  Nonlinear controlled plants with Bogdanov-Takens singularity may experience
surprising changes in their number of equilibria, limit cycles and/or their
stability types when the controllers slightly vary in the vicinity of critical
parameter varieties. Each such a change is called a local bifurcation. We
derive novel results with regards to truncated parametric normal form
classification of the generalized cusp plants. Then, we suggest effective
nonlinear bifurcation control law designs for precisely locating and accurately
controlling many different types of bifurcations for two measurable plants from
this family. The first is a general quadratic plant with a possible multi-input
linear controller while the second is a Z_2-equivariant general plant with
possible multi-input linear (Z_2-symmetry preserving) and quadratic
(symmetry-breaking) controllers. The bifurcations include from primary to
quinary bifurcations of either of the following types: saddle-node,
transcritical and pitchfork of equilibria, Z_2-equivariant bifurcations of
multiple limit cycles through Hopf, homoclinic, heteroclinic, saddle-node, and
saddle-connection, and finally their one-parameter symmetry breaking
bifurcations. Using our parametric normal form analysis, we propose a new
approach for efficient treatment of tracking and regulating engineering
problems with smooth manoeuvering possiblities. Due to the nonlinearity of a
ship maneuvering characteristic, there is a need for a controller design in a
ship steering system so that the ship follows a desired sea route. The results
in bifurcation control analysis are applied to two nonlinear ship course models
for such controller designs. Symbolic implementations in Maple and numerical
simulations in MATLAB confirm our theoretical results and accurate predictions.
