  We investigate linear-quadratic dynamical systems with energy preserving
quadratic terms. These systems arise for instance as Galerkin systems of
incompressible flows. A criterion is presented to ensure long-term boundedness
of the system dynamics. If the criterion is violated, a globally stable
attractor cannot exist for an effective nonlinearity. Thus, the criterion
represents a minimum requirement for a physically justified Galerkin model of
viscous fluid flows. The criterion is exemplified e.g. for Galerkin systems of
two-dimensional cylinder wake flow models in the transient and the
post-transient regime, the Lorenz system, and for physical design of a
Trefethen-Reddy Galerkin system. There are numerous potential applications of
the criterion, for instance system reduction and control of strongly nonlinear
dynamical systems.
