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The stability of special symmetric solutions of $\stackrel{˙}{x}\left(t\right)=\alpha f\left(x\left(t-1\right)\right)$ with small amplitudes. (English) Zbl 0704.34086
The differential delay equation $\stackrel{˙}{x}\left(t\right)=\alpha f\left(x\left(t-1\right)\right)$ where f is an odd ${C}^{3}$-map with ${f}^{\text{'}}\left(0\right)<0$ has a smooth primary branch of periodic solutions which bifurcates at ${\alpha }^{*}=-\pi /\left(2{f}^{\text{'}}\left(0\right)\right)$ 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 ${\alpha }^{*}$. It is shown that there is a ${C}^{1}$- map $\lambda$ : ]$\alpha$ ${}^{*}-ϵ,{\alpha }^{*}+ϵ\left[\to ℝ$ with $\lambda \left({\alpha }^{*}\right)=1$ such that $\lambda$ ($\alpha$) is the dominating eigenvalue of the linearized Poincaré map of ($\alpha$ f). Furthermore ${\lambda }^{\text{'}}\left({\alpha }^{*}\right)$ is calculated and shown that ${\lambda }^{\text{'}}\left({\alpha }^{*}\right)<0$. Thus bifurcation to the right $\left(\alpha >{\alpha }^{*}\right)$ decreases $\lambda$ ($\alpha$) and yields stable solutions, while backward bifurcation gives unstable solutions. The direction of the bifurcation depends on the sign of ${f}^{\text{'}\text{'}\text{'}}\left(0\right)$.
Reviewer: P.Dormayer
##### MSC:
 34K99 Functional-differential equations 34C25 Periodic solutions of ODE 34K20 Stability theory of functional-differential equations 34D20 Stability of ODE
##### Keywords:
differential delay equation; Poincaré map; bifurcation