Chaotic motion generated by delayed negative feedback. I. A transversality criterion

Abstract

We study the equation $$ x'(t) = g(x(t-1)) \tag{$g$} $$ for smooth functions $g:\mathbb{R} \rightarrow \mathbb{R} $ satisfying $ \xi g(\xi)<0 $ for $ \xi \neq 0, $ and the equation $$ x'(t) = b(t)x(t-1) \tag {$b$} $$ with a periodic coefficient $b:\mathbb{R} \rightarrow (-\infty,0)$. Equation $(b)$ generalizes variational equations along periodic solutions $y$ of equation $(g)$ in case $g'(\xi) < 0$ for all $\xi \in y(\mathbb{R} )$. We investigate the largest Floquet multipliers of equation $(b)$ and derive a characterization of vectors transversal to stable manifolds of Poincar\'e maps associated with slowly oscillating periodic solutions of equation $(g)$. The criterion is used in Part II of the paper in order to find $g$ and $y$ so that a Poincar\'e map has a transversal homoclinic trajectory, and a hyperbolic set on which the dynamics are chaotic.