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

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.