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Stable periodic motion of a system with state dependent delay. (English) Zbl 1034.34085
A system of functional-differential equations of the form $\frac{dx(t)}{dt}=v(\frac{c}2s(t-r)-w), \quad s(t)=\frac1c(| x(t-s(t))+w| +| x(t)+w| ),$ is considered, where $$c,w,r$$ are positive parameters and the response function $$v: {\mathbb R}\to {\mathbb R}$$ satisfies the negative feedback condition $$x\cdot v(x)<0, x\neq0$$, and is bounded, $$| v(x)| \leq b\; \forall x \in{\mathbb R}.\,$$ The existence and stability of a slowly oscillating periodic solution is derived for nonlinearities $$v$$ close to the step function $$-a\, sign (x), a\leq b$$, outside a small neighborhood $$(-\beta,\beta)$$ of $$0$$.$$\,$$ The result applies to examples with $$v(x)=-\arctan (\gamma x)$$ or $$-\tanh (\gamma x)$$, where $$\gamma>0$$ is large.

##### MSC:
 34K13 Periodic solutions to functional-differential equations