Hopf and Generalized Hopf Bifurcations in a Recurrent Autoimmune Disease Model

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This paper is concerned with bifurcation and stability in an autoimmune model, which was established to study an important phenomenon - blips arising from such models. This model has two equilibrium solutions, disease-free equilibrium and disease equilibrium. The positivity of the solutions of the model and the global stability of the disease-free equilibrium have been proved. In this paper, we particularly focus on Hopf bifurcation which occurs from the disease equilibrium. We present a detailed study on the use of center manifold theory and normal form theory, and derive the normal form associated with Hopf bifurcation, from which the approximate amplitude of the bifurcating limit cycles and their stability conditions are obtained. Particular attention is also paid to the bifurcation of multiple limit cycles arising from generalized Hopf bifurcation, which may yield bistable phenomenon involving equilibrium and oscillating motion. This result may explain some complex dynamical behavior in real biological systems. Numerical simulations are compared with the analytical predictions to show a very good agreement.

Original languageEnglish
Article number1650079
JournalInternational Journal of Bifurcation and Chaos
Issue number5
StatePublished - May 1 2016


  • Autoimmune disease model
  • Hopf bifurcation
  • center manifold
  • generalized Hopf bifurcation
  • limit cycle
  • normal form
  • stability


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