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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.
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 language | English |
|---|---|
| Publisher | Namur center for complex systems |
| Number of pages | 30 |
| Volume | 2 |
| Edition | 15 |
| Publication status | Published - 1 Jan 2015 |
Publication series
| Name | naXys Technical Report Series |
|---|---|
| Publisher | University of Namur |
| No. | 15 |
| Volume | 2 |
Keywords
- dynamical systems
- Normal forms method
- Control of chaos
- Hamiltonian control
- Symplectic maps
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Projects
- 1 Finished
PAI n°P7/19 - DYSCO: Dynamical systems, control and optimization (DYSCO)
WINKIN, J., Blondel, V., Vandewalle, J., Pintelon, R., Sepulchre, R., Vande Wouwer, A. & SARTENAER, A.
1/04/12 → 30/09/17
Project: Research
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.