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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.

34K13 Periodic solutions to functional-differential equations