Hopf bifurcation in numerical approximation of the sunflower equation

2006 ◽  
Vol 22 (1-2) ◽  
pp. 113-124 ◽  
Author(s):  
Chunrui Zhang ◽  
Baodong Zheng
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Approximation ◽  
Sunflower Equation

HOPF BIFURCATION ANALYSIS OF TWO SUNFLOWER EQUATIONS

2012 ◽  
Vol 05 (01) ◽  
pp. 1250001 ◽  
Author(s):  
JINGNAN WANG ◽  
WEIHUA JIANG
Keyword(s):  
Hopf Bifurcation ◽  
Global Hopf Bifurcation ◽  
Bifurcation Theorem ◽  
Local Hopf Bifurcation ◽  
Sunflower Equation ◽  
Nonlinear Relation ◽  
Bifurcation Direction ◽  
The Stability ◽  
Bifurcating Periodic Solution ◽  
Modified Equation

In this paper, two sunflower equations are considered. Using delay τ as a parameter and applying the global Hopf bifurcation theorem, we investigate the existence of global Hopf bifurcation for the sunflower equation. Furthermore, we analyze the local Hopf bifurcation of the modified equation with nonlinear relation about stem's increase, including the occurrence, the bifurcation direction, the stability and the approximation expression of the bifurcating periodic solution using the theory of normal form and center manifold. Finally, the obtained results of these two equations are compared, which finds that the result about the period of their bifurcating periodic solutions is obviously different, while the bifurcation direction and stability are identical.

Numerical Approximation of Hopf Bifurcation for Tumor-Immune System Competition Model with Two Delays

2013 ◽  
Vol 5 (2) ◽  
pp. 146-162
Author(s):  
Jing-Jun Zhao ◽  
Jing-Yu Xiao ◽  
Yang Xu
Keyword(s):  
Immune System ◽  
Hopf Bifurcation ◽  
Numerical Approximation ◽  
Center Manifold ◽  
Competition Model ◽  
Center Manifold Theory ◽  
Two Delays ◽  
Normal Form Method ◽  
The Stability ◽  
Manifold Theory

AbstractThis paper is concerned with the Hopf bifurcation analysis of tumor-immune system competition model with two delays. First, we discuss the stability of state points with different kinds of delays. Then, a sufficient condition to the existence of the Hopf bifurcation is derived with parameters at different points. Furthermore, under this condition, the stability and direction of bifurcation are determined by applying the normal form method and the center manifold theory. Finally, a kind of Runge-Kutta methods is given out to simulate the periodic solutions numerically. At last, some numerical experiments are given to match well with the main conclusion of this paper.

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