LAUSR

We consider a periodic function $$p:{\mathbb {R}}\rightarrow {\mathbb {R}}$$ of minimal period 4 which satisfies a family of delay differential equations 0.1 $$\begin{aligned} x'(t)=g(x(t-d_{\Delta }(x_t))),\quad \Delta \in {\mathbb {R}}, \end{aligned}$$ with a continuously differentiable function $$g:{\mathbb {R}}\rightarrow {\mathbb {R}}$$ and delay functionals $$\begin{aligned} d_{\Delta }:C([-2,0],{\mathbb {R}})\rightarrow (0,2). \end{aligned}$$ The solution segment $$x_t$$ in Eq. (0.1) is given by $$x_t(s)=x(t+s)$$ . For every $$\Delta \in {\mathbb {R}}$$ the solutions of Eq. (0.1) defines a semiflow of continuously differentiable solution operators $$S_{\Delta ,t}:x_0\mapsto x_t$$ , $$t\ge 0$$ , on a continuously differentiable submanifold $$X_{\Delta }$$ of the space $$C^1([-2,0],{\mathbb {R}})$$ , with codim $$X_{\Delta }=1$$ . At $$\Delta =0$$ the delay is constant, $$d_0(\phi )=1$$ everywhere, and the orbit $${{\mathcal {O}}}=\{p_t:0\le t<4\}\subset X_0$$ of the periodic solution is extremely stable in the sense that the spectrum of the monodromy operator $$M_0=DS_{0,4}(p_0)$$ is $$\sigma _0=\{0,1\}$$ , with the eigenvalue 1 being simple. For $$|\Delta |\nearrow \infty $$ there is an increasing contribution of variable, state-dependent delay to the time lag $$d_{\Delta }(x_t)=1+\cdots $$ in Eq. (0.1). We study how the spectrum $$\sigma _{\Delta }$$ of $$M_{\Delta }=DS_{\Delta ,4}(p_0)$$ changes if $$|\Delta |$$ grows from 0 to $$\infty $$ . A main result is that at $$\Delta =0$$ an eigenvalue $$\Lambda (\Delta )<0$$ of $$M_{\Delta }$$ bifurcates from $$0\in \sigma _0$$ and decreases to $$-\infty $$ as $$|\Delta |\nearrow \infty $$ . Moreover we verify the spectral hypotheses for a period doubling bifurcation from the periodic orbit $${{\mathcal {O}}}$$ at the critical parameter $$\Delta _{*}$$ where $$\Lambda (\Delta _{*})=-1$$ .