А.N. Каtulev, S.V. Jagolnikov

In the paper the method of investigation for stability of non-linear autonomous dynamic systems based on the application of Hamiltonian system for exposure of necessary and sufficient of stability for solution of non-linear autonomous systems of differential equations without introduction Lyapunov-s function, application of qualitative methods at reduction of right part to Gurvitz is given. Substantiation of the method is presented in form of theorems. On the basis these theorems algorithm for research of stability of autonomous nonlinear dynamic system is created. Developed method and algorithm are applied to research the stability of different non-linear autonomous dynamic systems with smooth and unsmooth continuous nonlinearities, solution stability homogeneous systems Volterra-s non-linear integral equations of two second kind when they may be reduced to the solution of the non-linear autonomous systems of usual differential equations for investigation of bifurcation set in the space of parameters of autonomous non-linear dynamic systems/ The method is also applicable for investigation of non-linear dynamic systems which are described by differential equations with particular derivations and for research of stability of linear autonomous systems. In the article solutions of practical tasks for stability investigation of different non-linear autonomous systems are given. They are illustration high efficiency of elaborated method. In this article also a task of research of structure stability of nonlinear dynamic system is given. Solution results confirm authenticity of method.