Lyapunov Coefficients For Hopf Bifurcations In Systems With Piecewise Smooth Nonlinlearity

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 subcriticality), but in general this differs from any fixed smoothening of the vector field. We focus on nonsmooth nonlinearities of the form u(i)vertical bar u(j)vertical bar, 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.