Slow and fast systems with Hamiltonian reduced problems

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.

Keywords

• Singular perturbations
• Hamiltonian system
• stroboscopy lemma
• nonstandard analysis