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Zbl 0704.34086
Dormayer, P.
The stability of special symmetric solutions of $\dot x(t)=\alpha f(x(t- 1))$ with small amplitudes. (English)
[J] Nonlinear Anal., Theory Methods Appl. 14, No.8, 701-715 (1990). ISSN 0362-546X

The differential delay equation $\dot x(t)=\alpha f(x(t-1))$ where f is an odd $C\sp 3$-map with $f'(0)<0$ has a smooth primary branch of periodic solutions which bifurcates at $\alpha\sp*=-\pi /(2f'(0))$ from the trivial solutions $x\equiv 0$. The paper gives an analysis of the stability properties of these solutions for small amplitudes, i.e. near the bifurcation point $\alpha\sp*$. It is shown that there is a $C\sp 1$- map $\lambda$ : ]$\alpha$ ${}\sp*-\epsilon,\alpha\sp*+\epsilon [\to {\bbfR}$ with $\lambda (\alpha\sp*)=1$ such that $\lambda$ ($\alpha$) is the dominating eigenvalue of the linearized Poincaré map of ($\alpha$ f). Furthermore $\lambda '(\alpha\sp*)$ is calculated and shown that $\lambda '(\alpha\sp*)<0$. Thus bifurcation to the right $(\alpha >\alpha\sp*)$ decreases $\lambda$ ($\alpha$) and yields stable solutions, while backward bifurcation gives unstable solutions. The direction of the bifurcation depends on the sign of $f'''(0)$.
[P.Dormayer]

MSC 2000:: *34K99 Functional-differential equations
34C25 Periodic solutions of ODE
34K20 Stability theory of functional-differential equations
34D20 Lyapunov stability of ODE

Keywords: differential delay equation; Poincaré map; bifurcation

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Zentralblatt MATH Berlin [Germany]