  In this paper, we study systems of time-invariant ordinary differential
equations whose flows are non-expansive with respect to a norm, meaning that
the distance between solutions may not increase. Since non-expansiveness (and
contractivity) are norm-dependent notions, the topology of $\omega$-limit sets
of solutions may depend on the norm. For example, and at least for systems
defined by real-analytic vector fields, the only possible $\omega$-limit sets
of systems that are non-expansive with respect to polyhedral norms (such as
$\ell^p$ norms with $p =1$ or $p=\infty$) are equilibria. In contrast, for
non-expansive systems with respect to Euclidean ($\ell^2$) norm, other limit
sets may arise (such as multi-dimensional tori): for example linear harmonic
oscillators are non-expansive (and even isometric) flows, yet have periodic
orbits as $\omega$-limit sets. This paper shows that the Euclidean linear case
is what can be expected in general: for flows that are contractive with respect
to any strictly convex norm (such as $\ell^p$ for any $p\not=1,\infty$), and if
there is at least one bounded solution, then the $\omega$-limit set of every
trajectory is also an omega limit set of a linear time-invariant system.
