We consider a periodic function \(p:{\mathbb {R}}\rightarrow {\mathbb {R}}\) of minimal period 4 which satisfies a family of delay differential equations

with a continuously differentiable function \(g:{\mathbb {R}}\rightarrow {\mathbb {R}}\) and delay functionals

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 \(\Lambda (\Delta _{*})=-1\).

我们考虑最小周期4的周期函数\（p：{\ mathbb {R}} \ rightarrow {\ mathbb {R}} \），它满足一族时滞微分方程

具有连续微分函数\（g：{\ mathbb {R}} \ rightarrow {\ mathbb {R}} \）和延迟函数

方程中的解段\（x_t \）。（0.1）由\（x_t（s）= x（t + s）\）给出。对于每个\（\ Delta \ in {\ mathbb {R}} \），方程式的解。（0.1）定义了一个连续可微子流形\（X _ {\ Delta} \上的一个连续可微解算子\（S _ {\ Delta，t}：x_0 \ mapsto x_t \），\（t \ ge 0 \）的半流程）的空间\（C ^ 1（[-2,0]，{\ mathbb {R}}）\），并带有codim \ {X _ {\ Delta} = 1 \）。在\（\ Delta = 0 \）处，延迟是恒定的，各处都是\（d_0（\ phi）= 1 \），并且轨道\（{{\ mathcal {O}}} = \ {p_t：0 \ le t <4 \} \子集X_0 \）周期性溶液是在这个意义上极其稳定，所述单值算子的谱\（M_0 = DS_ {0,4}（P_0）\）是\（\西格玛_0 = \ {0,1 \} \） ，用特征值1很简单。对于\（| \ Delta | \ nearrow \ infty \），方程中时间滞后\（d _ {\ Delta}（x_t）= 1 + \ cdots \）的变量变化，与状态有关的延迟增加。（0.1）。我们研究\（| \ Delta | \）从0增长时\（M _ {\ Delta} = DS _ {\ Delta，4}（p_0）\）的频谱\（\ sigma _ {\ Delta} \）的变化。到\（\ infty \）。主要结果是在\（\ Delta = 0 \）处的特征值\（\ Lambda（\ Delta）<0 \）为\（M _ {\ Delta} \）从\（0 \ in \ sigma _0 \）分叉，并减小为\（-\ infty \）为\（| \ Delta | \ nearrow \ infty \）。此外，我们验证频谱假设一段来自周期轨道分岔\（{{\ mathcal {ö}}} \）在关键参数\（\德尔塔_ {*} \） ，其中\（\拉姆达（\德尔塔_ {*}）=-1 \）。