Equações diferenciais, uma comparação entre regiões de estabilidade obtidas pelo metodo dos multiplicadores invariantes e pelo Teorema de Krasovskii

Abstract

Lyapunov stability theory is a fundamental tool in the qualitative analysis of nonlinear dynamical systems, allowing for the determination of solution behavior near equilibrium points without explicitly solving the equations. This work aims to study and compare two distinct methods for constructing Lyapunov functionals and analyzing stability: the Method of Invariant Multipliers and Krasovskii’s Theorem. The effectiveness of each approach is evaluated by applying them to three classic systems of physical and biological relevance: the Nonlinear Damped Mass-Spring System, the Lotka-Volterra Predation Model, and the Damped Pendulum Equation. For the mechanical systems, the Method of Invariant Multipliers proved to be superior, enabling the intuitive construction of energy functionals that demonstrate asymptotic stability and, in the case of the inverted pendulum, instability. Conversely, Krasovskii’s Theorem proved inconclusive for these systems due to the structure of their Jacobian matrices, which result in null principal minors. In the conservative Lotka-Volterra system, both methods failed to demonstrate asymptotic stability, corroborating the center nature of the equilibrium point. In this academic work, it is concluded that the Method of Multipliers is more robust for physical systems, while Krasovskii’s criterion presents significant restrictions that limits their applicability. Although the studies of more systems are necessary for a better conclusion.