On Boundedness of Quadratic Dynamics with Energy-Preserving Nonlinearity

TL;DR

This paper investigates the boundedness of quadratic dynamical systems with energy-preserving nonlinearities, confirming the sufficiency of a Lyapunov-based condition in 2D and revealing its limitations in 3D through a counterexample.

Contribution

It provides an independent proof of the necessary condition's validity in 2D and demonstrates its failure in 3D, highlighting a gap in current boundedness criteria.

Findings

The sufficient Lyapunov condition is valid in 2D systems.

A counterexample shows the necessary condition fails in 3D.

The results identify a theoretical gap in boundedness analysis.

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.