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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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\textit{P. Dormayer}, Nonlinear Anal., Theory Methods Appl. 14, No. 8, 701--715 (1990; Zbl 0704.34086)

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### References:

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