2025/10/07 by Lucía Alonso Mozo, Mozo, Lucía Alonso, Olivier Hénot +3
Physics and Astronomy · #37E05 #37G15 #37N25 #65G30 #92D25 #Dynamical Systems (math.DS) #FOS: Mathematics #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2510.06373
openalex publication_date 2025/10/07 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28
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⋅ 102 periodic orbits of periods p=1,… , 80, mostly unstable. For the predator-prey model, we rigorously detect over 80⋅ 104 periodic orbits of periods p=1,…, 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.