Homoclinic, Subharmonic and Superharmonic Bifurcations for a Pendulum with Periodically Varying Length

TL;DR

This paper analyzes the complex bifurcations in a pendulum with a periodically varying length, using Melnikov's method and averaging to identify homoclinic, subharmonic, and superharmonic behaviors, validated by numerical simulations.

Contribution

It introduces an analytical approach combining Melnikov's method and averaging to study bifurcations in a pendulum with variable length, providing new insights into its dynamic behavior.

Findings

Identification of bifurcations leading to complex oscillatory and rotational motions.

Analytical predictions are confirmed by numerical simulations.

The study enhances understanding of nonlinear dynamics in variable-length pendulums.

Abstract

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.