Skip to main content
Springer Nature Link
Log in

On the stability of the three classes of Newtonian three-body planar periodic orbits

  • Article
  • Published:
Save article
View saved research
Science China Physics, Mechanics & Astronomy Aims and scope Submit manuscript

Abstract

Currently, the fifteen new periodic orbits of Newtonian three-body problem with equal mass were found by Šuvakov and Dmitra šinović [Phys Rev Lett, 2013, 110: 114301] using the gradient descent method with double precision. In this paper, these reported orbits are checked stringently by means of a reliable numerical approach (namely the “Clean Numerical Simulation”, CNS), which is based on the arbitrary-order Taylor series method and data in arbitrary-digit precision with a procedure of solution verification. It is found that seven among these fifteen orbits greatly depart from the periodic ones within a long enough interval of time, and are thus most possibly unstable at least. It is suggested to carefully check whether or not these seven unstable orbits are the so-called “computational periodicity” mentioned by Lorenz in 2006. This work also illustrates the validity and great potential of the CNS for chaotic dynamic systems.

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

Access this article

Log in via an institution

Subscribe and save

Springer+
from $39.99 /Month
  • Starting from 10 chapters or articles per month
  • Access and download chapters and articles from more than 300k books and 2,500 journals
  • Cancel anytime
View plans

Buy Now

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

Explore related subjects

Discover the latest articles, books and news in related subjects, suggested using machine learning.

References

  1. Valtonen M, Karttunen H. The Three Body Problem. New York: Cambridge University Press, 2005

    Google Scholar 

  2. Moore C. Braids in classical dynamics. Phys Rev Lett, 1993, 70: 3675–3679

    Article  MathSciNet  MATH  ADS  Google Scholar 

  3. Broucke R, Boggs D. Periodic orbits in the planar general three body problem. Celest Mech, 1975, 11: 13–38

    Article  MathSciNet  MATH  ADS  Google Scholar 

  4. Hadjidemetriou J D, Christides T. Families of periodic orbits in the planar three-body problem. Celest Mech, 1975, 12: 175–187

    Article  MathSciNet  MATH  ADS  Google Scholar 

  5. Hadjidemetriou J D. Stability of periodic orbits in 3-body problem. Celest Mech, 1975, 12: 255–276

    Article  MathSciNet  MATH  ADS  Google Scholar 

  6. Broucke R. On relative periodic solutions of the planar general three body problem. Celest Mech, 1975, 12: 439–462

    Article  MathSciNet  MATH  ADS  Google Scholar 

  7. Hénon M. A family of periodic solutions of the planar three-body problem and their stability. Celest Mech, 1976, 13: 267–285

    Article  MATH  ADS  Google Scholar 

  8. Hénon M. Stability of interplay motions. Celest Mech, 1977, 15: 243–261

    Article  MATH  ADS  Google Scholar 

  9. Nauenberg M. Periodic orbits for three particles with finite angular momentum. Phys Lett A, 2001, 292: 93–99

    Article  MathSciNet  MATH  ADS  Google Scholar 

  10. Fejoz J, Chenciner A, Montgomery R. Rotating eights: I. The three families. Nonlinearity, 2005, 18: 1407–1424

    Article  MathSciNet  MATH  ADS  Google Scholar 

  11. Elipe A, Broucke R, Riaguas A. On the figure-8 periodic solutions in the three-body problem. Chaos Solitons Fractals, 2006, 30: 513–520

    Article  MathSciNet  MATH  ADS  Google Scholar 

  12. Nauenberg M. Continuity and stability of families of figure eight orbits with finite angular momentum. Celest Mech, 2007, 97: 1–15

    Article  MathSciNet  MATH  ADS  Google Scholar 

  13. Šuvakov M, Dmitrašinović V. Three classes of Newtonian three-body planar periodic orbits. Phys Rev Lett, 2013, 110: 114301

    Article  Google Scholar 

  14. Lorenz E N. Computational periodicity as observed in a simple system. Tellus-A, 2006, 58: 549–557

    Article  ADS  Google Scholar 

  15. Zeng Q G, Li J P, Chou J F. Computational uncertainty principle in nonlinear ordinary differential equations (ii): Theoretical analysis. Sci China Ser E-Technol Sci, 2001, 44: 55–74

    MathSciNet  MATH  Google Scholar 

  16. Wang P F, Li J P, Li Q. Computational uncertainty and the application of a high-performance multiple precision scheme to obtaining the correct reference solution of lorenz equations. Numer Algorithms, 2012, 59: 147–159

    Article  MathSciNet  MATH  Google Scholar 

  17. Liao S J. On the reliability of computed chaotic solutions of non-linear differential equations. Tellus-A, 2009, 61: 550–564

    Article  ADS  Google Scholar 

  18. Liao S J. On the numerical simulation of propagation of micro-level inherent uncertainty for chaotic dynamic systems. Chaos Solitons Fractals, 2013, 47: 1–12

    Article  MathSciNet  ADS  Google Scholar 

  19. Liao S J. Physical limit of prediction for chaotic motion of three-body problem. Commun Nonlinear Sci Numer Simul, 2014, 19: 601–616

    Article  MathSciNet  ADS  Google Scholar 

  20. Corliss G F, Chang Y F. Solving ordinary differential equations using Taylor series. ACM Trans Math Software, 1982, 8: 114–144

    Article  MathSciNet  MATH  Google Scholar 

  21. Blesa F, Barrio R, Lara M. VSVO formulation of the Taylor method for the numerical solution of ODEs. Comput Math Appl, 2005, 50: 93–111

    Article  MathSciNet  MATH  Google Scholar 

  22. Abadb A, Barrio R, Rodrguez M, et al. Uncertainty propagation or box propagation. Math Comput Model, 2011, 54: 2602–2615

    Article  Google Scholar 

  23. Oyanarte P. MP-A multiple precision package. Comput Phys Commun, 1990, 59: 345–358

    Article  Google Scholar 

  24. Kehlet B, Logg A. Quantifying the computability of the Lorenz system. 2013, arXiv: 1306.2782

    Google Scholar 

  25. Liao S J, Wang P F. On the mathematically reliable long-term simulation of chaotic solutions of lorenz equation in the interval [0, 10000]. Sci China-Phys Mech Astron, 2014, 57: 330–335

    Article  Google Scholar 

  26. Viswanath D. The fractal property of the lorenz attractor. Phys D, 2004, 190: 115–128

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ministry-of-Education Key Laboratory in Scientific Computing, Shanghai, 200240, China

    ShiJun Liao

  2. School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China

    XiaoMing Li & ShiJun Liao

  3. Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University (KAU), Jeddah, Saudi Arabia

    ShiJun Liao

Authors
  1. XiaoMing Li
    View author publications

    Search author on:PubMed Google Scholar

  2. ShiJun Liao
    View author publications

    Search author on:PubMed Google Scholar

Corresponding author

Correspondence to ShiJun Liao.

Additional information

Contributed by LIAO ShiJun (Associate Editor)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, X., Liao, S. On the stability of the three classes of Newtonian three-body planar periodic orbits. Sci. China Phys. Mech. Astron. 57, 2121–2126 (2014). https://doi.org/10.1007/s11433-014-5563-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue date:

  • DOI: https://doi.org/10.1007/s11433-014-5563-5

Keywords

Profiles

  1. ShiJun Liao View author profile