Singularly perturbed dynamics of the tippedisk

Area of Science:

• Mathematical Physics
• Mechanical Engineering
• Nonlinear Dynamics

Background:

• The tippedisk is a novel archetype for friction-induced instability.
• A reduced 3D ordinary differential equation model has been developed.
• Previous work included local stability analysis of spinning solutions.

Purpose of the Study:

• To perform a global analysis of the reduced tippedisk system.
• To investigate the role of singular perturbation theory in understanding the dynamics.
• To explain the inversion phenomenon through bifurcation analysis.

Main Methods:

• Singular perturbation theory applied to the reduced tippedisk model.
• Analysis of slow-fast dynamics and the resulting slow manifold.
• Investigation of homoclinic and Hopf bifurcations.

Main Results:

• Friction induces slow-fast dynamics and a two-dimensional slow manifold.
• A bifurcation scenario involving homoclinic and Hopf bifurcations explains disc inversion.
• A closed-form condition for the critical spinning speed was derived.

Conclusions:

• The tippedisk model provides a framework for analyzing global bifurcations in singularly perturbed systems.
• The study elucidates the mechanism behind the friction-induced inversion phenomenon.
• This work offers insights into nonlinear dynamics and stability analysis.