Manucharyan G. Development of numerical-analytical method of investigation of transition from regular to chaotic dynamics in nonlinear systems

The object of reserch is nonlinear dynamical systems with several equilibrium positions under external periodical excitation where the passing from regular to chaotic dynamics is possible. The goal of work is construction of effective numerical-analytical methods for determination of chaos onset in nonlinear systems with several equilibrium positions and investigation of passing to chaos in different models. Investigation is made by means of classical asymptotic methods, Pade and quasi-Pade approximants, methods by Runge-Kutta, Simpson, Newton. The theoretical result is creation of effective method for investigation of the passing to chaos in nonlinear systems which is based on criterion of homo- and heteroclinic trajectories and allows to determine the values of the governing parameters corresponding to chaos onset. The practical outcome is the creation of mathematical and program tools for numerical investigation of dynamical behavior of systems with homo- and heteroclinic structures. The scientificnovelty of obtained results is the following: - for the first time effective numerical-analytical method to determine the parameters of system corresponding to chaos onset for small values of dissipation in system is developed. The distinguishing features of proposed method are a construction of homo- and heteroclinic trajectories creation of which is a criterion of lower boundary of chaotic behavior domain and that that this method uses Pade and quasi-Pade appriximants; - new method of chaos investigation for the case when the dissipation is not small is proposed. It is based on the criterion of mutual instability of phase trajectories in domain of chaotic behavior; - the method of determination of dependences between parameters corresponding to chaos onset in nonlinear systems with several equilibrium positions by means of proposed methods is improved. Corresponding dependences are obtained for nonautonomous Duffing equation, equations of the pendulum with excited point of suspension, of the self-sustainedoscillation system Van der Pole-Duffing, of the motion of snap-through-truss and equation of oscillator with nonlinear dissipation characteristic under external periodical excitation. The results of analytical investigations and numerical experiment are used in scientific work of NTU "KhPI" and interesting in nonlinear dynamics of supple elements of constructions, for analysis of self-sustained oscillation system dynamics, for analysis of damping systems by means of dampers like snup-through-truss. Results can be used for the projection of modern equipment.