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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
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