quasiperiodic oscillations, different behaviors of synchronization of these
oscillators is observed which include phase-locking, phase trapping,
intermittent phase locking, and phase drifting. While during chaotic
oscillations, states of intermittent phase locking and phase drifting are
observed. The different phase dynamics observed during these state primarily
occur due to dissimilar spectral content of two locked frequencies. They also
proposed various statistical measures (see Fig. 25), such as the phase locking
value ($PLV$), correlation coefficient ($r$), and relative mean frequency
($\Delta\omega$) to quantitatively characterize the synchronization behavior
between the acoustic pressure and the heat release rate fluctuations during
self-excited states of thermoacoustic instability. These measures can help in
detecting the boundaries of different states of synchronization observed
during the quasiperiodicity route to chaos.

Weng et al. weng2020synchronization developed a nonlinearly coupled damped
oscillator model to study the coupled behavior of acoustic pressure
($p^{\prime}$) and heat release rate ($\dot{q^{\prime}}$) fluctuations in a
laminar premixed Rijke tube burner. The governing equations are given below:

$\ddot{p}+\zeta_{1}\dot{p}(t)+\omega_{p^{\prime}}^{2}p(t)=C_{pq}(1-q(t-\tau_{2})^{2})\dot{p}(t)$
(13)
$\ddot{q}+\zeta_{2}\dot{q}(t)+\omega_{\dot{q}^{\prime}}^{2}q(t)=C_{qp}(p(t-\tau_{1})^{3}-1)$
(14)

where $\tau_{1}=2\pi/\omega_{p^{\prime}}$,
$\tau_{2}=2\pi/\omega_{\dot{q}^{\prime}}$, and $C_{qp}$ (or $C_{pq}$)
indicates the coupling strength between $p^{\prime}$ and $\dot{q}^{\prime}$
(or $\dot{q}^{\prime}$ and $p^{\prime}$). $\omega_{p^{\prime}}$,
$\omega_{\dot{q}^{\prime}}$ are the angular frequency of the acoustic and the
heat release rate fluctuations, and $\zeta_{1}$ and $\zeta_{2}$ are damping
terms. This model is able to capture the experimentally observed
quasiperiodicity route to chaos (discussed in Sec. III.4.2) by Kabiraj et al.
kabiraj2012route and the state of strange non-chaos identified by Premraj et
al. premraj2020strange. Furthermore, this model also qualitatively replicates
the synchronization behavior between the acoustic pressure and heat release
rate fluctuations observed by Mondal et al. mondal2017synchronous, as shown in
Fig. 25.

Figure 25:  (a) The variation of the phase locking value ($PLV$), (b) the
correlation coefficient ($r$), and (c) the relative mean frequency
($\Delta\omega$) between the acoustic pressure and heat release rate
fluctuations for different regimes of quasiperiodicty route to chaos, when
location of the flame ($x_{f}$) inside the premixed laminar Rijke tube burner
is varied as control parameter. Different regions of synchronization are
indicated as phase locking (I and V), intermittent phase locking (II and IV),
and phase drifting (III). Reproduced with permission from Mondal et al.
mondal2017synchronous.

In the preceding sections, we have discussed various behaviors and
characteristics exhibited by a general nonlinear oscillator and elaborated the
existence of such dynamical behavior in Rijke tube systems. We established the
vast potential of Rijke tubes in experimentally verifying complex dynamical
behaviors commonly reported in literature through nonlinear oscillator models.
Next, we move our focus towards various prediction and control strategies
devised to warn the undesired impending thermoacoustic oscillations or to
suppress them after their onset.

##  VI Control and prediction strategies for thermoacoustic instability

As we discussed in the previous section, thermoacoustic instabilities have
been observed in the form of large amplitude self-sustained oscillations in
the acoustic field of the combustor. We showed that such oscillations occur
via Hopf bifurcation in the Rijke-type thermoacoustic systems. The presence of
thermoacoustic instabilities is undesirable in practical systems as they cause
severe vibrations leading to heavy structural damage and loss in the
performance of the engine. Therefore, it is necessary to keep the system away
from the regime of operation of thermoacoustic instability. There have been
several studies dedicated to develop control strategies that can mitigate and
forewarn thermoacoustic instability. Due to the simplistic nature and ease of
handling, Rijke-type thermoacoustic systems remain the primary choice for many
researchers to experiment or model novel control methodologies that can
mitigate or predict thermoacoustic instabilities mondal2021mitigation;
zou2021quenching. In this section, we will summarize traditional as well as
recently discovered control methodologies based on synchronization theory to
suppress thermoacoustic instability in Rijke tube systems. We will also
discuss recent developments in early warning technologies to forewarn critical
transition to thermoacoustic instabilities in rate-dependant experiments on
Rijke tube systems.

