scholarly journals Stable periodic solutions for delay equations with positive feedback - a computer-assisted proof

2006 ◽  
Vol 14 (4) ◽  
pp. 721-736 ◽  
Author(s):  
A. Aschwanden ◽  
◽  
A. Schulze-Halberg ◽  
D. Stoffer
Keyword(s):  
Periodic Solutions ◽  
Positive Feedback ◽  
Delay Equations ◽  
Computer Assisted ◽  
Computer Assisted Proof ◽  
Stable Periodic

scholarly journals Determinants and periodic solutions of delay equations

2005 ◽  
Vol 411 ◽  
pp. 356-363
Author(s):  
M.C. Crabb ◽  
A.J.B. Potter
Keyword(s):  
Periodic Solutions ◽  
Delay Equations

Hopf Bifurcation Induced by Delay Effect in a Diffusive Tumor-Immune System

2018 ◽  
Vol 28 (11) ◽  
pp. 1850136 ◽  
Author(s):  
Ben Niu ◽  
Yuxiao Guo ◽  
Yanfei Du
Keyword(s):  
Immune System ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Center Manifold ◽  
Immune Cell ◽  
Tumor Treatment ◽  
Delay Effect ◽  
Stable Periodic ◽  
The Stability ◽  
Bifurcation And Stability

Tumor-immune interaction plays an important role in the tumor treatment. We analyze the stability of steady states in a diffusive tumor-immune model with response and proliferation delay [Formula: see text] of immune system where the immune cell has a probability [Formula: see text] in killing tumor cells. We find increasing time delay [Formula: see text] destabilizes the positive steady state and induces Hopf bifurcations. The criticality of Hopf bifurcation is investigated by deriving normal forms on the center manifold, then the direction of bifurcation and stability of bifurcating periodic solutions are determined. Using a group of parameters to simulate the system, stable periodic solutions are found near the Hopf bifurcation. The effect of killing probability [Formula: see text] on Hopf bifurcation values is also discussed.

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