## The stability of special symmetric solutions of $$\dot x(t)=\alpha f(x(t- 1))$$ with small amplitudes.(English)Zbl 0704.34086

The differential delay equation $$\dot x(t)=\alpha f(x(t-1))$$ where f is an odd $$C^ 3$$-map with $$f'(0)<0$$ has a smooth primary branch of periodic solutions which bifurcates at $$\alpha^*=-\pi /(2f'(0))$$ 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 $$\alpha^*$$. It is shown that there is a $$C^ 1$$- map $$\lambda$$ : ]$$\alpha$$ $${}^*-\epsilon,\alpha^*+\epsilon [\to {\mathbb{R}}$$ with $$\lambda (\alpha^*)=1$$ such that $$\lambda$$ ($$\alpha$$) is the dominating eigenvalue of the linearized Poincaré map of ($$\alpha$$ f). Furthermore $$\lambda '(\alpha^*)$$ is calculated and shown that $$\lambda '(\alpha^*)<0$$. Thus bifurcation to the right $$(\alpha >\alpha^*)$$ decreases $$\lambda$$ ($$\alpha$$) and yields stable solutions, while backward bifurcation gives unstable solutions. The direction of the bifurcation depends on the sign of $$f'''(0)$$.
Reviewer: P.Dormayer

### MSC:

 34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument) 34C25 Periodic solutions to ordinary differential equations 34K20 Stability theory of functional-differential equations 34D20 Stability of solutions to ordinary differential equations

### Keywords:

differential delay equation; Poincaré map; bifurcation
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### References:

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