. M. and Wiggins S., The Mathematical Foundations of Mixing. The Linked Twist Map as a Paradigm in Applications. Micro to Macro, Fluids to Solids, Cambridge Monogr. Appl. Comput. Math. 22, Cambridge University Press, Cambridge, 2006.

  • [31]

    Torres P. J., Mathematical Models with Singularities. A Zoo of Singular Creatures, Atlantis Briefs Differ. Equ. 1, Atlantis Press, Amsterdam, 2015.

Purchase article
Get instant unlimited access to the article.
$42.00
Log in
Already have access? Please log in.


or
Log in with your institution

Journal + Issues

Advanced Nonlinear Studies (ANS)  is aimed at publishing scholarly articles on nonlinear problems, particularly those involving Differential and Integral  Equations, Dynamical Systems, Calculus of Variations, and related areas. It will also publish novel and interesting applications of these areas to problems in biology,  engineering,  materials sciences,  physics and other  sciences.

Search