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

2016 ◽  
Vol 26 (03) ◽  
pp. 1650040 ◽  
Author(s):  
Heping Jiang ◽  
Jiao Jiang ◽  
Yongli Song
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Differential Equations ◽  
Functional Differential Equations ◽  
Normal Form Theory ◽  
Hopf Bifurcation Point ◽  
Predator Prey Model ◽  
Saddle Node ◽  
Retarded Functional Differential Equations ◽  
Functional Differential

In this paper, we firstly employ the normal form theory of delayed differential equations according to Faria and Magalhães to derive the normal form of saddle-node-Hopf bifurcation for the general retarded functional differential equations. Then, the dynamical behaviors of a Leslie–Gower predator–prey model with time delay and nonmonotonic functional response are considered. Specially, the dynamical classification near the saddle-node-Hopf bifurcation point is investigated by using the normal form and the center manifold approaches. Finally, the numerical simulations are employed to support the theoretical results.

STABILITY AND HOPF BIFURCATION FOR AN ECO-EPIDEMIOLOGY MODEL WITH HOLLING-III FUNCTIONAL RESPONSE AND DELAYS

2008 ◽  
Vol 01 (03) ◽  
pp. 377-389 ◽  
Author(s):  
WEI ZOU ◽  
JIEHUA XIE ◽  
ZUOLIANG XIONG
Keyword(s):  
Hopf Bifurcation ◽  
Functional Differential Equations ◽  
Positive Equilibrium ◽  
Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Retarded Functional Differential Equations ◽  
Boundary Equilibrium ◽  
Functional Differential ◽  
Holling Iii Functional Response

In this paper, a system of retarded functional differential equations is proposed as a predator-prey model with disease in the prey. The invariance of non-negativity, nature of boundary equilibrium and global stability are analyzed. It also shows that positive equilibrium is locally asymptotically stable when time delay τ = τ1 + τ2 is suitable small, while a loss of stability by a Hopf bifurcation can occur around the positive equilibrium as the delays increase.

scholarly journals Stability and Bifurcation for a Delayed Diffusive Two-Zooplankton One-Phytoplankton Model with Two Different Functions

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-24
Author(s):  
Xin-You Meng ◽  
Li Xiao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Normal Form ◽  
Center Manifold ◽  
Functional Differential Equations ◽  
Bifurcation Parameter ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Functional Differential

In this paper, a diffusion two-phytoplankton one-zooplankton model with time delay, Beddington–DeAnglis functional response, and Holling II functional response is proposed. First, the existence and local stability of all equilibria of such model are studied. Then, the existence of Hopf bifurcation of the corresponding model without diffusion is given by taking time delay as the bifurcation parameter. Next, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are investigated by using the normal form theory and center manifold theorem. Furthermore, due to the local bifurcation theory of partial functional differential equations, Hopf bifurcation of the model is investigated by considering time delay as the bifurcation parameter. The explicit formulas to determine the properties of Hopf bifurcation are given by the method of the normal form theory and center manifold theorem for partial functional differential equations. Finally, some numerical simulations are performed to check out our theoretical results.

Hopf analysis of a differential-algebraic predator–prey model with Allee effect and time delay

2015 ◽  
Vol 08 (03) ◽  
pp. 1550041 ◽  
Author(s):  
Xue Zhang ◽  
Qingling Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Normal Form ◽  
Allee Effect ◽  
Functional Differential Equations ◽  
Local Analysis ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Predator Prey Model ◽  
Predator Model

A differential-algebraic prey–predator model with time delay and Allee effect on the growth of the prey population is investigated. Using differential-algebraic system theory, we transform the prey–predator model into its normal form and study its dynamics in terms of local analysis and Hopf bifurcation. By analyzing the associated characteristic equation, it is observed that the model undergoes a Hopf bifurcation at some critical value of time delay. In particular, we study the direction of Hopf bifurcation and the stability of bifurcated periodic solutions, and an explicit algorithm is given by applying the normal form theory and the center manifold reduction for functional differential equations. Finally, numerical simulations supporting the theoretical analysis are also included.

Normal Forms for Partial Neutral Functional Differential Equations with Applications to Diffusive Lossless Transmission Line

2020 ◽  
Vol 30 (02) ◽  
pp. 2050028 ◽  
Author(s):  
Chuncheng Wang
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Differential Equations ◽  
Transmission Line ◽  
Normal Forms ◽  
Functional Differential Equations ◽  
Neutral Functional Differential Equations ◽  
Line Equation ◽  
Bifurcation Problems ◽  
Functional Differential

A class of partial neutral functional differential equations are considered. For the linearized equation, the semigroup properties and formal adjoint theory are established. Based on these results, we develop two algorithms of normal form computation for the nonlinear equation, and then use them to study Hopf bifurcation problems of such equations. In particular, it is shown that the normal forms, derived from these two different approaches, for the Hopf bifurcation are exactly the same. As an illustration, the diffusive lossless transmission line equation where a Hopf singularity occurs is studied.

scholarly journals Equivariant Hopf Bifurcation in a Ring of Identical Cells with Delay

2009 ◽  
Vol 2009 ◽  
pp. 1-34 ◽  
Author(s):  
Dejun Fan ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Functional Differential Equations ◽  
Hopf Bifurcations ◽  
Normal Form Theory ◽  
Form Theory ◽  
Functional Differential ◽  
Delay Differential ◽  
Equivariant Hopf Bifurcation

A kind of delay neural network withnelements is considered. By analyzing the distribution of the eigenvalues, a bifurcation set is given in an appropriate parameter space. Then by using the theory of equivariant Hopf bifurcations of ordinary differential equations due to Golubitsky et al. (1988) and delay differential equations due to Wu (1998), and combining the normal form theory of functional differential equations due to Faria and Magalhaes (1995), the equivariant Hopf bifurcation is completely analyzed.

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