Skip to main content
Springer Nature Link
Log in

A new shooting algorithm for the search of periodic orbits

  • Original Article
  • Published:
Artificial Life and Robotics Aims and scope Submit manuscript

Abstract

The periodic orbit theory gives the basic framework to study a quantum and classical correspondence. In this paper, we firstly report that we have found the existence of a certain surface, which we call the devil’s staircase surface. Secondly, taking the advantage of some intriguing properties of this surface, we propose a new method to exhaustively search for periodic orbits in the anisotropic Kepler problem. Our method fully takes into account of an intriguing property of the initial value problem of the anisotropic Kepler problem, and it reduces the two-dimensional search into the one-dimensional search. Using this method, all of the periodic orbits up to the length \(2N=20\) (altogether 19284 distinct periodic orbits) have been successfully obtained, which exceeds the world record of 76 periodic orbits up to \(2N=10\).

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+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

Notes

  1. In [2], we restricted ourselves to a symmetric orbit search and showed that a simple shooting with full attention to the scaling property of the AKP supplies better convergence than a traditional search [1] based on the Hill’s method in the restricted 3-body problem. The quantum physics issue was also briefly introduced. In this report, we treat a two-dimensional PO search which requires further considerations.

  2. Due to the symmetry of the AKP Hamiltonian, distinct codes related by certain transformations may express essentially the same periodic orbits as analyzed in detail in [3]. Therefore, equivalent codes are classified into a class. The representative code for each class is the one which wins the highest \(\zeta\) (Eq. (7)) among the class. Then, one compares the representatives and assigns identification numbers to them in the decreasing order of their \(\zeta\)s.

  3. While we can construct numbers for the future(\(\zeta ^f\)) and past(\(\zeta ^p\)) [3], we use only \(\zeta ^f\) in this paper, and we suppress the superscript \(f\).

  4. We show the initial value space limited to the fundamental region \((X_0,U_0) \in [0,2] \times [0,1]\).

  5. The number \(\tilde{\zeta }\) for the target-PO is also calculated by the cyclic binary code with 49 binary digits, just as we calculate the \(\zeta (X_0,U_0)\) by 49 binary digits.

  6. The Poincaré section is defined by \(y=0\), and \(y_i=0\). But, for \(y_f\), there remains inevitably finite \(y_f\), because of the finite \(\Delta t\) in the integration procedure. We have applied a linear interpolation to improve the final time decision within \(\Delta t =10^{-4}\), and used supplemental integration within the last \(\Delta t\) to determine the final Poincaré section.

References

  1. Gutzwiller MC (1971) Periodic orbits and classical quantization conditions. J Math Phys 12:343–358

    Article  Google Scholar 

  2. Kubo K, Shimada T (2009) AKP energy levels by a simple shooting scheme for a periodic orbit. Artif Life Robot 14:557–561

    Article  Google Scholar 

  3. Gutzwiller MC (1977) Bernoulli sequences and trajectories in the anisotropic Kepler problem. J Math Phys 18:806–823

    Article  MathSciNet  Google Scholar 

  4. Devaney RL (1978) Collision orbits in the anisotropic Kepler problem. Invent Math 45:221–251

    Article  MATH  MathSciNet  Google Scholar 

  5. Gutzwiller MC (1989) Multifractal measures and stability islands in the anisotropic Kepler problem. Phys D 38:160–171

    Article  MATH  MathSciNet  Google Scholar 

  6. Peitgen HO, Jürgens H, Saupe D (1992) Chaos and fractals: new frontiers of science. Springer, New York, pp 220–228

  7. Gutzwiller MC (1981) Periodic orbits in the anisotropic Kepler problem. In: Proceedings of classical mechanics and dynamical systems. Marcel Dekker, New York, pp 69–90

  8. Kubo K, Shimada T (2014) Periodic orbit theory revisited in the anisotropic Kepler problem. Prog Theor Exp Phys 023A0:1–19

  9. Gutzwiller MC (1988) From classical to quantum mechanics with hard chaos. J Phys Chem 92:3154–3163

    Article  Google Scholar 

  10. Gutzwiller MC (1980) Classical quantization of a Hamiltonian with ergodic behavior. Phys Rev Lett 45:150–153

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, School of Science and Technology, Meiji University, 1-1-1 Higashimita, Tama, Kawasaki, Kanagawa, 214-8571, Japan

    Keita Sumiya & Tokuzo Shimada

  2. Institut fuer Physik, Technische Universitaet Ilmenau, Weimarer Strasse 25, 98693, Ilmenau, Germany

    Kazuhiro Kubo

Authors
  1. Keita Sumiya
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Kazuhiro Kubo
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Tokuzo Shimada
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Tokuzo Shimada.

Additional information

This work was presented in part at the 19th International Symposium on Artificial Life and Robotics, Beppu, Oita, 22–24 January, 2014.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sumiya, K., Kubo, K. & Shimada, T. A new shooting algorithm for the search of periodic orbits. Artif Life Robotics 19, 262–269 (2014). https://doi.org/10.1007/s10015-014-0166-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10015-014-0166-9

Keywords