  We study systems with periodically oscillating parameters that can give way
to complex periodic or non periodic orbits. Performing the long time limit, we
can define ergodic averages such as Lyapunov exponents, where a negative
maximal Lyapunov exponent corresponds to a stable periodic orbit. By this,
extremely complicated periodic orbits composed of contracting and expanding
phases appear in a natural way. Employing the technique of $\epsilon$-uncertain
points, we find that values of the control parameters supporting such periodic
motion are densely embedded in a set of values for which the motion is chaotic.
When a tiny amount of noise is coupled to the system, dynamics with positive
and with negative non-trivial Lyapunov exponents are indistinguishable. We
discuss two physical systems, an oscillatory flow inside a duct and a dripping
faucet with variable water supply, where such a mechanism seems to be
responsible for a complicated alternation of laminar and turbulent phases.
