  In an acoustic cavity with a heat source, such as a flame in a gas turbine,
the thermal energy of the heat source can be converted into acoustic energy,
which may generate a loud oscillation. If uncontrolled, these nonlinear
acoustic oscillations, also known as thermoacoustic instabilities, can cause
large vibrations up to structural failure. Numerical and experimental studies
showed that thermoacoustic oscillations can be chaotic. It is not yet known,
however, how to minimise such chaotic oscillations. We propose a strategy to
analyse and minimise chaotic acoustic oscillations, for which traditional
stability and sensitivity methods break down. We investigate the acoustics of a
nonlinear heat source in an acoustic resonator. First, we propose covariant
Lyapunov analysis as a tool to calculate the stability of chaotic acoustics
making connections with eigenvalue and Floquet analyses. We show that covariant
Lyapunov analysis is the most general flow stability tool. Second, covariant
Lyapunov vector analysis is applied to a chaotic system. The time-averaged
acoustic energy is investigated by varying the heat-source parameters.
Thermoacoustic systems can display both hyperbolic and non-hyperbolic chaos, as
well as discontinuities in the time-averaged acoustic energy. Third, we embed
sensitivities of the time-averaged acoustic energy in an optimisation routine.
This procedure achieves a significant reduction in acoustic energy and
identifies the bifurcations to chaos.
  The analysis and methods proposed enable the reduction of chaotic
oscillations in thermoacoustic systems by optimal passive control. The
techniques presented can be used in other unsteady fluid-dynamics problems with
virtually no modification.
