Bifurcation control for a ship maneuvering model with nonsmooth nonlinearities

TL;DR

This paper analyzes bifurcations in a ship maneuvering model with nonsmooth nonlinearities, using analytical and numerical methods to determine local bifurcation types and their dependence on control parameters.

Contribution

It introduces an analytical approach to study bifurcations in nonsmooth ship models and provides detailed analysis of bifurcation locations and types under proportional control.

Findings

Supercritical Andronov--Hopf bifurcations are typical.

Bifurcation locations depend on control and design parameters.

Numerical continuation reveals stable and unstable states.

Abstract

We consider a widely used form of models for ship maneuvering, whose nonlinearities entail continuous but nonsmooth second-order modulus terms. For such models bifurcations of straight motion are not amenable to standard center manifold reduction and normal forms. Based on a recently developed analytical approach, we nevertheless determine the character of local bifurcations when stabilizing the straight motion course with standard proportional control. For a specific model class we perform a detailed analysis of the linearization to determine the location of these bifurcations in the control parameter space and its dependence on selected design parameters. By computing the analytically derived characteristic parameters, we find that `safe' supercritical Andronov--Hopf bifurcations are typical. Through numerical continuation we provide a more global bifurcation analysis, which identifies the arrangement and relative location of stable and unstable equilibria and periodic orbits.