Localizing EP sets in dissipative systems and the self-stability of bicycles

TL;DR

This paper introduces a method to locate complex exceptional points in dissipative systems, particularly applied to bicycle stability, revealing how these points influence design for robustness and self-stability.

Contribution

The authors propose a novel approach to localize high-order exceptional points in dissipative systems, with specific application to bicycle self-stability analysis.

Findings

Explicit localization of EP sets in bicycle models

Scaling laws for robust bicycle design derived from EP analysis

Agreement with experimental bicycle stability data

Abstract

Sets in the parameter space corresponding to complex exceptional points have high codimension and by this reason they are difficult objects for numerical localization. However, complex EPs play an important role in the problems of stability of dissipative systems where they are frequently considered as precursors to instability. We propose to localize the set of complex EPs using the fact that the minimum of the spectral abscissa of a polynomial is attained at the EP of the highest possible order. Applying this approach to the problem of self-stabilization of a bicycle we find explicitly the EP sets that suggest scaling laws for the design of robust bikes that agree with the design of the known experimental machines