Bifurcation analysis of the problem of a "rubber" ellipsoid of revolution rolling on a plane

TL;DR

This paper analyzes the stability and bifurcations of an ellipsoid of revolution rolling on a plane, providing a comprehensive classification of its motion trajectories based on parameters and initial conditions.

Contribution

It offers a detailed bifurcation analysis and classification of the rolling ellipsoid's trajectories, extending understanding of its dynamic behavior.

Findings

Identified permanent rotations and their stability

Performed a bifurcation analysis of the system

Classified possible trajectories based on parameters

Abstract

This paper is concerned with the problem of an ellipsoid of revolution rolling on a horizontal plane under the assumption that there is no slipping at the point of contact and no spinning about the vertical. A reduction of the equations of motion to a fixed level set of first integrals is performed. Permanent rotations corresponding to the rolling of an ellipsoid in a circle or in a straight line are found. A linear stability analysis of permanent rotations is carried out. A complete classification of possible trajectories of the reduced system is performed using a bifurcation analysis. A classification of the trajectories of the center of mass of the ellipsoid depending on parameter values and initial conditions is performed.