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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