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 $$2N=20$$ 2 N = 20 (altogether 19284 distinct periodic orbits) have been successfully obtained, which exceeds the world record of 76 periodic orbits up to $$2N=10$$ 2 N = 10 .