  For an elastic system that is non-conservative but autonomous, subjected for
example to time-independent loading by a steadily flowing fluid (air or water),
a dangerous bifurcation, such as a sub-critical bifurcation, or a cyclic fold,
will trigger a dynamic jump to one or more remote stable attractors. When there
is more than one candidate attractor, the one onto which the structure settles
can then be indeterminate, being sensitive to infinitesimally small variations
in starting conditions or parameters. In this paper we develop and study an
archetypal model to explore the nonlinear dynamic interactions between
galloping at an incipient sub-critical Hopf bifurcation of a structure with
shell-like buckling behaviour that is gravity-loaded to approach a sub-critical
pitch-fork bifurcation. For the fluid forces, we draw on the aerodynamic
coefficients determined experimentally by Novak for the flow around a bluff
body of rectangular cross-section. Meanwhile, for the structural component, we
consider a variant of the propped-cantilever model that is widely used to
illustrate the sub-critical pitchfork: within this model a symmetry-breaking
imperfection makes the behaviour generic. The compound bifurcation
corresponding to simultaneous galloping and buckling is the so-called
Takens-Bodganov Cusp. We make a full unfolding of this codimension-3
bifurcation for our archetypal model to explore the adjacent phase-space
topologies and their indeterminacies.
