Integrable symplectic maps with a polygon tessellation

TL;DR

This work uncovers a new class of integrable symplectic maps in the McMillan-Hénon form that admit polygonal invariants and perfect tessellations of the plane/torus. By allowing piecewise linear forces with arithmetic quasiperiodicity or periodic discontinuities, the authors automate the discovery of integrable regimes via a rotation-number analysis along symmetry lines, revealing global mode-locking and a previously undocumented integrable diffusion. The study links these tessellating invariants to both classical tiling theory and MC-illman–Suris mappings, and introduces a smoothening procedure to extend exact integrability toward quasi-integrable dynamics with potential practical applications, including accelerator physics. The results broaden the scope of integrable discrete dynamics and provide a framework for classifying a vast landscape of polygonal, tile-based phase spaces with discrete symmetries.

Abstract

The identification of integrable dynamics remains a formidable challenge, and despite centuries of research, only a handful of examples are known to date. In this article, we explore a special form of area-preserving (symplectic) mappings derived from the stroboscopic Poincare cross-section of a kicked rotator. Notably, Suris' theorem constrains the integrability within this category of mappings, outlining potential scenarios with analytic invariants of motion. In this paper, we challenge the assumption of the analyticity of the invariant, by exploring piecewise linear transformations on a torus and associated systems on the plane, incorporating arithmetic quasiperiodicity and discontinuities. By introducing a new automated technique, we discovered previously unknown scenarios featuring polygonal invariants that form perfect tessellations and, moreover, fibrations of the plane/torus. In this way, this work reveals a novel category of planar tilings characterized by discrete symmetries that emerge from the invertibility of transformations and are intrinsically linked to the presence of integrability. Our algorithm relies on the analysis of the Poincare rotation number and its piecewise monotonic nature for integrable cases, contrasting with the noisy behavior in the case of chaos, thereby allowing for clear separation. Some of the newly discovered systems exhibit the peculiar behavior of integrable diffusion, marked by infinite and quasi-random hopping between tiles while being confined to a set of invariant segments. Finally, through the implementation of a smoothening procedure, all mappings can be generalized to quasi-integrable scenarios with smooth invariant motion, thereby opening doors to potential practical applications.

Figures (19)

• Figure 1: Top row illustrates a regular piecewise linear force function composed of three segments (left), along with its periodic (middle) and arithmetically quasiperiodic (right) unwrapping from the torus (mod $L$), depicted with a cyan square. The bottom plot displays the corresponding force on a torus, providing visual references for the introduced notations.
• Figure 2: Phase space portrait for the map $\mathcal{M}_\mathrm{b1}^\mathrm{tor}$ along with the corresponding unwrapping of the invariant to $\mathbb{R}^2$. Various second symmetry lines illustrate possible configurations of the map: periodic force with discontinuities (per), or, arithmetically quasiperiodic force allowing for the central cell (cc) and central nodes (cn). Throughout this paper, we employ the following color scheme to represent invariant level sets in phase space: red corresponds to isolating invariants/separatrices, while blue and black indicate phase space orbits within layers with nonlinear (amplitude-dependent) and linear (mode-locked) dynamics, respectively. The solid and dashed green lines in all diagrams correspond to the second, $p=f(q)/2$, and first symmetry lines, $p=q$.
• Figure 3: Phase space portraits for the mappings on a torus, $\mathcal{M}_{\mathrm{d}1}^\mathrm{tor}$ and $\mathcal{M}_{\mathrm{a}1}^\mathrm{tor}$ (top row), along with their corresponding unwrapping with arithmetical quasiperiodicity (middle row) and periodic unwrapping with discontinuities (bottom row). The left plot in the middle, for the case (d.1), is non-integrable; chaotic trajectories are depicted with points in varying shades of gray. All other systems maintain integrability, showcasing sample orbits represented by series of connected dots.
• Figure 4: Schematic illustration of the global mode-locking around the central cell [cc] or central node [cn] for different values of $\nu_P$. Tiles of the same color are grouped in chains of islands, and the arrows indicate the order in which particular tiles are visited.
• Figure 5: Top row displays the replication of invariant level sets traced out by sample trajectories for the unwrappings of (a.1), Fig. \ref{['fig:Dynamics']}. The bottom row illustrates integrable diffusion for the top-right plot, depicting $10^4$ and $10^5$ iterations obtained by tracking, with a brown square having a side length of $2\,L$ for reference.