abstr7
We give an example of a smooth function
g: R ---> Rwith only one extremum, with
sign g(x) = - \sign g(-x) for x < > 0,
and the following properties:
The delay equation x'(t) = g(x(t-1)) has an
unstable periodic solution and a solution with phase curve
transversally homoclinic to the orbit of the periodic solution.
The complicated motion arising from this structure, and its robustness
under perturbation of g, are described in terms of a
Poincaré map. The example is minimal in the sense
that the condition g' < 0 (under which there would be no extremum)
excludes complex solution behavior.
Based on numerical observations, we discuss the role of the
erratic solutions in the set of all solutions.