Locating the Sets of Exceptional Points in Dissipative Systems and the Self-Stability of Bicycles

Abstract

Sets in the parameter space corresponding to complex exceptional points (EP) have high codimension, and by this reason, they are difficult objects for numerical location. However, complex EPs play an important role in the problems of the stability of dissipative systems, where they are frequently considered as precursors to instability. We propose to locate the set of complex EPs using the fact that the global 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.

Keywords

• exceptional points in classical systems
• coupled systems
• non-holonomic constraints
• nonconservative forces
• stability optimization
• spectral abscissa
• swallowtail
• bicycle self-stability