Hopf Bifurcation for Implicit Neutral Functional Differential Equations

1993 ◽  
Vol 36 (3) ◽  
pp. 286-295 ◽  
Author(s):  
Tomasz Kaczynski ◽  
Huaxing Xia
Keyword(s):  
Hopf Bifurcation ◽  
Differential Equations ◽  
Functional Differential Equations ◽  
Neutral Functional Differential Equations ◽  
Bifurcation Theorem ◽  
Functional Differential

AbstractAn analog of the Hopf bifurcation theorem is proved for implicit neutral functional differential equations of the form F(xt, D′(xt, α), α) = 0. The proof is based on the method of S1-degree of convex-valued mappings. Examples illustrating the theorem are provided.

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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