Lyapunov coefficients for Hopf bifurcations in systems with piecewise smooth nonlinearity

TL;DR

This paper extends the concept of Lyapunov coefficients to analyze Hopf bifurcations in systems with piecewise smooth nonlinearities, providing explicit formulas and insights into bifurcation behavior in nonsmooth systems.

Contribution

It derives explicit formulas for Lyapunov coefficients in nonsmooth systems, generalizing bifurcation analysis beyond smooth vector fields.

Findings

Derived formulas for Lyapunov coefficients in nonsmooth systems

Identified differences from smoothed vector field bifurcations

Applied results to a model of a shimmying wheel

Abstract

Motivated by models that arise in controlled ship maneuvering, we analyze Hopf bifurcations in systems with piecewise smooth nonlinear part. In particular, we derive explicit formulas for the generalization of the first Lyapunov coefficient to this setting. This generically determines the direction of branching (super- versus sub-criticality), but in general this differs from any fixed smoothing of the vector field. We focus on nonsmooth nonlinearities of the form $u_i|u_j|$, but our results are formulated in broader generality for systems in any dimension with piecewise smooth nonlinear part. In addition, we discuss some codimension-one degeneracies and apply the results to a model of a shimmying wheel.