On a constructive method of search for rotary and oscillatory modes in autonomous dynamical systems

L. A. Klimina ; B. Ya. Lokshin

Russian journal of nonlinear dynamics, Tome 13 (2017) no. 1, pp. 25-40

Résumé

An autonomous dynamical system with one degree of freedom with a cylindrical phase space is studied. The mathematical model of the system is given by a second-order differential equation that contains terms responsible for nonconservative forces. A coefficient $\alpha$ at these terms is supposed to be a small parameter of the model. So the system is close to a Hamiltonian one. In the first part of the paper, it is additionally supposed that one of nonconservative terms corresponds to dissipative or to antidissipative forces, and coefficient $b$ at this term is a varied parameter. The Poincaré – Pontryagin approach is used to construct a bifurcation diagram of periodic trajectories with respect to the parameter $b$ for sufficiently small values of $\alpha$. In the second part of the paper, a system with nonconservative terms of general form is studied. Two supplementary systems of special form are constructed. Results of the first part of the paper are applied to these systems. Comparison of bifurcation diagrams for these supplementary systems has allowed deriving necessary conditions for the existence of periodic trajectories in the initial system for sufficiently small $\alpha$. The third part of the paper contains an example of the study of periodic trajectories of one system, which, for zero value of the small parameter, coincides with a Hamiltonian system $H_0$. It is proved that there exist periodic trajectories which do not satisfy the Poincaré – Pontryagin sufficient conditions for emergence of periodic trajectories from trajectories of the system $H_0$.

Détail

Keywords: autonomous dynamical system, Poincaré – Pontryagin approach, sufficient conditions for the existence of periodic trajectories, bifurcation diagram, necessary conditions for the existence of periodic trajectories.