Skip to main content
SpringerLink
Log in

Self-resonant bifurcations of the Sitnikov family and the appearance of 3D isolas in the restricted three-body problem

  • Original Article
  • Published:
Celestial Mechanics and Dynamical Astronomy Aims and scope Submit manuscript

Abstract

The association of the Sitnikov family with families of multiple three-dimensional periodic orbits is studied. In particular, the families consisting of three-dimensional periodic orbits bifurcating from self-resonant orbits of the Sitnikov family at double, triple and quadruple period of the bifurcation orbit are considered. The branch families close upon themselves and remain 3D up to their terminations having two common members with the Sitnikov family. By varying the mass parameter we also study the evolution of some of the computed families and find that they become isolas and disappear gradually in three-dimensions by shrinking to point size.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Alfaro J.M., Chiralt C.: Invariant rotational curves in Sitnikov’s problem. Celest. Mech. Dyn. Astron. 55, 351–367 (1993)

    Article  ADS  MATH  Google Scholar 

  • Belbruno E., Llibre J., Ollé M.: On the families of periodic orbits which bifurcate from the circular Sitnikov motions. Celest. Mech. Dyn. Astron. 60, 99–129 (1994)

    Article  ADS  MATH  Google Scholar 

  • Dellwo D., Keller H.B., Matkowsky B.J., Reiss E.L.: On the birth of isolas. SIAM J. Appl. Math. 42, 956–963 (1981)

    Article  MathSciNet  Google Scholar 

  • Douskos C., Kalantonis V., Markellos P., Perdios E.: On Sitnikov-like motions generating new kinds of 3D periodic orbits in the R3BP with prolate primaries. Astrophys. Space Sci. 337, 99–106 (2012)

    Article  ADS  MATH  Google Scholar 

  • Dvorak R.: Numerical results to the Sitnikov-problem. Celest. Mech. Dyn. Astron. 56, 71–80 (1993)

    Article  ADS  Google Scholar 

  • Faruque S.B.: Solution of the Sitnikov problem. Celest. Mech. Dyn. Astron. 87, 353–369 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Hagel J.: A new analytic approach to the Sitnikov problem. Celest. Mech. Dyn. Astron. 53, 267–292 (1992)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Hagel J., Lhotka C.: A high order perturbation analysis of the Sitnikov problem. Celest. Mech. Dyn. Astron. 93, 201–228 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Kalantonis V.S., Perdios E.A., Perdiou A.E.: The Sitnikov family and the associated families of 3D periodic orbits in the photogravitational RTBP with oblateness. Astrophys. Space Sci. 315, 323–334 (2008)

    Article  ADS  Google Scholar 

  • Kovács T., B.: Transient chaos in the Sitnikov problem. Celest. Mech. Dyn. Astron. 105, 289–304 (2009)

    Article  ADS  MATH  Google Scholar 

  • Lara L.J., Buendía A.E.: Symmetries and bifurcations in the Sitnikov problem. Celest. Mech. Dyn. Astron. 79, 97–117 (2001)

    Article  ADS  MATH  Google Scholar 

  • Llibre J., Ortega R.: On the families of periodic orbits of the Sitnikov problem. SIAM J. Appl. Dyn. Syst. 7, 561–576 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Mac Millan W.D.: An integrable case in the restricted problem of three bodies. Astron. J. 27, 11–13 (1913)

    Article  Google Scholar 

  • Pavanini, G.: Sopra una nuova categoria di solutioni periodiche nel problema dei tre corpi. Ann. Math. Serie III, Tomo XIII, 179–202 (1907)

  • Perdios E.A.: The manifold of families of 3D periodic orbits associated to Sitnikov motions in the restricted three-body problem. Celest. Mech. Dyn. Astr. 99, 85–104 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Perdios E.A., Kalantonis V.S., Douskos C.N.: Straight-line oscillations generating three-dimensional motions in the photogravitational restricted three-body problem. Astrophys. Space Sci. 314, 199–208 (2008)

    Article  ADS  MATH  Google Scholar 

  • Perdios E.A., Markellos V.V.: Stability and bifurcations of Sitnikov motion. Celest. Mech. 42, 187–200 (1988)

    MathSciNet  ADS  Google Scholar 

  • Robin I.A., Markellos V.V.: Numerical determination of three-dimensional periodic orbits generated from vertical self-resonant satellite orbits. Celest. Mech. 21, 395–434 (1980)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Robin I.A., Markellos V.V.: The mechanism of branching of three-dimensional periodic orbits from the plane. In: Markellos, V.V., Kozai, Y. (eds) Dynamical Trapping and Evolution in the Solar System, pp. 213–224. D.Reidel, Dordrecht (1983)

    Chapter  Google Scholar 

  • Ruzza S., Lhotka C.: High order normal form construction near the elliptic orbit of the Sitnikov problem. Celest. Mech. Dyn. Astron. 111, 449–464 (2011)

    Article  ADS  Google Scholar 

  • Sidorenko V.V.: On the circular Sitnikov problem: the alternation of stability and instability in the family of vertical motions. Celest. Mech. Dyn. Astron. 109, 367–384 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  • Sitnikov K.: The existence of oscillatory motions in the three-body problem. Dokl. Akad. Nauk. 133, 303–306 (1960)

    MathSciNet  Google Scholar 

  • Szebehely V.: Theory of Orbits. Academic Press, Orlando (1967)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Engineering Sciences, University of Patras, 26500, Patras, Greece

    E. A. Perdios & V. S. Kalantonis

Authors
  1. E. A. Perdios
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. S. Kalantonis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. S. Kalantonis.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Perdios, E.A., Kalantonis, V.S. Self-resonant bifurcations of the Sitnikov family and the appearance of 3D isolas in the restricted three-body problem. Celest Mech Dyn Astr 113, 377–386 (2012). https://doi.org/10.1007/s10569-012-9424-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10569-012-9424-0

Keywords

Search

Navigation