TY - JOUR
T1 - Hopf and Generalized Hopf Bifurcations in a Recurrent Autoimmune Disease Model
AU - Zhang, Wenjing
AU - Yu, Pei
N1 - Publisher Copyright:
© 2016 World Scientific Publishing Company.
PY - 2016/5/1
Y1 - 2016/5/1
N2 - 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.
AB - 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.
KW - Autoimmune disease model
KW - Hopf bifurcation
KW - center manifold
KW - generalized Hopf bifurcation
KW - limit cycle
KW - normal form
KW - stability
UR - http://www.scopus.com/inward/record.url?scp=84973277715&partnerID=8YFLogxK
U2 - 10.1142/S0218127416500796
DO - 10.1142/S0218127416500796
M3 - Article
AN - SCOPUS:84973277715
VL - 26
JO - International Journal of Bifurcation and Chaos
JF - International Journal of Bifurcation and Chaos
SN - 0218-1274
IS - 5
M1 - 1650079
ER -