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The stability of special symmetric solutions of x ˙(t)=αf(x(t-1)) with small amplitudes. (English) Zbl 0704.34086
The differential delay equation x ˙(t)=αf(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 α * =-π/(2f ' (0)) from the trivial solutions x0. The paper gives an analysis of the stability properties of these solutions for small amplitudes, i.e. near the bifurcation point α * . It is shown that there is a C 1 - map λ : ]α * -ϵ,α * +ϵ[ with λ(α * )=1 such that λ (α) is the dominating eigenvalue of the linearized Poincaré map of (α f). Furthermore λ ' (α * ) is calculated and shown that λ ' (α * )<0. Thus bifurcation to the right (α>α * ) decreases λ (α) 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:
34K99Functional-differential equations
34C25Periodic solutions of ODE
34K20Stability theory of functional-differential equations
34D20Stability of ODE