The control strategies developed for suppression of thermoacoustic
instabilities have been classified as passive and active
lieuwen2005combustion; huang2009dynamics. Passive control strategies aim at
evading the occurrence of thermoacoustic instability by introducing
modifications in hardware design of the components such as combustor geometry,
fuel injection system or using acoustic dampers such as resonators or liners
to remove acoustic energy from the system richards2003passive; zhao2015review;
schadow1992combustion; candel2002combustion; lieuwen2005combustion;
culick2006unsteady. In contrast, active control strategies are based on
interrupting the coupling between the acoustic field and the heat release rate
field of the combustor through external perturbations, leading to the decay of
thermoacoustic oscillations in the system mcmanus1993review;
poinsot2017prediction. Active controls are further classified into open-loop
and closed-loop control (or commonly known as feedback control) depending on
whether the control strategy is independent or dependent on the system
response, respectively dowling2005feedback. In the following subsections, we
will discuss control strategies, such as feedback control and open-loop
control, developed for the mitigation of limit cycle oscillations in a single
Rijke-type thermoacoustic system.

###  VI.1 Feedback control

Feedback control strategy involves the usage of external energy attenuators to
perturb the inlet flow field of the Rijke tube in response to the behavior of
a dynamical variable measured from the system heckl1988active;
mcmanus1993review; lang1987active. To elaborate, we supply a portion of the
information of the system to its input through controllers (Fig. 26a); thus,
both the input and the output of the system are dependent on each other. The
controller adjusts the phase and the gain of the output signal before feeding
it back into the input. Hence, this strategy involves three crucial steps
dowling2005feedback; illingworth2010advances. The acquisition of the signal of
thermoacoustic instability from the system using sensors such as a pressure
transducer or a thermal sensor. This signal is then fed in to a controller,
where the signal is processed and supplied to an actuator. Actuators use this
processed information to alter the inlet conditions of the system, thus,
changing the coupling between the acoustic field and the heat release rate
fluctuations, causing the mitigation of thermoacoustic instabilities.
Actuation devices used in thermoacoustic systems are loudspeakers that perturb
the acoustic velocity or the acoustic pressure field and fuel valves that
change the heat release rate field in the Rijke tube burners li2016feedback;
bl; blonbou2000active; vaudrey2003time; rubio2015nonlinear;
zalluhoglu2016delayed. Delayed feedback has been used as a common methodology
to suppress limit cycle oscillations in general oscillator systems
atay1998van; reddy2000dynamics; xu2003effects starting from the early
implementation of feedback control in a laminar flame Rijke tube burner by
Dines dines1984active, Ffowcs ffowcs1984review and Heckl heckl1988active.
Recently, such methods have been rigorously studied in thermoacoustic
literature to mitigate limit cycle oscillations lang1987active;
heckl1988active; candel1992combustion; mcmanus1993review; dowling2005feedback;
illingworth2010advances; dines1984active; lieuwen2005combustion.

Figure 26: Schematic diagrams representing the application of (a) closed-loop
(or feedback) control and (b) open-loop control to mitigate limit cycle
oscillations in a thermoacoustic system. Adapted with permission from Dowling
and Morgans dowling2005feedback.

###  VI.2 Open-loop control via asynchronous quenching

As we discussed in Sec. V.2, the application of external periodic forcing to
an oscillator can entrain the frequency of self-excited oscillations with the
forcing during the onset of forced synchronization. Further, we discussed that
the forced synchronization can occur through the locking-route or the
suppression-route (asynchronous quenching) depending on whether the frequency
detuning is small or large dewan1972harmonic; balanov2009synchronization.
External periodic forcing has been commonly used in practice to suppress self-
sustained oscillations in hydrodynamically unstable flows
staubli1987entrainment and wakes taira2018phase, ionization waves
ohe1974asynchronous, oscillatory reactions fjeld1974relaxed, transmission
electrical lines shimura1967analysis, etc., through the phenomenon known as
asynchronous quenching. In order to achieve asynchronous quenching, the
oscillator must be forced at a frequency far from its natural frequency
dewan1972harmonic. In thermoacoustic literature, this method of forcing is
referred to as open-loop control mcmanus1993review; candel2002combustion;
evesque2003self.

