Rotation Numbers and Geometric Invariants in Bicycle Dynamics

TL;DR

The paper analyzes planar bicycle dynamics through the rotation-number function $\rho_{\Gamma}(R)$ associated with a closed front track $\Gamma$ and bicycle length $R$, proving that mode-locking plateaus occur only at integer values and that $\rho_{\Gamma}(R)$ is real-analytic off resonance. It introduces two geometric invariants, $ {\underline{R}}(\Gamma)$ and $ {\overline{R}}(\Gamma)$, and shows that for star-shaped curves these invariants coincide, yielding a sharp monodromy transition from hyperbolic to elliptic at the common length. The study combines projectivized $SU(1,1)$ dynamics with Riccati equations and rotation-number theory, and uses Magnus expansion to derive asymptotics and isoperimetric-type inequalities that connect geometry of $\Gamma$ to bicycle dynamics. These insights advance understanding of geometric invariants in monodromy classifications and provide precise criteria for transition behavior in the bicycle system.

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.

Figures (1)

• Figure 1: Rotation numbers $\rho_b(R)$, $b=2$, as a function of $R$.

Theorems & Definitions (46)

• Lemma 2.1
• Lemma 2.2
• proof
• Definition 2.3
• Lemma 2.4
• Corollary 2.5
• proof
• Lemma 2.6
• Lemma 2.7
• proof