Normal Form of Saddle-Node-Hopf Bifurcation in Retarded Functional Differential Equations and Applications

Jiang, Heping; Jiang, Jiao; Song, Yongli

doi:10.1142/s0218127416500401

Cited by 6 publications

(7 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In order to calculate the value of the coefficient B i𝑗 , i, 𝑗 = 1, 2, 3, 4, 5, we introduce Jiang et al [34], where…”

Section: 𝛼+𝛾𝑦(T−1)+𝑦(t−1) 2 − Bz(t) Andmentioning

confidence: 99%

Delayed nonmonotonic immune response in HIV infection system

Wang,

Wang

2024

Math Methods in App Sciences

View full text Add to dashboard Cite

In this paper, we study a delayed HIV infection model with nonmonotonic immune response and perform stability and bifurcation analysis. Our results show that the delayed HIV infection system with nonmonotonic immune response has bistability and stable periodic solution appear. We find that both the uninfected and immune‐free equilibria are globally asymptotically stable under certain conditions which are not affected by time delay. However, the time delay makes one immune equilibrium always unstable for and also makes another immune equilibrium appear stability switches; meanwhile, the system will exhibit local Hopf bifurcation, global Hopf bifurcation, and saddle‐node‐Hopf bifurcation. Numerical simulations are carried out to verify our results.

show abstract

“…In order to calculate the value of the coefficient B i𝑗 , i, 𝑗 = 1, 2, 3, 4, 5, we introduce Jiang et al [34], where…”

Section: 𝛼+𝛾𝑦(T−1)+𝑦(t−1) 2 − Bz(t) Andmentioning

confidence: 99%

Delayed nonmonotonic immune response in HIV infection system

Wang,

Wang

2024

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…Delay differential equations exhibit very richer bifurcation phenomena, including Hopf bifurcation (Hale and Verduyn Lunel [62], Hassard et al [63], Faria and Magalhães [49], Guo and Wu [60]), Bautin bifurcation (Bi and Ruan [13], Ion [66]), Bogdanov-Takens bifurcation (Faria and Magalhães [50], Xiao and Ruan [125]), Fold-Hopf (zero-Hopf) bifurcation (Choi and LeBlanc [27], Guo et al [59], Jiang et al [67], Jiang and Wang [68], Wu and Wang [122]), Hopf-Hopf bifurcation (Campbell and Belair [25], Bruno and Bélair [21], Wu and Wang [124]), and triple zero singularities (Campbell and Yuan [26], LeBlanc [77]). In applications it is usually interesting but difficult to show that a specific biological or physical model undergoes any of these bifurcations in particular the degenerate ones.…”

Section: Tumor Cells T(t)mentioning

confidence: 99%

Nonlinear dynamics in tumor-immune system interaction models with delays

Ruan¹

2021

Discrete &Amp; Continuous Dynamical Systems - B

View full text Add to dashboard Cite

In this paper, we review some recent results on the nonlinear dynamics of delayed differential equation models describing the interaction between tumor cells and effector cells of the immune system, in which the delays represent times necessary for molecule production, proliferation, differentiation of cells, transport, etc. First we consider a tumor-immune system interaction model with a single delay and present results on the existence and local stability of equilibria as well as the existence of Hopf bifurcation in the model when the delay varies. Second we investigate a tumor-immune system interaction model with two delays and show that the model undergoes various possible bifurcations including Hopf, Bautin, Fold-Hopf (zero-Hopf), and Hopf-Hopf bifurcations. Finally we discuss a tumor-immune system interaction model with three delays and demonstrate that the model exhibits more complex behaviors including chaos. Numerical simulations are provided to illustrate the nonlinear dynamics of the delayed tumor-immune system interaction models. More interesting issues and questions on modeling and analyzing tumor-immune dynamics are given in the discussion section.

show abstract

“…In this section, we use the center manifold theory and normal form method [6,23] to study Hopf-zero bifurcations. The normal form of a Hopf-zero bifurcation for a general delay-differential equations has been given in the following two papers: one is for a saddlenode-Hopf bifurcation [11], and the other is for a steady-state Hopf bifurcation [20].…”

Section: Normal Form For Hopf-zero Bifurcationmentioning

confidence: 99%

Hopf-zero bifurcation of Oregonator oscillator with delay

2018

View full text Add to dashboard Cite

In this paper, we study the Hopf-zero bifurcation of Oregonator oscillator with delay. The interaction coefficient and time delay are taken as two bifurcation parameters. Firstly, we get the normal form by performing a center manifold reduction and using the normal form theory developed by Faria and Magalhães. Secondly, we obtain a critical value to predict the bifurcation diagrams and phase portraits. Under some conditions, saddle-node bifurcation and pitchfork bifurcation occur along M and N, respectively; Hopf bifurcation and heteroclinic bifurcation occur along H and S, respectively. Finally, we use numerical simulations to support theoretical analysis.

show abstract

Normal Form of Saddle-Node-Hopf Bifurcation in Retarded Functional Differential Equations and Applications

Cited by 6 publications

References 31 publications

Delayed nonmonotonic immune response in HIV infection system

Delayed nonmonotonic immune response in HIV infection system

Nonlinear dynamics in tumor-immune system interaction models with delays

Hopf-zero bifurcation of Oregonator oscillator with delay

Contact Info

Product

Resources

About