During open-loop control, external periodic forcing is used to perturb limit
cycle oscillations in a thermoacoustic system at different amplitudes and
frequencies. Unlike feedback control, we do not need any input from the
combustor dynamics to drive the actuator in open-loop controls. The external
perturbations either affect the flow field incoming to the system or affect
the acoustic field developed in the system (Fig. 26b). At appropriate values
of the forcing parameters, external perturbations interrupt the coupling
between the acoustic field and the heat release rate field, thereby quenching
thermoacoustic instabilities in the system.

Recently, an approach based on forced synchronization (Sec. V.2) of limit
cycle oscillator has been used to explain the mitigation of thermoacoustic
instabilities through open-loop controls mondal2019forced; guan2019openSymp;
guan2018strange; skene2022phase; guan2019control; roy2020mechanism;
sahay2020dynamics. In Fig. 27, we show the forced response of limit cycle
oscillations in the acoustic pressure field of a horizontal Rijke tube for
different parameters of periodic forcing generated through loudspeakers (Sahay
et al. 2021). In this figure, the distribution of different colours indicates
the relative change in the amplitude of forced pressure fluctuations,
$p_{rms}^{\prime}$, against the amplitude of these oscillations in the
unforced state, $p_{0}^{\prime}$ (i.e., $\Delta
p_{rms}^{\prime}/p_{0}^{\prime}$, where $\Delta
p_{rms}^{\prime}=p_{0}^{\prime}-p_{rms}^{\prime}$) and a V-shaped plot
signifies the Arnold tongue (i.e., forced synchronization boundary).

We notice that the introduction of forcing significantly affects the amplitude
of limit cycle oscillations in the system. When forcing is applied close to
the natural frequency of the oscillator, we notice the occurrence of resonance
amplification of synchronized limit cycle oscillations (i.e., synchronance) in
the Arnold tongue, where the growth of the amplitude is greater than twice the
amplitude of limit cycle oscillations in the unforced state, i.e., $\Delta
p_{rms}^{\prime}/p_{0}^{\prime}<-1$ mondal2019forced; guan2019forced;
sahay2020dynamics. In contrast, when forcing is introduced at a frequency
lower than the natural frequency of limit cycle oscillations (see for
$f_{f}/f_{f0}<1$), we observe the quenching of limit cycle oscillations in the
system due to asynchronous quenching minorsky1967comments. The maximum
suppression of limit cycle oscillations (i.e., $\Delta
p_{rms}^{\prime}/p_{0}^{\prime}\rightarrow 1$) is observed along the Arnold
tongue, i.e., at forcing parameters required for achieving forced
synchronization in the system (Fig. 27). Asynchronous quenching is not
observed for $f_{f}/f_{f0}>1$ in the horizontal Rijke tube system
mondal2019forced; sahay2020dynamics; however, asynchronous quenching of limit
cycle oscillations has been reported for both the sides of $f_{n0}$ in a
laminar premixed flame Rijke tube burner by Guan et al. guan2019openSymp.
Furthermore, Roy et al. roy2020mechanism provided physical reasons behind
asynchronous quenching of limit cycle oscillations by analysing the forced
response of coupled acoustic pressure and heat release rate fluctuations in
the system. They found that these oscillations are locked at -90 degrees
during the state of asynchronous quenching; as a result, the driving of the
acoustic field by the heat release rate field is very low which in turn leads
to suppression of acoustic oscillations in the system.

Figure 27: The amplitude response (shown as the fractional change in the
amplitude $\Delta p_{rms}^{\prime}/p_{0}^{\prime}$ indicated by the colormap)
and synchronization properties (indicated by black lines) of limit cycle
oscillations, observed experimentally in a horizontal Rijke tube oscillator.
Here, $p_{rms}^{\prime}$ and $p_{0}^{\prime}$ are the root-mean-square (rms)
values of acoustic pressure fluctuations in the forced and the unforced state.
$\~{A}_{f}$ and $f_{f}/f_{n}$ are normalized forcing amplitude and forcing
frequency. Adapted with permission from Sahay et al. sahay2020dynamics.

