Abstract Stability analysis is a crucial tool for studying dynamical systems in mathematics, physics, and engineering. Lyapunov functions provide a framework for determining the stability of these systems, but constructing them is often challenging for non-linear systems. This paper focuses on using sum-of-squares (SOS) methods to construct quadratic Lyapunov functions for proving the local stability of finite-dimensional systems. These systems are designed to share certain properties of shear flows and are therefore described by second-order polynomials. Additionally, the system matrix of the linear part is non-normal, which presents challenges in constructing a Lyapunov function. Two established SOS-based algorithms from the literature (SOS1 and SOS2) are compared based on the size of the provable region of attraction (ROA), computational time, and the maximum allowable degrees of freedom of the system. Furthermore, this paper introduces a novel modification to the SOS2 algorithm, termed SOS2m, which aims to provide a larger ROA than the original SOS2 method with minimal additional computational cost. This new SOS2m method is shown to strike an effective balance between accuracy and computational efficiency. The presented methods, while demonstrated on shear flow models, are extendable to other dynamical systems described by polynomial functions. A further motivation for this research is to advance analysis methods for investigating subcritical laminar-turbulent transition and to develop methodologies for estimating permissible perturbation levels. To demonstrate their effectiveness, the three algorithms (SOS1, SOS2, and SOS2m) are applied to a simplified model of laminar-turbulent transition and truncated finite-dimensional reduced-order models of Poiseuille and Couette flows.
Péter Nagy (2026) studied this question.