Publications [#246470] of Daniel J. Gauthier
Papers Published
- Illing, L; Gauthier, DJ, Hopf bifurcations in time-delay systems with band-limited feedback,
Physica D: Nonlinear Phenomena, vol. 210 no. 3-4
(October, 2005),
pp. 180-202, Elsevier BV [doi]
(last updated on 2019/07/17)Abstract:
We investigate the steady-state solution and its bifurcations in time-delay systems with band-limited feedback. This is a first step in a rigorous study concerning the effects of AC-coupled components in nonlinear devices with time-delayed feedback. We show that the steady state is globally stable for small feedback gain and that local stability is lost, generically, through a Hopf bifurcation for larger feedback gain. We provide simple criteria that determine whether the Hopf bifurcation is supercritical or subcritical based on the knowledge of the first three terms in the Taylor-expansion of the nonlinearity. Furthermore, the presence of double-Hopf bifurcations of the steady state is shown, which indicates possible quasiperiodic and chaotic dynamics in these systems. As a result of this investigation, we find that AC-coupling introduces fundamental differences to systems of Ikeda-type [K. Ikeda, K. Matsumoto, High-dimensional chaotic behavior in systems with time-delayed feedback, Physica D 29 (1987) 223-235] already at the level of steady-state bifurcations, e.g. bifurcations exist in which limit cycles are created with periods other than the fundamental "period-2" mode found in Ikeda-type systems. © 2005 Elsevier B.V. All rights reserved.