###  VI.3 Mitigation of thermoacoustic instabilities through mutual coupling

In Sec. V.1, we saw that coupling two or more oscillators can either
synchronize their oscillations or mitigate them through the amplitude death
phenomenon. During amplitude death, all oscillators reach the same steady
state zou2021quenching. Similarly, coupling a system to itself via self
feedback can also quench limit cycle oscillations in the single system
srikanth2021self. Various coupling schemes have been developed to mitigate
self-excited oscillations in a system of coupled oscillators zou2021quenching;
saxena2012amplitude. Here, we will discuss the application of the amplitude
death phenomenon in quenching limit cycle oscillations in a system of coupled
thermoacoustic oscillators.

Practical gas turbine engines such as can or can-annular combustors often
consists of a ring array of multiple combustion units (known as “cans”),
working simultaneously to provide the required thrust walsh2004gas. Hence,
these combustors tend to interact with each other and are therefore coupled
through components such as the plenum chamber, the turbine stage, and cross-
fire tubes farisco2017thermo. Most of the traditional active and passive
control strategies discussed before to mitigate thermoacoustic instability are
expensive and are devised for an isolated combustion system. However, the
application of these strategies to simultaneously mitigate thermoacoustic
instabilities in a system with multiple combustors is yet to be explored in
detail.

Recent developments in the suppression of thermoacoustic instabilities in
single or multiple thermoacoustic systems rely on implementing different
schemes of coupling between oscillators. Towards this purpose, both
experimental and theoretical studies have been performed on a single Rijke
tube oscillator srikanth2021self and a system of two coupled Rijke tube
oscillators biwa2015amplitude; dange2019oscillation; sahay2020dynamics;
thomas2018effecta; thomas2018effectb; sahay2020dynamics;
srikanth2021dynamical. For suppressing limit cycle in a single system through
self-delayed feedback, the acoustic field of the system is fed back into the
system after a finite time delay, which is experimentally achieved by using a
single coupling tube biwa2016suppression; lato2019passive; srikanth2021self.
For coupled Rijke oscillators, suppression of low amplitude limit cycle
oscillations is experimentally achieved by connecting the oscillators using
one biwa2015amplitude; dange2019oscillation or two hyodo2020suppression
coupling tubes. On the other hand, suppression of high amplitude limit cycle
oscillations is achieved by adding a frequency mismatch between the
oscillators dange2019oscillation. Both time delay and dissipative coupling
schemes have been used to mitigate limit cycle oscillations in thermoacoustic
oscillators. The coupled behavior of such oscillators have been shown to be
similar to that observed in Van der Pol oscillators biwa2015amplitude and
Stuart-Landau oscillators premraj2021effect. Thus, the possibility of
mitigation of limit cycle oscillations developed in single and multiple
thermoacoustic systems through the mechanism of mutual coupling has emerged as
a promising and cost-effective methodology.

Figure 28 shows the acoustic response of the horizontal Rijke tube oscillator
when limit cycle oscillations of different amplitudes are self-coupled through
a single connecting tube of different lengths, $L$ srikanth2021self. Low
amplitude limit cycle oscillations can be easily quenched through the method
of self-coupling for a larger range of length of the coupling tube, while the
suppression of large amplitude limit cycle oscillations is observed only in a
narrow range of the length of the coupling tube. Srikanth et al.
srikanth2021self also found that the quenching of limit cycle oscillations is
easier if the diameter of the coupling is larger. They also found that the
transition from limit cycle oscillations to steady state depends on the nature
of the inherent bifurcation of the oscillator. To elaborate, this transition
is explosive and hysteric for an oscillator exhibiting subcritical Hopf
bifurcation in the uncoupled state, whereas, it is continuous for an
oscillator undergoing supercritical Hopf bifurcation.

Figure 28: The amplitude response ($\Delta p_{rms}^{\prime}/p_{0}^{\prime}$)
of limit cycle oscillations to self-coupling using different lengths of the
coupling tube ($L$), normalized by the length of duct of the Rijke tube
($L_{duct}$) observed experimentally in a horizontal Rijke tube oscillator.
The diameter of the coupling tube is fixed at 8 mm. Adapted with permission
from Srikanth et al. srikanth2021dynamical.

