Heteroclinic bifurcation analysis of the tippedisk through the use of Melnikov theory

Abstract

The Tippedisk is a mechanical-mathematical archetype for a peculiar friction-induced inversion phenomenon that occurs when an unbalanced disk is spun rapidly about an in-plane axis, with the center of gravity rising counterintuitively as the orientation of the disk inverts. To understand the qualitative behavior of the tippedisk, a nonlinear analysis is performed, revealing the singularly perturbed structure of the system equations. Application of singular perturbation theory shows that the long-term behavior is dominated by a two-dimensional slow manifold, on which the asymptotic dynamics takes place. Moreover, Melnikov theory is used to derive a closed form approximation of a heteroclinic bifurcation, which allows general statements to be made about the dynamic behavior of the tippedisk. MSC2010 numbers: 70E18, 70K20, 70E50.