Proving the existence of long periodic orbits in 1D maps using interval Newton method and backward shooting

In this paper, we present a method that can be used to find very long periodic orbits for one-dimensional maps. This new approach is a combination of the interval Newton method and the shooting technique. We also describe how to use this approach to find better approximation of position of computer generated pseudo-periodic orbit. As an example, we find very long periodic orbits for the logistic map and we calculate the Lyapunov exponents of these orbits. Finally, we investigate the performance of this method used for finding all short period cycles.