A Practical Guide to Rigorously Locate Periodic Orbits in Discrete Dynamics
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arXiv
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| Format: | Preprint |
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2025
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| author | Mozo, Lucía Alonso Hénot, Olivier Lappicy, Phillipo |
| author_facet | Mozo, Lucía Alonso Hénot, Olivier Lappicy, Phillipo |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_06373 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Practical Guide to Rigorously Locate Periodic Orbits in Discrete Dynamics Mozo, Lucía Alonso Hénot, Olivier Lappicy, Phillipo Dynamical Systems 37E05, 37G15, 37N25, 65G30, 92D25 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. |
| title | A Practical Guide to Rigorously Locate Periodic Orbits in Discrete Dynamics |
| topic | Dynamical Systems 37E05, 37G15, 37N25, 65G30, 92D25 |
| url | https://arxiv.org/abs/2510.06373 |