scholarly journals Normal Forms for Retarded Functional Differential Equations with Parameters and Applications to Hopf Bifurcation

1995 ◽  
Vol 122 (2) ◽  
pp. 181-200 ◽  
Author(s):  
T. Faria ◽  
L.T. Magalhaes
Keyword(s):  
Hopf Bifurcation ◽  
Differential Equations ◽  
Normal Forms ◽  
Functional Differential Equations ◽  
Retarded Functional Differential Equations ◽  
Functional Differential

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.

Normal forms for periodic retarded functional differential equations

1997 ◽  
Vol 127 (1) ◽  
pp. 21-46 ◽  
Author(s):  
Teresa Faria
Keyword(s):  
Differential Equations ◽  
Normal Forms ◽  
Linear Part ◽  
Functional Differential Equations ◽  
Retarded Functional Differential Equations ◽  
Functional Differential

This paper addresses the computation of normal forms for periodic retarded functional differential equations (FDEs) with autonomous linear part. The analysis is based on the theory previously developed for autonomous retarded FDEs. Adequate nonresonance conditions are derived. As an illustration, the Bogdanov–Takens and the Hopf singularities are considered.

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.

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