Lyapunov coefficients for Hopf bifurcations in systems with piecewise smooth nonlinearity

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 smoothening of the vector field. We focus on non-smooth 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.Comment: 39 pages, 10 figure

Verzweigungsanalyse von Systemen mit stückweise glatten Nichtlinearitäten und deren Anwendungen

In the first part of this thesis, Andronov-Hopf bifurcations in systems with piecewise smooth nonlinearity are analyzed, which are motivated by models that arise in controlled ship maneuvering. In particular, within this nonsmooth setting, we derive explicit formulas for the generalization of the first Lyapunov coefficient, which determines the direction of branching. In addition, we show that, in general, this differs from any fixed smoothing of the vector field. Specifically, we focus on nonsmooth nonlinearities of the form the product of one variable times the absolute value of another variable. However, our results are formulated in broader generality for systems in any dimension with piecewise smooth nonlinear part. Furthermore, other bifurcations occur in systems with the aforementioned nonsmooth nonlinearity. In particular, we perform an analysis of normal forms with piecewise smooth nonlinear part for coefficients for Bogdanov-Takens points, and compare these results with the corresponding smooth version. After the most theoretical part of this manuscript, we consider a particular ship maneuvering model, and show that the previously developed outcomes apply in actual systems. We determine the criticality of Hopf bifurcations that arises in stabilizing the straight motion of a marine craft model. For such a 3 degree of freedom system of ship motion with yaw damping and yaw restoring control, we present a detailed study of the possibilities for stabilizing the straight motion and the resulting nonlinear effects. To facilitate the analysis, we consider a combination of rudder and propeller forces into an effective thruster force. We identify the existence, location and geometry of the stability boundary in terms of the controls, including the dependence on the propeller diameter and the thruster position. We find numerically that “safe” supercritical Hopf bifurcations are typical and, by means of numerical continuation, we provide a global bifurcation analysis, which identifies the arrangement and relative location of stable and unstable equilibria and periodic orbits. We illustrate the resulting stable ship motions in Earth-fixed coordinates and present some direct numerical simulations