Delayed Feedback Control of a Delay Equation at Hopf Bifurcation

Autor(es)

Fiedler, Bernold ; Oliva, Sergio Muniz

• Tipo de documento: Artigo Científico
• Fonte: Journal of Dynamics and Differential Equations; v. 28, n. 3-4, p. 35-pg., 2016-09-01.

Resumo

We embark on a case study for the scalar delay equation with odd nonlinearity f, real nonzero parameters , and three positive time delays . We assume supercritical Hopf bifurcation from in the well-understood single-delay case . Normalizing , branches of constant minimal period are known to bifurcate from eigenvalues at , for any nonnegative integer k. The unstable dimension is k, at the local branch k. We obtain stabilization of such branches, for arbitrarily large unstable dimension k. For the branch k of constant period persists as a solution, for any and . Indeed the delayed feedback term controlled by b vanishes on branch k: the feedback control is noninvasive there. Following an idea of Pyragas, we seek parameter regions of controls and delays such that the branch k becomes stable, locally at Hopf bifurcation. We determine rigorous expansions for in the limit of large k. The only two regions which we were able to detect, in this setting, required delays near 1, controls b near , and were of very small area of order . Our analysis is based on a 2-scale covering lift for the frequencies involved. (AU)

• Processo FAPESP: 08/55516-3 - Sistemas dinâmicos não lineares em espaços de dimensão infinita
• Beneficiário: Alexandre Nolasco de Carvalho
• Modalidade de apoio: Auxílio à Pesquisa - Temático