A new shooting algorithm for the search of periodic orbits


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 \(2N=10\).

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  1. 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. 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. 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. 4.

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

  5. 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. 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.


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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.

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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

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  • Quantum chaos
  • Anisotropic Kepler problem
  • Devil’s staircase
  • Periodic orbit theory