Homoclinic,subharmonic, and superharmonic bifurcations for a pendulum with periodically varying length

Dynamic behavior of a weightless rod with a point mass sliding along the rod axis according to periodic law is studied. This is the simplest model of child’s swing. Melnikov’s analysis is carried out to find bifurcations of homoclinic, subharmonic oscillatory, and subharmonic rotational orbits. For the analysis of superharmonic rotational orbits, the averaging method is used and stability of obtained approximate solution is checked. The analytical results are compared with numerical simulation results.