Determination of Stability Domains for Nonlinear Dynamical Systems Using the Weighted Residuals Method

Finding a suitable estimation of stability domain around stable equilibrium points is an important issue in the study of nonlinear dynamical systems. This paper intends to apply a set of analytical-numerical methods to estimate the region of attraction for autonomous nonlinear systems. In mechanical and structural engineering, autonomous systems could be found in large deformation problems or control of structures. In order to have an appropriate estimation of stability domain, some suitable Lyapunov functions are calculated by satisfying the modified Zubov's partial differential equation in a finite area around the asymptotically stable equilibrium point. To achieve this, the techniques of Collocation, Galerkin, Least squares, Moments and Sub-domain are applied. Furthermore, a number of numerical examples are solved by the suggested techniques and Zubov's construction procedure. In most cases, the proposed approaches compared with Zubov’s scheme give a better estimation stability domain.