Global bifurcation of oscillatory periodic solutions of delay differential equations (Q1284667)

This work provides a global bifurcation result for periodic solutions to the following general class of delay differential equations \[ \varepsilon\cdot\dot{x}(t)=f(x_t) , t>0,\tag{1} \] where \( \varepsilon >0\) is a small parameter, \(f:C([-M,0];\mathbb{R}^1)\to \mathbb{R}^1\) is a continuous function, \(M > 0\) is a real number. For every function \(y:\mathbb{R}^1 \to \mathbb{R}^1\) the ``translates'' \(y_t:[-M,0] \to \mathbb{R}^1\) are defined by \(y_t(s)=y(t+s)\), where \(-M \leq s \leq 0\). Equation (1) includes equations with constant delay (e.g. \(f(\phi)=g(\phi(-M))\), \(\phi\)\(\in\)\( C([-M;0];\mathbb{R}^1)\) , for some \(g:\mathbb{R}^1 \to \mathbb{R}^1\)) or state dependent delays (e.g. \(f(\phi)=g(\phi(-r(\phi(0))))\), for some \(r:\mathbb{R}^1\to[0;\infty)\)).