In Fig. 29, we show the dynamical behavior of a system of two identical Rijke
tube oscillators coupled using a single connecting tube dange2019oscillation.
A two-parameter bifurcation diagram is shown between the amplitude of acoustic
pressure fluctuations ($p_{0}$) in the isolated oscillator and the length of
the connecting tube ($L$). We observe a similar behavior of amplitude
suppression as discussed in Fig. 28 for a system of self-coupled Rijke tube
oscillator. Amplitude death can be easily achieved by coupling low amplitude
limit cycle oscillations, while it is difficult quench large amplitude limit
cycle oscillations in two identical Rijke tube oscillators. As we discussed in
Fig. 21, the coupling of large amplitude limit cycle oscillations results in
phase-flip bifurcation on increasing the length of the coupling tube in the
system.

Figure 29: Two-parameter bifurcation diagram between the isolated amplitude of
each oscillator ($p_{0}$) and the length of the coupling tuhe ($L$)
highlighting the dependency of the coupling and the system parameters on the
occurrence of amplitude death in a system of coupled identical Rijke tube
oscillators. Colors for AP, IP, and AD indicate regions of anti-phase
synchronization, in-phase synchronization, and amplitude death, respectively.
Reproduced with permission from Dange et al. dange2019oscillation.

In order to quench high amplitude limit cycle oscillations in two Rijke tube
oscillators, Dange et al. dange2019oscillation introduced frequency detuning
in the system (see Fig. 30). Frequency detuning can be introduced by varying
the natural frequency of one of the oscillators while keeping the natural
frequency of the other oscillator a constant. In a Rijke tube system, the
frequency of acoustic oscillations is varied by changing the length of the
duct, where the frequency is inversely proportional to the length of the duct
(i.e., $f_{n}\propto 1/L$). In Fig. 30c,d, we notice that the mutual
interaction between detuned Rijke tube oscillators having high amplitude of
limit cycle oscillations facilitates a small suppression of oscillations.
Increasing the detuning between the oscillators gradually increases the
suppression of limit cycle oscillations (Fig. 30a), and at sufficiently large
detuning, the state of amplitude death is observed in the system (Fig. 30e).
Further increase in the detuning engenders the state of partial amplitude
death (Fig. 30f), where large amplitude limit cycle oscillations are restored
in one Rijke tube oscillator while the other retains in a state of nearly
suppressed periodic oscillations. These results are qualitatively similar to
the occurrence of amplitude death and partial amplitude death in non-identical
diffusively and time delay coupled weakly nonlinear oscillators atay2003total.

Figure 30: Variation of (a) relative suppression in pressure amplitude and (b)
frequency of each oscillator with frequency detuning in a pair of Rijke tube
oscillators. Introduction of frequency detuning in the coupled systems
engendered (c, d) enhanced suppression, (e) amplitude death and (f) partial
amplitude death with an increase in the frequency detuning between the
oscillators. Reproduced with permission from Dange et al.
dange2019oscillation.

Srikanth et al. srikanth2021dynamical theoretically studied the effect of
variation in coupling parameters (Fig. 31a) and system parameters (Fig. 31c)
on the occurrence of amplitude death in two time-delay coupled Rijke tube
oscillators given in equations (11). In Fig. 31a, we notice that amplitude
death occurs for a specific range of coupling delay ($\tau_{c}$), and the size
of the amplitude death islands decreases with an increase in the value of
coupling strength ($K_{\tau c}$). In Fig 31c, we observe that the occurrence
of amplitude death is highly dependent on the heater power ($\overline{W}$),
and thus the amplitude of limit cycle oscillations in the uncoupled state;
increase in the amplitude, narrows the amplitude death regions and eventually
suppresses them completely. Furthermore, they srikanth2021dynamical found that
the transition between amplitude death and oscillatory state (i.e., limit
cycle oscillations) depends on the nature of the bifurcation of the isolated
oscillator. To elaborate, if an oscillator exhibits subcritical Hopf
bifurcation during the transition from steady state to limit cycle
oscillations in its uncoupled state, the coupled behavior of two such
oscillators show explosive transition with a hysteresis region as observed for
subcritical Hopf bifurcation (see Fig. 31b,d). Arrows 1 and 2 in Fig. 31a,c
indicate that the transition from in-phase synchronization to anti-phase
synchronization in coupled Rijke tube oscillators happens either through
amplitude death region or via phase-flip bifurcation, respectively. Thomas et
al. thomas2018effectb investigated the effect of Gaussian white noise on the
occurrence of amplitude death in the model of coupled Rijke tube oscillators
and showed that the abrupt transition from the oscillatory state to the steady
state becomes continuous due to prebifurcation noise amplification
surovyatkina2005prebifurcation.

