A Practical Guide to Rigorously Locate Periodic Orbits in Discrete Dynamics

Abstract

Periodic orbits are important objects of discrete dynamical systems, but finding them is not always easy. We present a self-contained introductory account, aimed at non-experts, to prove their existence and study their stability using the aid of the computer. The method consists in three main steps. First, we reformulate the problem of identifying a $p$-periodic orbit as a root-finding problem. Second, we find a numerical approximation of the root (i.e. a periodic orbit candidate). Third, we verify rigorously the contraction of a quasi-Newton operator near this approximation, which guarantees the existence of a unique root (i.e. periodic orbit). The neighbourhood of contraction is a ball centered at the approximation, whose radius yields a rigorous a posteriori error bound on the numerical approximation. To illustrate the effectiveness of this method, we implement it in two examples: the well-known logistic map and a discretization of a predator prey model. For the logistic map, we prove the existence of more than $80\cdot 10^2$ periodic orbits of periods $p=1,\ldots , 80$, mostly unstable. For the predator-prey model, we rigorously detect over $80\cdot 10^4$ periodic orbits of periods $p=1,\ldots, 10$, mostly unstable as well. This confirms well-known dynamical features such as period-doubling bifurcations and the emergence of increasingly complex orbit structures as the parameter changes.