Functional equations satisfied by families of linear bounded
operators related to Cauchy problems imply that the trajectories of the
families exhibit surprising regularity properties. It turns out that
these trajectories tend to be the more regular the higher the order of
the Cauchy problem is. The talk is devoted to differences between
trajectories of semigroups of operators (related to the first order Cauchy
problem) and those of cosine operator families (related to the second order
Cauchy problem). In particular we show that in contrast to the rich theory
of singular perturbations of semigroups, there are no objects to study in
the theory of singular perturbations of cosine families. Similarly, while
there is plenty of examples of asymptotically stable semigroups
(i.e., semigroups {S(t),t≥0} for which the limit
lim_{t→∞}S(t) exists in the strong topology), there is only one
cosine family with this property, and this is the trivial family with all
operators being the identity operator.