Rotation Numbers and Geometric Invariants in Bicycle Dynamics
Abstract
We study planar bicycle dynamics via the rotation number function associated with a closed front track and bicycle length R. We prove that mode-locking plateaus occur only at integer rotation numbers and that the rotation number function is real-analytic off resonance. From the rotation number function we introduce two new geometric invariants: the critical B-length (right end of the first plateau) and the turning B-length (left end of the maximal monotone interval). We prove that, for a star-shaped curve, these invariants coincide, yielding a sharp transition of the bicycle monodromy: hyperbolic for R below the critical B-length and elliptic for R above it. The proofs combine projectivized SU(1,1) dynamics with Riccati equations and rotation-number theory.
Keywords
Cite
@article{arxiv.2511.06031,
title = {Rotation Numbers and Geometric Invariants in Bicycle Dynamics},
author = {Diantong Li and Qiaoling Wei and Meirong Zhang and Zhe Zhou},
journal= {arXiv preprint arXiv:2511.06031},
year = {2025}
}
Comments
21 pages