High-order control for symplectic maps

• Control for symplectic maps in a neighborhood of an elliptic equilibria.
• Normal form for maps using Lie transforms.
• Effective numerical algorithm for the control problem for maps.
• Enlarging the size of the stability domain of a 2D Hénon type map.

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 Hénon type.