SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2847-2886, January 2020.
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 $u_i|u_j|$, 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.

中文翻译：

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分段光滑非线性系统Hopf分支的Lyapunov系数。

SIAM应用动力系统杂志，第19卷，第4期，第2847-2886页，2020年1月
。受受控船舶操纵中产生的模型的影响，我们分析了具有分段光滑非线性部分的系统中的Hopf分叉。特别是，我们导出了将第一个Lyapunov系数推广到该设置的显式公式。这通常确定分支的方向（超临界与次临界），但通常与矢量场的任何固定平滑化不同。我们将重点放在形式为$ u_i | u_j | $的非光滑非线性上，但是对于具有分段光滑非线性部分的任何维数的系统，我们的结果都用更广泛的一般性来表述。另外，我们讨论了一些余维一简并把结果应用于摆线轮模型。