High-order control for symplectic maps

TL;DR

This paper develops a normal form algorithm using Lie transforms to control symplectic maps near elliptic equilibria, enhancing stability domains without interpolating with a flow, supported by numerical Hénon map examples.

Contribution

Introduces a novel normal form algorithm for symplectic maps that avoids flow interpolation, with quantitative estimates and control strategies to enlarge stable regions.

Findings

Normal form algorithm effectively controls symplectic maps.

Quantitative estimates demonstrate the asymptotic nature of the transformation.

Control terms can significantly increase the stable domain size.

Abstract

We revisit the problem of introducing an a priori control for devices that can be modeled via a symplectic map in a neighborhood of an elliptic equilibrium. Using a technique based on Lie transform methods we produce a normal form algorithm that avoids the usual step of interpolating the map with a flow. The formal algorithm is completed with quantitative estimates that bring into evidence the asymptotic character of the normal form transformation. Then we perform an heuristic analysis of the dynamical behavior of the map using the invariant function for the normalized map. Finally, we discuss how control terms of different orders may be introduced so as to increase the size of the stable domain of the map. The numerical examples are worked out on a two dimensional map of H\'enon type.