On Boundedness of Quadratic Dynamics with Energy-Preserving Nonlinearity

TL;DR

The paper revisits boundedness for quadratic dynamics with an energy-preserving nonlinearity and analyzes the Schlegel-Noack conditions via a trapping-region SDP. An independent proof shows that the necessary condition for boundedness holds in two dimensions, while a three-dimensional counterexample demonstrates its failure in higher dimensions, revealing a theoretical gap. The results indicate a crucial role for system dimension in boundedness analysis and motivate the development of less conservative, dimension-aware criteria for energy-preserving quadratic models. Overall, the work clarifies the limitations of current necessary conditions and points toward future refinement of boundedness tests.

Abstract

Boundedness is an important property of many physical systems. This includes incompressible fluid flows, which are often modeled by quadratic dynamics with an energy-preserving nonlinearity. For such systems, Schlegel and Noack proposed a sufficient condition for boundedness utilizing quadratic Lyapunov functions. They also propose a necessary condition for boundedness aiming to provide a more complete characterization of boundedness in this class of models. The sufficient condition is based on Lyapunov theory and is true. Our paper focuses on this necessary condition. We use an independent proof to show that the condition is true for two dimensional systems. However, we provide a three dimensional counterexample to illustrate that the necessary condition fails to hold in higher dimensions. Our results highlight a theoretical gap in boundedness analysis and suggest future directions to address the conservatism.