abstr5
We study the equation
x'(t) = g(x(t-1)) (g)
for smooth functions g: R ---> R satisfying
x g(x )< 0 for x < > 0,
and the equation
x'(t) = b(t)x(t-1) (b)
with a periodic coefficient b taking negative values.
Eq. (b) generalizes variational equations along periodic solutions y of
eq. (g) in case g'(x) < 0 for all x in y(R).
We investigate the largest Floquet multipliers of eq. (b) and derive
a characterization of vectors transversal to stable manifolds
of Poincaré maps associated with slowly oscillating periodic solutions
of eq. (g). The criterion is used in part II of the paper in order
to find g and y so that a Poincaré map has a transversal homoclinic trajectory,
and a hyperbolic set on which the dynamics are chaotic.