论文标题
lyapunov系数用于具有平滑非线性的系统中的HOPF分叉
Lyapunov coefficients for Hopf bifurcations in systems with piecewise smooth nonlinearity
论文作者
论文摘要
由受控船舶操纵的模型激励，我们分析了具有光滑非线性部分的系统中的HOPF分叉。特别是，我们为第一个Lyapunov系数的概括为此设置提供了明确的公式。这通常确定分支的方向（超级临界性），但通常这与矢量场的任何固定平滑性不同。我们专注于表单$ u_i | u_j | $的非平滑非线性，但是我们的结果以更广泛的通用性为任何维度的更广泛的通用性配制，任何分段平滑的非线性部分。此外，我们讨论了一些编态度 - 归化性，并将结果应用于颤抖的轮模型。
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.
