Construction of Lyapunov functions from first integrals for stochastic stability analysis

Abstract

This paper explores the application of first integrals in constructing Lyapunov functions for stability analysis of dynamical systems in stochastic domains. A key advantage of using first integrals is their ability to embed system-specific structural and physical information, distinguishing the resulting Lyapunov functions from generic positive definite functions with no intrinsic connection to the system. However, since first integrals do not inherently satisfy Lyapunov conditions, additional constraints—often with direct physical interpretations—must be introduced to ensure positive definiteness and suitable monotonic behavior. The method is demonstrated on three mechanical systems subjected to parametric noise: a nonlinear aeroelastic single-degree-of-freedom oscillator, a spherical pendulum with two first integrals, and a gyroscope with three first integrals.