  An inverted flag has its trailing edge clamped and exhibits dynamics distinct
from that of a conventional flag, whose leading edge is restrained. We perform
nonlinear simulations and a global stability analysis of the inverted-flag
system for a range of Reynolds numbers, flag masses and stiffnesses. Our global
stability analysis is based on a linearisation of the fully-coupled
fluid-structure system of equations. The calculated equilibria are steady-state
solutions of the fully-coupled nonlinear equations. By implementing this
approach, we (i) explore the mechanisms that initiate flapping, (ii) study the
role of vortex shedding and vortex-induced vibration (VIV) in large-amplitude
flapping, and (iii) characterise the chaotic flapping regime. For point (i), we
identify a deformed-equilibrium state and show through a global stability
analysis that the onset of flapping is due to a supercritical Hopf bifurcation.
For large-amplitude flapping, point (ii), we confirm the arguments of Sader et
al. (2016) that for a range of parameters this regime is a VIV. We also show
that there are other flow regimes for which large-amplitude flapping persists
and is not a VIV. Specifically, flapping can occur at low Reynolds numbers
($<50$), albeit via a previously unexplored mechanism. Finally, with respect to
point (iii), chaotic flapping has been observed experimentally for Reynolds
numbers of $O(10^4)$, and here we show that chaos also persists at a moderate
Reynolds number of 200. We characterise this chaotic regime and calculate its
strange attractor, whose structure is controlled by the above-mentioned
deformed equilibria and is similar to a Lorenz attractor. These results are
contextualised with bifurcation diagrams that depict the different equilibria
and various flapping regimes.
