Rotation Numbers and Geometric Invariants in Bicycle Dynamics

TL;DR

This paper investigates the mathematical properties of bicycle dynamics using rotation numbers, revealing new geometric invariants and their implications for the system's transition between hyperbolic and elliptic behaviors.

Contribution

It introduces two novel geometric invariants derived from the rotation number function and characterizes their role in the bicycle monodromy transition.

Findings

Mode-locking plateaus occur only at integer rotation numbers.

The rotation number function is real-analytic off resonance.

For star-shaped curves, the invariants coincide, indicating a sharp transition in monodromy.

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.