The authors develop a new method to compute long unstable periodic points of chaotic discrete dynamical systems. A problem of major interest is to understand the behavior of systems through unstable long periodic orbits since they carry much information on the dynamics. Unfortunately, long unstable periodic orbits are not visible in numerical simulations.
Since a periodic point \(p\) of period \(m\) of the system \((X,f)\) is a fixed point of the map \(f^m\) \((f^m(p)= p)\) the problem of the location of the long period orbit can be approximately solved using the standard Newton method. To this end, we need a starting point \(p_0\) near \(p\). But in some cases this is difficult to get.
Instead it is possible to compute pseudo periodic orbits of the system with local errors near the machine precision by some numerical methods and then apply a periodic shadowing theorem on the existence of a periodic orbit if it has been computed a pseudo one previously. This result was developed by the authors in [Six Lectures on Dynamical Systems, World Scientific, Singapore, 163-211 (1996; Zbl 0915.39008)]. The numerical method they use is a multiple back and forth shooting method adequate to the problem.
In the paper, the whole procedure is described for chaotic maps \(f:\mathbb{R}\to \mathbb{R}\) but can be easily extended to maps on \(\mathbb{R}^n\) where other problems appear such as the existence of expanding and contracting components.
As an illustration, the authors apply the method to a problem in dimension one, the logistic map and in dimension two, the Hénon map. They prove the existence of periodic orbits of period 41 and 100,003 respectively.