  Dynamical systems theory has emerged as an interdisciplinary area of research
to characterize the complex dynamical transitions in real-world systems.
Various nonlinear dynamical phenomena and bifurcations have been discovered
over the decades using different reduced-order models of oscillators. Different
measures and methodologies have been developed theoretically to detect,
control, or suppress the nonlinear oscillations. However, obtaining such
phenomena experimentally is often challenging, time-consuming, and risky,
mainly due to the limited control of certain parameters during experiments.
With this review, we aim to introduce a paradigmatic and easily configurable
Rijke tube oscillator to the dynamical systems community. The Rijke tube is
commonly used by the combustion community as a prototype to investigate the
detrimental phenomena of thermoacoustic instability. Recent investigations in
such Rijke tubes have utilized various methodologies from dynamical systems
theory to better understand the occurrence of thermoacoustic oscillations,
their prediction and mitigation, both experimentally and theoretically. The
existence of various dynamical behaviors has been reported in single as well as
coupled Rijke tube oscillators. These behaviors include bifurcations, routes to
chaos, noise-induced transitions, synchronization, and suppression of
oscillations. Various early warning measures have been established to predict
thermoacoustic instabilities. Therefore, this review paper consolidates the
usefulness of a Rijke tube oscillator in terms of experimentally discovering
and modeling different nonlinear phenomena observed in physics; thus,
transcending the boundaries between the physics and the engineering
communities.
