A case study in bifurcation and stability analysis is presented, in which
reduced dynamical system modelling yields substantial new global and predictive
information about the behaviour of a complex system. The first smooth pathway,
free of pathological and persistent degenerate singularities, is surveyed
through the parameter space of a nonlinear dynamical model for a
gradient-driven, turbulence-shear flow energetics in magnetized fusion plasmas.
Along the route various obstacles and features are identified and treated
appropriately. An organizing centre of low codimension is shown to be robust,
several trapped singularities are found and released, and domains of
hysteresis, threefold stable equilibria, and limit cycles are mapped.
Characterization of this rich dynamical landscape achieves unification of
previous disparate models for plasma confinement transitions, supplies valuable
intelligence on the big issue of shear flow suppression of turbulence, and
suggests targeted experimental design, control and optimization strategies.Comment: 21 pages, 12 figures, 34 postscript figure file