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Regular and Chaotic Solutions of the Sitnikov Problem near the 3/2 Commensurability

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Abstract

Regular solutions at the 3/2 commensurability are investigated forSitnikov’s problem. Utilizing a rotating coordinate system and theaveraging method, approximate analytical equations are obtained for thePoincare sections by means of Jacobian elliptic functions and 3πperiodicsolutions are generated explicitly. It is revealed that the system exhibitsheteroclinic orbits to saddle points. It is also shown that chaotic regionemerging from the destroyed invariant tori, can easily be seen for certaineccentricities. The procedure of the current study provides reliable answersfor the long-time behavior of the system near resonances.

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References

  • Alekseev, V. M.: 1968a, 1968b, 1969, 'Quasirandom dynamical systems I, II, III', Math. USSR Sbornik 5, 73-128; 6, 505–560; 7, 1-43.

  • Alfaro, J. M. and Chiralt, C.:1993, 'Invariant rotational curves in Sitnikov's problem', Celes. Mech. 55, 351–367.

    Google Scholar 

  • Belbruno, E., Llibre, J. and Ollé, M.: 1994,'On the families of periodic orbits which bifurcate from the circular Sitnikov motions', Celes. Mech. 60, 99–129.

    Google Scholar 

  • Berdichevsky, V., Özbek, O. and Kim, W. W.: 1994,'Thermodynamics of Duffing's oscillator', ASME J. Appl. Mech. 61, 670–675.

    Google Scholar 

  • Conley, C. C.:1969,'On the ultimate behavior of orbits with respect to an unstable critical point I, Oscillating, Asymptotic and Capture Orbits', J. Diff. Eq. 5, 136–158.

    Google Scholar 

  • Davis, H. T.: 1962, Introduction to Nonlinear Differential and Integral Equations, Dover Publications.

  • Guckenheimer, J. and Holmes, P.: 1983, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, Berlin.

    Google Scholar 

  • Hagel, J.: 1992, 'A new analytic approach to the Sitnikov problem', Celes. Mech. 53, 267–292.

    Google Scholar 

  • Heck, A.: 1993, Introduction to Maple, Springer-Verlag, Berlin.

    Google Scholar 

  • Jie Liu and Yi-Sui Sun: 1990, 'On the Sitnikov problem', Celes. Mech. 49, 285–302.

    Google Scholar 

  • Moser, J.: 1973, Stable and Random Motions in Dynamical Systems,Princeton University Press, Princeton.

    Google Scholar 

  • Sitnikov, K. A.: 1960, 'Existence of oscillatory motion for the three-body problem', Dokl. Akad. Nauk. USSR 133, No. 2, 303–306.

    Google Scholar 

  • Stumpff, K.: 1965, Himmelsmechanik, Band II, VEB, Berlin.

    Google Scholar 

  • Wiggins, S.: 1990, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, Berlin.

    Google Scholar 

Download references

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Authors and Affiliations

  1. Mechanical Engineering Department, Applied Mechanics Division, Iran

    M. A. Jalali

  2. Mechanical Engineering Department, Aerospace Division, Sharif University of Technology, P.O. Box 11365, 9567,Azadi Ave., Tehran, Iran

    S. H. Pourtakdoust

Authors
  1. M. A. Jalali
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  2. S. H. Pourtakdoust
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Jalali, M.A., Pourtakdoust, S.H. Regular and Chaotic Solutions of the Sitnikov Problem near the 3/2 Commensurability. Celestial Mechanics and Dynamical Astronomy 68, 151–162 (1997). https://doi.org/10.1023/A:1008216128436

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  • DOI: https://doi.org/10.1023/A:1008216128436

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