A Homoclinic Loop Generated by Variable Delay

Abstract

For $$\frac{\pi }{2}<\alpha <\frac{5\pi }{2}$$ the zero solution of the linear equation $$\begin{aligned} x'(t)=-\alpha x(t-1) \end{aligned}$$ is hyperbolic with two-dimensional unstable space. For $$\alpha $$ close to $$\frac{5\pi }{2}$$ we replace the constant delay $$1$$ by a state-dependent delay $$\begin{aligned} d_U:C\supset U\rightarrow (0,2), \end{aligned}$$ with $$d_U(\phi )=1$$ on a neighbourhood of $$0\in C=C([-2,0],{\mathbb {R}})$$ , in such a way that the nonlinear equation $$\begin{aligned} x'(t)=-\alpha x(t-d_U(x_t)) \end{aligned}$$ has a homoclinic solution $$x=h$$ , $$\begin{aligned} h(t)\rightarrow 0\quad \text {as}\quad |t|\rightarrow \infty , 0\ne h_t=h(t+\cdot )\in C. \end{aligned}$$ On the solution manifold $$\begin{aligned} X=\{\phi \in U\cap C^1([-2,0],{\mathbb {R}}):\phi '(0)=-\alpha \phi (-d_U(\phi ))\}. \end{aligned}$$ the nonlinear equation defines a semiflow whose flowline $$\begin{aligned} {\mathbb {R}}\ni t\mapsto h_t\in X \end{aligned}$$ is a minimal intersection of the unstable manifold at equilibrium with the local stable manifold.