Construction of approximate invariants for non-integrable Hamiltonian systems

Abstract

We present a method to construct high-order polynomial approximate invariants (AI) for non-integrable Hamiltonian dynamical systems, and apply it to modern ring-based particle accelerators. Taking advantage of a special property of one-turn transformation maps in the form of a square matrix, AIs can be constructed order-by-order iteratively. Evaluating AI with simulation data, we observe that AI's fluctuation is actually a measure of chaos. Through minimizing the fluctuations with control knobs in accelerators, the stable region of long-term motions could be enlarged.

4 pages, 6 figures, accepted by Phys. Rev. Accel. Beams on July 2nd, 2025

References in corpus (4)

• Detecting chaos in particle accelerators through the frequency map analysis method
• The Reversibility Error Method (REM): a new, dynamical fast indicator for planetary dynamics
• Extracting Dynamical Frequencies from Invariants of Motion in Finite-Dimensional Nonlinear Integrable Systems
• Online regularization of Poincaré map of storage rings with Shannon entropy

Cited by in corpus (1)

• Sextupole reduction via chaos suppression at the National Synchrotron Light Source II