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Hopf bifurcation calculations for scalar neutral delay differential equations. (English) Zbl 1116.34057
The paper studies Hopf bifurcations of scalar neutral delay differential equations of the form $$\dot x-a\dot x(t-\tau)=L(\gamma)x_t+f(x_t,\gamma),$$ where $\gamma$ is the bifurcation parameter and $\vert a\vert <1$. The characteristic equation has the form $$\lambda(1-\exp(-\lambda\tau))-b-c\exp(-\lambda\tau)=0.$$ First the paper presents the stability charts depending on the coefficients of the characteristic equation. Then the paper presents a sequence of formulas leading to the criticality coefficient determining the stability of the emerging periodic solutions in the center direction. The normal form computation is done with the technique introduced by {\it T. Faria} and {\it L. T. Magalh√£es} [J. Differ. Equ. 122, No. 2, 181--200 (1995; Zbl 0836.34068)]. Two examples are included. Both have a delay $\tau=1$ and a linear part of the form $\gamma x(t-1)$. The nonlinear parts are $\gamma x(t)x(t-1)$ and $\gamma x(t)x(t-1)^2$, respectively.

34K18Bifurcation theory of functional differential equations
34K40Neutral functional-differential equations
34K17Transformation and reduction of functional-differential equations and systems; normal forms
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