Having discussed various studies on the mitigation of thermoacoustic
instabilities in Rijke-tube oscillators utilizing different approaches from
synchronization theory, we next discuss studies that propose various early
warning signals for critical transitions in Rijke tube systems.

Figure 31: Two-parameter bifurcation plots between (a) coupling delay
($\tau_{c}$) and coupling strength ($K_{\tau c}$) and (c) coupling delay
($\tau_{c}$) and heater power ($\overline{W}$) obtained from a model of two
time-delay coupled horizontal Rijke tube oscillators. (b, d) One-parameter
bifurcation plot between the root-mean-square value of acoustic pressure
fluctuations ($p^{\prime}_{rms}$) in the coupled system with a change in
values of $K_{\tau c}$ and $\overline{W}$ for a fixed value of $\tau_{c}$,
respectively. We note that an increase in $\overline{W}$ correspondingly
increases the amplitude of limit cycle oscillations in the system. Arrows in
(a) and (c) indicate the routes from in-phase to anti-phase states of
synchronization. Adapted with permission from Srikanth et al.
srikanth2021dynamical.

###  VI.4 Early warning signals (Precursors)

As discussed in Sec. III, many dynamical systems undergo abrupt transitions
(also called tipping or critical transitions) as the control parameter is
varied. In such systems, an early warning for the occurrence of these
transitions is necessary to avoid the consequences that arise after their
onset. In some cases where the variation of control parameter is continuous,
the rate at which such a parameter is varied greatly affects the performance
of early warning measures for such critical transitions. In this section, we
discuss various early warning measures developed in Rijke tube systems to
detect the occurrence of critical transitions.

In thermoacoustic systems, different measures have been invented to obtain
early warning signals that predict the occurrence of thermoacoustic
instability pavithran2021predicting. These measures exhibit a drastic change
in their values prior to the onset of such instability. Tracking the changes
in the measure can help us provide early warning for the impending
instability. Thus, we can prevent the system from reaching the state of
thermoacoustic instability and thereby evade its consequences. This approach
goes along the saying, ‘prevention is better than cure’.

Figure 32: (a) Time series of the acoustic pressure exhibiting subcritical
transition to limit cycle and the effectiveness of early warning signals such
as (b) variance and (c) lag-1 autocorrelation in identifying the onset of the
impending thermoacoustic instability. The onset of such instability is marked
using a solid black vertical line, whereas the dotted line represents the time
step from which the measures are calculated. Adapted with permission from
Gopalakrishnan et al. gopalakrishnan2016early.

Gopalakrishnan et al. gopalakrishnan2016early applied the knowledge of
critical slowing down on approaching the tipping point in a horizontal Rijke
tube to predict the occurrence of subcritical Hopf bifurcation in the system.
The phenomenon of critical slowing down is associated with the loss of
stability of the system as the control parameter approaches the bifurcation
point. It also indicates the slow recovery rate of the system to the external
perturbations introduced close to a critical transition strogatz1994nonlinear.
Their study obtained early warning signals from variance and lag-1
autocorrelation of the acoustic pressure data from experiments and from the
model of the Rijke tube (Fig. 32). Figure 32(a) shows the acoustic pressure
signal obtained from the horizontal Rijke tube as the heater power is varied
continuously at a fixed rate. The onset of limit cycle oscillations in the
system is observed at $t=644s$. The measures, variance and lag-1
autocorrelation, are calculated for a moving window. Gopalakrishnan et al.
gopalakrishnan2016early observed an increase in the variance (Fig. 32b) and a
decrease in lag-1 autocorrelation (Fig. 32c) well before the onset of
thermoacoustic instability. Furthermore, they suggested that the variance of
the signal is more robust to external noise imposed by the loudspeaker in
predicting the onset of thermoacoustic instability when compared to
autocorrelation function.

