Online regularization of Poincaré map of storage rings with Shannon entropy

Shannon entropy, as a chaos indicator, is used for online Poincaré map regularization and dynamic aperture optimization in the National Synchrotron Light Source-II (NSLS-II) ring. Although various chaos indicators are widely used in studying nonlinear dynamical systems, including modern particle accelerators, it is the first time to use a measurable one in a real-world machine for online nonlinear optimization. Poincaré maps, constructed with the turn-by-turn beam trajectory readings from beam position monitors, are commonly used to observe the nonlinearity in ring-based accelerators. However, such observations typically only provide a qualitative interpretation. We analyze their entropy to quantify the chaos in measured Poincaré maps. After some canonical transformations on the Poincaré maps, not only can the commonly used nonlinear characterizations be extracted, but more importantly, the chaos can be quantitatively calibrated with Shannon entropy, and then used as the online optimization objectives.

💡 Research Summary

This paper introduces a novel, measurement‑based chaos indicator—Shannon entropy—to enable real‑time regularization of Poincaré maps and online nonlinear optimization of a storage ring, specifically the National Synchrotron Light Source‑II (NSLS‑II). While many chaos diagnostics such as Frequency Map Analysis (FMA), maximal Lyapunov exponents, and resonance driving term extraction are standard in accelerator simulations, they are impractical for online tuning because they require extensive data, precise frequency determination, or heavy computational effort. The authors instead exploit turn‑by‑turn beam position monitor (BPM) data to construct Poincaré maps, then apply a sequence of canonical transformations that map linear motion onto a single point, thereby guaranteeing zero entropy for an ideal, regular orbit.

First, the one‑turn transport matrix M is used together with measured Twiss parameters to convert raw coordinates into action‑angle variables (J, φ). A parallel‑shift transformation φₙ → φₙ − n·2πν removes the deterministic phase advance, collapsing a perfectly linear torus to a fixed point. For nonlinear dynamics, a normal‑form transformation based on a type‑2 generating function introduces correction terms Cₘ sin(mφ+φₘ₀) that minimize the variance of the transformed action ¯J. After these transformations, the Poincaré map appears as an “island” whose radial size reflects action‑angle smearing and whose angular offset from the linear reference encodes the tune‑shift‑with‑amplitude (TSWA) effect.

Entropy is then estimated not by a continuous probability density (which would be ill‑defined for the limited number of turns) but by discretizing the transformed (ˆJ, ˆφ) plane into a fine mesh (ΔˆJ = 1 × 10⁻⁷ m, Δˆφ = 1°). Each occupied mesh cell defines a macro‑state with probability pᵢ = Nᵢ/N, where Nᵢ is the number of points in cell i and N is the total number of turns. The Shannon entropy S = –∑ pᵢ log₂ pᵢ quantifies how many distinct macro‑states the trajectory explores; a purely regular motion yields S ≈ 0, while chaotic diffusion populates many cells and raises S. The mesh size is chosen to balance BPM resolution against the need to resolve the island’s shape.

The method is validated experimentally on NSLS‑II. Six families of harmonic sextupoles located in dispersion‑free straight sections serve as tunable knobs. For each sextupole setting, the beam is excited to four amplitudes using a fast transverse “pinger” and the corresponding Poincaré maps are recorded by a pair of BPMs positioned in a magnet‑free straight. The objective function is the average normalized entropy
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