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


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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[1] Dormayer, P., Exact formulae for periodic solutions of \(ẋ(t + 1) = α(-x(t) + bx\^{}\{3\}(t))\), Z. angew. math. phys., 37, 765-775, (1986) · Zbl 0631.34080
[2] Dormayer, P., Smooth bifurcation of symmetric periodic solutions of functional differential equations, J. diff. eqns, 82, 109-155, (1989) · Zbl 0694.34058
[3] Hadeler, K.P., Effective computation of periodic orbits and bifurcation diagrams in delay equations, Num. math., 34, 457-467, (1980) · Zbl 0419.34070
[4] Kaplan, J.L.; Yorke, J.A., Ordinary differential equations which yield periodic solutions of differential delay equations, J. math. analysis applic., 48, 317-324, (1974) · Zbl 0293.34102
[5] Walther, H.Q., A theorem on the amplitudes of periodic solutions of differential delay equations with application to bifurcation, J. diff. eqns, 29, 394-404, (1978)
[6] Walther, H.Q., Bifurcation from periodic solutions in functional differential equations, Math. Z., 182, 269-289, (1983) · Zbl 0488.34066
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