A recent study by Pavithran and Sujith pavithran2021effect examined the impact
of the rate of change of control parameter on the performance of different
early warning signals of thermoacoustic instability in a horizontal Rijke
tube. Various early warning measures, such as lag-1 autocorrelation ($AC$),
variance ($VAR$), skewness ($SKEW$), kurtosis ($K$), and Hurst exponent ($H$),
were chosen for the investigation (Fig. 33). During the onset of
thermoacoustic instability, the root mean square value of the acoustic
pressure fluctuations indicates a significant growth in the amplitude (Fig.
33a,g,m). Autocorrelation (Fig. 33b,h,n) and variance (Fig. 33c,i,o) tend to
increase much before the actual transition on approaching the onset of
thermoacoustic instability due to critical slowing down scheffer2009early;
dakos2012methods. Skewness which also tends to increase on approaching the
tipping point (Fig. 33d,j,p) does not have any relation with critical slowing
down scheffer2009early. As the system approaches the transition to
thermoacoustic instability, the skewness of the distribution changes from
being negative to positive. Kurtosis ($K$) exhibits a value of 3 for a normal
distribution; however, it does not show any perceivable trend during the onset
of thermoacoustic instability (Fig. 33e,k,q).

The Hurst exponent ($H=2-D$, $D$ is the fractal dimension) computes the
correlations in the time series hurst1951long; mandelbrot1977fractals. When
the value of $H$ is greater than 0.5, the signal is said to be persistent, and
if it is less than 0.5, the signal is anti-persistent. When $H=0.5$, the
signal is uncorrelated. Hence, as the system transitions from a regime of
uncorrelated oscillations to a state of periodic high amplitude oscillations,
the Hurst exponent decreases from 0.5 to zero, serving as a potential measure
to detect the transition to thermoacoustic instability with changing fractal
characteristics of the pressure signal (Fig. 33f,l,r). Comparing the behavior
of each of the aforementioned early warning measures at different rates of
change of the control parameter, Pavithran and Sujith pavithran2021effect
noticed that measures such as lag-1 autocorrelation and Hurst exponent are
able to predict the transition well before the tipping point for both slow and
high rates; thus, providing adequate warning time for control actions (Fig.
33).

Although these measures provide warnings about an upcoming critical
transition, they cannot provide information about what bifurcation is to be
expected. As the system approaches the tipping point, its dynamical behavior
can be simplified into a limited number of possible “normal forms”. This, in
turn, provide information about the new state (i.e., oscillatory or steady
state) that may occur after the tipping point. Towards this, Bury et al.
bury2021deep proposed a deep learning algorithm that provides early warning
signals by using information about normal forms and scaling behavior of the
dynamics near tipping points in a horizontal Rijke tube system.

Lee et al. lee2020input proposed a framework for performing input-output
system identification near Hopf bifurcation. By using the data from the steady
state behavior, they were able to predict the location and the criticality of
Hopf bifurcation in a laminar Rijke tube burner and in a model of Duffing-Van
der Pol oscillator perturbed with additive white noise. This novel methodology
does not involve the crossing of threshold values (unlike other measure
discussed before) and can be applied to various other dynamical systems that
exhibit Hopf bifurcation (or systems that can be reduced to Stuart-Landau
equations). Premraj et al. premraj2021dragon investigated the occurrence of a
catastrophic transition, such as flame blowout, in a premixed laminar Rijke
tube burner. During flame blowout, the flame ceases to exist in the combustor
as the time scales of flow fluctuations become much larger than the reaction
timescales in the system. After the flame blowout, the amplitude of the
acoustic pressure fluctuations in the system drops to a very low value.
Premraj et al. premraj2021dragon noticed that the occurrence of flame blowout
is preceded by the existence of extreme events in the acoustic field of the
system and found the presence of special kind of extreme events called the
dragon-king extreme events just prior to flame blowout. Thus, the early
warning to flame blowout can be provided by identifying the dragon-king
extreme events in the acoustic field of the system.

Early warning measures have been investigated extensively in many systems
including Rijke tubes and to date remain a topic of immense attention due to
its potential applications. The readers are guided towards a recent review on
critical transitions and early warning signals by Pavithran et al.
pavithran2021critical for a more elaborate discussion.

