On the rate of convergence of periodic solutions in perturbed autonomous systems as the perturbation vanishes

2007/09/27 by Oleg Makarenkov, Makarenkov, Oleg, Paolo Nistri +1
Mathematics · #34A10 #34C24 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:34A10 #msc:34C24

paper · pdf · doi:10.48550/arxiv.0709.4431

To Appear in Commun. Pure Appl. Anal

arxiv created 2007/09/27 · arxiv updated 2009/12/01

Abstract

We consider an autonomous system in Rⁿ having a limit cycle x₀ of period T>0 which is nondegenerate in a suitable sense. We then consider the perturbed system obtained by adding to the autonomous system a T-periodic, not necessarily differentiable, term whose amplitude tends to 0 as a small parameter e>0 tends to 0. Assuming the existence of a T-periodic solution x_e of the perturbed system and its convergence to x₀ as e->0, the paper establishes the existence of d_e->0 as e->0 such that ‖x_e(t+d_e)-x₀(t)‖<=eM for some M>0 and any e>0 sufficiently small.

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