Hopf Bifurcation Analysis in a Mackey–glass System

Abstract

The dynamics of a Mackey–Glass equation with delay are investigated. We prove that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. Explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived, using the theory of normal form and center manifold. Global existence of periodic solutions are established using a global Hopf bifurcation result due to Wu [1998] and a Bendixson criterion for higher dimensional ordinary differential equations due to Li and Muldowney [1994].

Bifurcation Analysis of a Nfde Arising From Multiple-Delay Feedback Control

“…We would like to mention that extensive investigations have been carried out on the bifurcation of delay differential equations, we refer the readers to [Faria & Magalhaes, 1995a, 1995bYu & Chen, 2004;Wang & Zou, 2005;Yuan & Wei, 2006;Wei & Fan, 2007;Chen & Wu, 2008;Taborda et al, 2009;Wijeratne et al, 2009;Yan & Li, 2009] and the references cited therein. Also there are a few papers on the bifurcation of neutral differential equations, see [Weedermann, 2006;Wang & Wei, 2008] and the references cited therein.…”

Bifurcation analysis in single-species population model with delay

“…In 1994 Velasco-Hernendez used (1.2) with B 1 (N ) for the vector population equation in a model for changes disease, in 1997 Mackey and Glass used (1.2) with B(N ) = B 2 (N ) to model blood cell production (see [6,11]). …”

Organization and identification of solutions in the time-delayed Mackey-Glass model

“…As the delay further increases, a sequence of bifurcations generates oscillations with higher periods and eventually aperiodic behavior. 19 Over the years, the dynamics of the MG model has been investigated numerically, [20][21][22][23][24][25] and it has been used to generate high-dimensional chaotic signals. 26 The MG model has also been used as a toy model to study chaotic synchronization in delayed systems.…”