Figure 33: Variation of (a,g,m) the rms amplitude of the pressure data
obtained from a horizontal Rijke tube and the early warning measures such as
(b,h,n) lag-1 autocorrelation, (c,i,o) variance, (d,j,p) skewness, (e,k,q)
kurtosis, and (f,l,r) Hurst exponent applied to identify the impending
thermoacoustic instability with respect to the control parameter, heater
power. Each column corresponds to varying rate of change of the control
parameter. The dashed vertical line indicates the onset of limit cycle
oscillations in the system and the red arrow points at the quasistatic Hopf
point ($\mu$). Adapted with permission from Pavithran and Sujith
pavithran2021effect.

##  VII  Conclusions

In the present review, we introduced the Rijke tube oscillator as a novel
oscillator to the nonlinear dynamics community. Towards this purpose, We
systematically presented the potential applications of the Rijke tube
oscillator in obtaining experimental verification of various dynamical
phenomena that are observed in general paradigmatic oscillators. We showed
that, depending on the operating conditions, the onset of limit cycle
oscillations in a Rijke tube can happen either through subcritical or
supercritical Hopf bifurcation. We also observed the occurrence of secondary
bifurcations to various dynamical phenomena such as quasiperiodic, period-$k$,
chaotic, and strange nonchaotic oscillations, along with the presence of
different routes to chaos in the Rijke tube system. We further noticed the
existence of different noise-induced transitions such as coherence resonance,
stochastic bifurcation, and subcritical excitation to limit cycle oscillations
in the subthreshold and bistable regimes of the system operation. We examined
the mutual synchronization and forced synchronization properties of Rijke tube
oscillators and summarised different states of synchronization witnessed in
such systems. We further discussed the application of different concepts from
synchronization theory to control and mitigate the limit cycle oscillations
using the phenomena of amplitude death, partial amplitude death, and
asynchronous quenching in the Rijke tube systems. Finally, we presented the
use of a Rijke tube system to experimentally verify various early warning
measures proposed theoretically for the catastrophic transitions.

We hope that this review paves way for the dynamical systems community to
utilize the Rijke tube oscillator for performing both experimental and
theoretical studies in the future. In this paper, we have restricted our
discussion on the application of Rijke tube oscillators to investigate the
dynamical behavior of a single and two coupled oscillators, for the variation
of both systems parameters and coupling parameters. However, we know that with
an increase in the number of oscillators in a system (or a network) or change
in their coupling structures between the oscillators (e.g., local, non-local,
and global couplings or star, line and ring topologies), we can observe many
complex dynamical phenomena resulting from the coupling of oscillators
manoj2021experimental; wickramasinghe2013spatially. These phenomena include
clustering, splay states, bare minimum chimera, weak chimera, etc. in minimal
number of oscillators, where the number of oscillators is less ($\approx$ 10).
Different modeling studies in the past showed the occurrence of these
phenomena in small networks of oscillators. In the future, we can construct a
minimal network of Rijke tube oscillators and through appropriate coupling
mechanisms, we can experimentally validate theoretically discovered phenomena.
Furthermore, we have also discussed the possibility of oscillation quenching
via amplitude death in a system coupled oscillators. However, in some systems,
such suppression of oscillations is undesirable. Recently, through various
theoretical studies zou2021quenching, restoring of oscillations in a system is
shown by adding a processing delay factor in the coupling term of oscillators.
Developing a controlled experiment on a network of Rijke tube oscillators to
revoke the oscillations from the death state by varying the feedback factor in
the coupling of oscillators is another potential future scope.

In practical systems, the control parameter of a system does not change quasi-
statically; however, it varies continuously in time with different rates. Many
studies have been performed investigating rate-dependent tipping (bifurcation)
on a single system. Such investigations can be extended to multiple coupled
oscillators. Moreover in a system of two coupled oscillators, the effect of
the rate of change of control parameters in one system on the the tipping
behavior of the other can be studied theoretically and validated
experimentally using coupled Rijke tube systems. Furthermore, the presence of
noise on the coupled behavior of Rijke tube oscillators, in terms of their
transition from steady state to limit cycle oscillations or the occurrence of
different dynamical states, can also be included under future investigations.

## Acknowledgements

We grateful to the IoE initiative (SB/2021/0845/AE/MHRD/002696), and the J. C.
Bose Fellowship (No. JCB/2018/000034/SSC) from the Department of Science and
Technology (DST) for the financial support.