## Differential and Integral Equations

### 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.

#### Article information

Source
Differential Integral Equations, Volume 8, Number 6 (1995), 1407-1452.

Dates
First available in Project Euclid: 15 May 2013

https://projecteuclid.org/euclid.die/1368638174

Mathematical Reviews number (MathSciNet)
MR1329849

Zentralblatt MATH identifier
0827.34059

Subjects
Primary: 58F13
Secondary: 34K15 39B12: Iteration theory, iterative and composite equations [See also 26A18, 30D05, 37-XX]

#### Citation

Lani-Wayda, Bernhard; Walther, Hans-Otto. Chaotic motion generated by delayed negative feedback. I. A transversality criterion. Differential Integral Equations 8 (1995), no. 6, 1407--1452. https://projecteuclid.org/euclid.die/1368638174