The peaking phenomenon and singular perturbations
Résumé
We study the asymptotic behaviour, when the parameter ε tends to 0, of a class of singularly perturbed triangular systems x˙ = f(x, y), y˙ = G(y, ε). We assume that all solutions of the second equation tend to zero arbitrarily fast when ε tends to 0. We assume that the origin of equation x˙ = f(x, 0) is globally asymptotically stable. Some states of the second equation may peak to very large values, before they rapidly decay to zero. Such peaking states can destabilize the first equation. The paper introduces the concept of instantaneous stability, to measure the fast decay to zero of the solutions of the second equation, and the concept of uniform infinitesimal boundedness to measure the effects of peaking on the first equation. Whe show that all the solutions of the triangular system tend to zero when ε → 0 and t → +∞. Our results are formulated in both classical mathematics and nonstandard analysis.