High-order control for symplectic maps

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Abstract

We revisit the problem of control for devices that can be modeled via a symplectic
map in a neighbourhood 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 behaviour 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.
Original languageEnglish
PublisherNamur center for complex systems
Number of pages30
Volume2
Edition15
Publication statusPublished - 1 Jan 2015

Publication series

NamenaXys Technical Report Series
PublisherUniversity of Namur
No.15
Volume2

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Keywords

  • dynamical systems
  • Normal forms method
  • Control of chaos
  • Hamiltonian control
  • Symplectic maps

Cite this

Sansottera, M., Giorgilli, A., & Carletti, T. (2015). High-order control for symplectic maps. (15 ed.) (naXys Technical Report Series; Vol. 2, No. 15). Namur center for complex systems.