On long-term boundedness of Galerkin models

Abstract

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.