Stability in Lyapunov systems

The stability of the stationary point of a Lyapunov system [Malkin IG, Some Problems in the Theory of Non-linear Oscillations. Moscow: Gostekhizd; 1956.], which describes the perturbed motion of a dynamical system with two degrees of freedom, is investigated. It is assumed that the characteristic equation of the first approximation of the system has two pairs of pure imaginary roots and that the quadratic part of the integral is not sign-definite. Both the non-resonance case, as well as cases of lower order (second-, third- and fourth-order) resonances are considered. The necessary and sufficient conditions for stability are given in cases when the problem is solved by a combination of the first non-linear terms of normal form.