# zbMATH — the first resource for mathematics

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.

##### 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
Full Text: