Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 06, pp. 1-19.
Title: Slow and fast systems with Hamiltonian reduced problems
Authors: Maamar Benbachir (Univ. de Bechar, Algerie)
Karim Yadi (Univ. Aboubekr Belkaid, Algerie)
Rachid Bebbouchi (Univ. des Sciences et de la Technologie, Algerie)
Abstract:
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.