New method to examine the stability of equilibrium points for a class of nonlinear dynamical systems

The Lyapanov’s linearization method (Lyapanov’s indirect method) fails to determine the stability of the equilibrium points for nonlinear systems if the linearized system is marginally stable. This problem makes a big restriction in the application of the theorem. In this study, we propose a new theorem to overcome this problem for a large class of nonlinear systems that are restricted to autonomous systems with continuously differentiable vector field. The theorem is proved based on classical proof of the Lyapanov’s linearization method, but according to this theorem the Jacobean matrix is evaluated in a region around the equilibrium point rather than the point itself. The asymptotic stability and/or the instability of the equilibrium point is determined by evaluating the eigenvalues of the Jacobean matrix in this region. In some cases, where the linearization method fails, the new theorem can be simply used. Some examples are presented to illustrate the theorem and to make a comparison with the cases where the Lyapanov’s linearization method fails.