High-order control for symplectic maps

M. Sansottera, A. Giorgilli, T. Carletti

Research output: Contribution to journalArticle

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 Hénon type.
Original language English 1-15 15 Physica D: Nonlinear Phenomena 316 https://doi.org/10.1016/j.physd.2015.10.012 Published - 15 Feb 2016

estimates

Keywords

• Linearization
• Normal form
• Perturbation theory
• Resonances
• Symplectic maps

Cite this

title = "High-order control for symplectic maps",
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{\'e}non type.",
keywords = "Linearization, Normal form, Perturbation theory, Resonances, Symplectic maps",
author = "M. Sansottera and A. Giorgilli and T. Carletti",
year = "2016",
month = "2",
day = "15",
doi = "10.1016/j.physd.2015.10.012",
language = "English",
volume = "316",
pages = "1--15",
journal = "Physica D",
issn = "0167-2789",
publisher = "Elsevier",

}

In: Physica D: Nonlinear Phenomena, Vol. 316, 15.02.2016, p. 1-15.

Research output: Contribution to journalArticle

TY - JOUR

T1 - High-order control for symplectic maps

AU - Sansottera, M.

AU - Giorgilli, A.

AU - Carletti, T.

PY - 2016/2/15

Y1 - 2016/2/15

N2 - 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.

AB - 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.

KW - Linearization

KW - Normal form

KW - Perturbation theory

KW - Resonances

KW - Symplectic maps

UR - http://www.scopus.com/inward/record.url?scp=84947967649&partnerID=8YFLogxK

U2 - 10.1016/j.physd.2015.10.012

DO - 10.1016/j.physd.2015.10.012

M3 - Article

VL - 316

SP - 1

EP - 15

JO - Physica D

JF - Physica D

SN - 0167-2789

ER -