A Practical Guide to Rigorously Locate Periodic Orbits in Discrete Dynamics

TL;DR

The paper presents a practical, computer-assisted framework to rigorously locate periodic orbits in discrete dynamical systems by recasting the search as finding zeros of $F$, computing a numerical root, and then validating a contraction of the quasi-Newton operator with $A\approx DF(\bar{x})^{-1}$ to obtain a rigorous error ball around the true orbit. It introduces an a posteriori method using interval arithmetic to bound $Y$, $Z_1$, and $Z_2$, ensuring existence, local uniqueness, and stability of the $p$-periodic points, and extends the technique to track period-doubling bifurcations uniformly across a parameter range via Chebyshev interpolation and a uniform contraction theorem. The approach is demonstrated on a logistic map and a discretized predator–prey map, yielding thousands of stable and unstable periodic orbits across multiple periods and providing rigorous confirmation of period-doubling cascades. These results showcase a robust, scalable workflow for certifying periodic structures and bifurcations in discrete dynamical systems with direct applications to mathematical biology and beyond.

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.

Figures (2)

• Figure 1: Left: The well-known bifurcation diagram for the logistic map containing one period-$p$ orbit for each $p=2,\ldots,80$ described in githubGitHubLuciaalonsomozoCAPs. Right: The number of periodic points (of periods up to $p=80$) increases with $\mu$. Blue (resp. magenta) denotes stable (resp. unstable) points.
• Figure 2: Left and middle: Regions of the parameter plane $(\beta, \kappa)$ associated with stable and unstable periodic points. Note the curves that separate regions of period $p$ from period $2p$, associated to period-doubling bifurcations; these are rigorously constructed in \ref{['sec:perdob']}. These curves form the so-called shrimp structures, see GallasStoopGlasskryzhevich2021bistability. Right: Heatmap that depicts if there are periodic points of periods up to a given value $p$ in the parameter plane $(\beta, \kappa)$. Here, we do not account for how many orbits of the same period may exist; we simply plot the existence of periodic points for all periods up to a fixed period $p$, which determines the shade of red.

Theorems & Definitions (5)

• Definition 1
• Theorem 1
• Remark 1
• Remark 2
• Theorem 2