Electron. J. Diff. Equ., Vol. 2010(2010), No. 06, pp. 1-19.

Slow and fast systems with Hamiltonian reduced problems

Maamar Benbachir, Karim Yadi, Rachid Bebbouchi

Slow and fast systems are characterized by having some of the derivatives multiplied by a small parameter $\epsilon$. We study systems of reduced problems which are Hamiltonian equations, with or without a slowly varying parameter. Tikhonov's theorem gives approximate solutions for times of order 1. Using the stroboscopic method, we give approximations for time of order $1/\epsilon$. More precisely, the variation of the total energy of the problem, and the eventual slow parameter, are approximated by a certain averaged differential equation. The results are illustrated by some numerical simulations. The results are formulated in classical mathematics and proved within internal set theory which is an axiomatic approach to nonstandard analysis.

Submitted June 1, 2009. Published January 13, 2010.
Math Subject Classifications: 34D15, 34E18, 70H09, 03H05.
Key Words: Singular perturbations; Hamiltonian system; stroboscopy lemma; nonstandard analysis.

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Maamar Benbachir
Université de Béchar, BP 417
Route de Kenadsa, 08000 Béchar, Algérie
email: mbenbachir2001@yahoo.fr
Karim Yadi
Laboratoire des Systémes Dynamiques et Applications
Université Aboubekr Belkaïd
BP 119, 13000 Tlemcen Algérie
email: yadikdz@yahoo.fr, k_yadi@mail.univ-tlemcen.dz
Rachid Bebbouchi
Faculté de Mathématiques
Université des Sciences et de la Technologie
Houari Boumediéne, Alger Bab ezzouar, Algérie
email: rbebbouchi@hotmail.com

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