  We consider the unsteady regimes of an acoustically-driven jet that forces a
recirculating flow through successive reflections on the walls of a square
cavity. The specific question being addressed is to know whether the system can
sustain states of low-dimensional chaos when the acoustic intensity driving the
jet is increased, and, if so, to characterise the pathway and underlying
physical mechanisms. We adopt two complementary approaches, both based on data
extracted from numerical simulations: (i) We first characterise successive
bifurcations through the analysis of leading frequencies. Two successive phases
in the evolution of the system are singled out in this way, both leading to
potentially chaotic states. The two phases are separated by a drastic
simplification of the dynamics that immediately follows the emergence of
intermittency. The second phase also features a second intermediate state where
the dynamics is simplified due to frequency-locking. (ii) Nonlinear time series
analysis enables us to reconstruct the attractor of the underlying dynamical
system, and to calculate its correlation dimension and leading Lyapunov
exponent. Both these quantities bring confirmation that the state preceding the
dynamic simplification that initiates the second phase is chaotic. Poincar\'e
maps further reveal that this chaotic state in fact results from a dynamic
instability of the system between two non-chaotic states respectively observed
at slightly lower and slightly higher acoustic forcing.
