Solutions of \(\dot x(t)=-g(x(t-1))\) approach the Kaplan-Yorke orbits for odd sigmoid \(g\). (English) Zbl 0823.34068

If \(g\) is an odd and differentiable function with positive derivative, then every bounded solution of (1) \(x'(t) = - g(x(t - 1))\) satisfies \(\lim_{t \to \infty} [x(t) + x(t - 2)] = 0\) and all periodic solutions of (1) are of Kaplan-Yorke type. Solutions of this type are oscillatory periodic solutions of period 4 and they satisfy \(x'(t) = - g(y(t))\), \(y'(t) = g(x(t))\) where \(y(t) = x(t - 1)\). Further the convergence of the trajectories of (1) to the orbit of a Kaplan-Yorke solution is discussed and the stability of Kaplan-Yorke solutions is investigated.


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
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