Machine-assisted discovery of integrable symplectic mappings

TL;DR

This work presents a machine-assisted framework to automatically discover area-preserving 2D symplectic maps with polygon invariants, significantly expanding the catalog of integrable polygon mappings. By restricting to piecewise-linear forces and exploiting polygonal invariants, the authors extract and classify over 100 new integrable families, including classical McMillan–Suris cases and links to discrete Painlevé equations. The methodology combines trajectory-shape analysis, vertex-counting of orbit polygons, and rigorous McMillan-type integrability checks, supplemented by a detailed algorithm and pseudo-code; it also demonstrates pathways to near-integrable dynamics through smoothing and discrete perturbation theory. The results offer both theoretical insights into the geometry of invariant sets and practical avenues for applications in near-integrable systems and accelerator physics, where polygonal invariants can inform stable focusing schemes and perturbative analyses. Overall, the paper provides a comprehensive framework for discovering, verifying, and applying polygon-based integrable maps, along with a roadmap toward integrating these discrete models with more general 2D symplectic dynamics.

Abstract

We present a new automated method for finding integrable symplectic maps of the plane. These dynamical systems possess a hidden symmetry associated with an existence of conserved quantities, i.e. integrals of motion. The core idea of the algorithm is based on the knowledge that the evolution of an integrable system in the phase space is restricted to a lower-dimensional submanifold. Limiting ourselves to polygon invariants of motion, we analyze the shape of individual trajectories thus successfully distinguishing integrable motion from chaotic cases. For example, our method rediscovers some of the famous McMillan-Suris integrable mappings and discrete Painlevé equations. In total, over 100 new integrable families are presented and analyzed; some of them are isolated in the space of parameters, and some of them are families with one parameter (or the ratio of parameters) being continuous or discrete. At the end of the paper, we suggest how newly discovered maps are related to a general 2D symplectic map via an introduction of discrete perturbation theory and propose a method on how to construct smooth near-integrable dynamical systems based on mappings with polygon invariants.

Figures (24)

• Figure 2: Two constant level sets $\mathcal{K}(p,q)=\mathrm{const}$ for the chaotic Hénon map, $f(p) = a\,p + b\,p^2$. The black curve shows a closed trajectory encompassing the origin and the gray curves show a set of islands (i.e. under iterations, points hop from island to island covering the closed gray curves labeled i1 -- i5). While the first (reflection) symmetry is quite obvious from the figure, we will focus on the second one. The upper branch of the blue curve, $\Phi(q)$, is vertically equidistant from the lower branch, $\Phi^{-1}(q)$. Island i3 satisfies the second symmetry in the same manner as does the blue curve. Islands i1 and i2 satisfy the symmetry in a sense that they are vertical reflections of islands i5 and i4, respectively (e.g. the lower part of i1 is equidistant from the upper part of i5 and the upper part of i1 is equidistant from the lower part of i5). The dashed and solid green lines are the first and second symmetry lines.
• Figure 3: Integer mappings with linear force function and polygon invariants $\alpha$, $\beta$ and $\gamma$ (we use the same naming convention as Ref. cairns2014piecewisecairns2016conewise). Note that the invariant triangle was modified in order to satisfy both symmetries. Each figure shows one invariant polygon, with each colored region $i$ being mapped to region $i+1$. Here $n$ is the period and $\nu$ is the rotation number of the map, which represents the average increase in the angle per map iteration. The dashed and solid dark green lines show the first and second symmetry lines.
• Figure 4: Constant level sets of invariant for 1-piece map in the MH form, $\mathcal{K}(p,q)=p^2-k\,p\,q+q^2 = \mathrm{const}$ . For $|k|<2$ the level sets are ellipses, for $|k|>2$ the level sets are hyperbolas and for $|k|=2$ they are parallel lines. The dashed and solid green lines are the first and second symmetry lines.
• Figure 5: Fundamental polygons of the first kind for different values of rotation number $\nu$. Each plot shows the invariant ellipse $\mathcal{K}(p,q)=p^2-k\,p\,q+q^2 = \mathrm{const}$ (black) and two invariant fundamental polygons inscribed into it (blue and red). The dashed and solid green lines correspond to the first and second symmetry lines.
• Figure 6: Integer mappings with 2-piece force function and polygon invariants D -- H (named after cairns2014piecewise). Each figure shows one invariant polygon, with each colored region $i$ being mapped to the region $i+1$. Here $n$ and $\nu$ are the period and rotation number of the map. The solid and dashed dark green lines show the second and first symmetry lines.