  Varying one of the governing parameters of a dynamical system may lead to a
critical transition, where the new stable state is undesirable. In some cases,
there is only a limited range of the bifurcation parameter that corresponds to
that unwanted attractor, while the system runs problem-less otherwise. In this
study, we present experimental results regarding a thermoacoustic system
subject to two consecutive and mirrored supercritical Hopf bifurcations: the
system exhibits high amplitude thermoacoustic limit cycles for intermediate
values of the bifurcation parameter. Changing quickly enough the bifurcation
parameter, it was possible to dodge the unwanted limit cycles. A low-order
model of the complex thermoacoustic system was developed, in order to describe
this interesting transient dynamics. It was afterward used to assess the risk
of exceeding an oscillation amplitude threshold as a function of the rate of
change of the bifurcation parameter.
