On numerical bifurcation analysis of periodic motions of autonomous Hamiltonian systems with two degrees of freedom

Abstract

Abstract In this work we consider bifurcation problem for natural families of periodic motions of autonomous Hamiltonian systems with two degrees of freedom. While there exists a well-developed analytical approach to this problem, it is limited to small neighborhoods of known equilibria and stationary solutions. To explore the bifurcations of periodic motions for all admissible values of the problem’s parameters it is necessary to employ numerical methods. We propose an approach combining analytical and numerical computation of the natural families with numerical bifurcation analysis. We obtain the so-called base solutions either analytically or numerically for particular values of the problem’s parameters and then employ a numerical method to continue the base solutions to the borders of their existence domains and to identify bifurcation points. Linear orbital stability domains are also obtained in course of the continuation. To illustrate the proposed approach we analyze bifurcation of periodic motions emanating from Cylindrical precession of a dynamically-symmetric satellite on a circular orbit.