# Dynamics of coupled thermoacoustic oscillators
under asymmetric forcing: Experiments and theoretical modeling

Ankit Sahay<EMAIL_ADDRESS>Department of Aerospace Engineering,
Indian Institute of Technology Madras, Chennai 600036, India  Amitesh Roy
Department of Aerospace Engineering, Indian Institute of Technology Madras,
Chennai 600036, India  Samadhan A. Pawar  Department of Aerospace Engineering,
Indian Institute of Technology Madras, Chennai 600036, India  R. I. Sujith
Department of Aerospace Engineering, Indian Institute of Technology Madras,
Chennai 600036, India

###### Abstract

Quenching of limit cycle oscillations (LCO), either through mutual coupling or
external forcing, has attracted wide attention in several fields of science
and engineering. However, the simultaneous utilization of these coupling
schemes in quenching of LCO has rarely been studied despite its practical
applicability. We study the dynamics of two thermoacoustic oscillators
simultaneously subjected to mutual coupling and asymmetric external forcing
through experiments and theoretical modeling. We investigate the forced
response of both identical and non-identical thermoacoustic oscillators for
two different amplitudes of LCO. Under mutual coupling alone, identical
thermoacoustic oscillators display the occurrence of partial amplitude death
and amplitude death, whereas under forcing alone, asynchronous quenching of
LCO is observed at non-resonant conditions. When the oscillators are
simultaneously subjected to mutual coupling and asymmetric forcing, we observe
a larger parametric region of oscillation quenching than when the two
mechanisms are utilized individually. This enhancement in the region of
oscillation quenching is due to the complementary effect of amplitude death
and asynchronous quenching. However, a forced response of coupled non-
identical oscillators shows that the effect of forcing is insignificant on
synchronization and quenching of oscillations in the oscillator which is not
directly forced. Finally, we qualitatively capture the experimental results
using a reduced-order theoretical model of two coupled Rijke tubes which are
coupled through dissipative and time-delay coupling and asymmetrically forced.
We believe that these findings offer fresh insights into the combined effects
of mutual and forced synchronization in a system of coupled nonlinear
oscillators.

††preprint: APS/123-QED

##  I  Introduction

Coupled interacting nonlinear oscillators appear extensively in physical
systems around us [1]. Depending upon the nature of coupling, a population of
interacting oscillators can synchronize starting from an initially incoherent
or desynchronized state. Such synchronized oscillators can exhibit an
emergence of ordered patterns as observed in several examples in nature, such
as flocking of bird or flashing of light by fireflies [2]. However,
synchronization of oscillations may be detrimental sometimes, and may need to
be desynchronized [3] or quenched [4]. Here, we study the practical
application of synchronization theory in attaining control of self-excited
oscillations in a system of coupled prototypical thermoacoustic oscillators. A
thermoacoustic oscillator refers to a confined combustion system (for example,
gas turbine combustor and rocket engines). In such a system, positive feedback
between the acoustic pressure oscillations and the heat release rate
oscillations leads to the generation of large amplitude, self-sustained
periodic oscillations in the acoustic field. The occurrence of these large
amplitude oscillations is referred to as thermoacoustic instability (TAI). The
presence of such instabilities can inflict considerable damage to mechanical
and structural components used in gas turbine and rocket engines [5, 6, 7].
Complex interactions between the acoustic field, turbulent flow field, and
heat release rate field has recently led to the widespread use of complex
systems approach to understand the phenomenon of thermoacoustic instability
[6, 8].

In general, there are two types of interactions leading to synchronization of
oscillators: mutual and forced [7]. In the former, oscillators mutually
interact through bidirectional coupling, leading to the adjustment of phases
and frequencies of both the oscillators to a common state of mutual
synchronization. Oscillation quenching attained through the phenomenon of
amplitude death (AD) in mutually coupled systems is an exciting prospect that
has been shown to work for various systems [9, 10, 11, 12]. Amplitude death in
a population of strongly coupled oscillators refers to a situation where all
the oscillators pull each other off of their oscillatory state into the same
stable fixed point, leading to complete cessation of oscillations. If the
mutual coupling is not strong enough, the situation of partial amplitude death
(PAD) may arise, where some oscillators retain their oscillatory behavior
while oscillations are ceased in others [13]. On the other hand, in the case
of forced synchronization, an independent master system (an external force)
drives a slave system (driven oscillator); thus, forming a unidirectional
master-slave system [14]. In this type of coupling, the driven system adjusts
its phase and frequency to that of the external forcing during the state of
synchronization. Depending upon the frequency and the amplitude of forcing,
the natural oscillations can be phase-locked to forcing, and in some cases,
the self-excited oscillations can be completely suppressed through the
phenomenon of asynchronous quenching [15, 16].

Quite a few numerical studies have been conducted to investigate the
simultaneous effect of forcing in a system of coupled oscillators. In a system
of two weakly coupled Van der Pol oscillators, Battelino [17] observed that
when each oscillator is externally forced, and a constant phase difference is
present between the forcing signals applied to each of the oscillators, the
system exhibits three-frequency quasiperiodicity. As the coupling strength and
forcing amplitude are increased, three-frequency quasiperiodicity is first
replaced by a two-frequency quasiperiodic regime, and subsequently by phase-
locked periodic and chaotic regimes. The transition between the phase-locked
region and the two- and three-frequency quasiperiodic regions takes place
through saddle-node bifurcations [18, 19]. The external force first destroys
the regime of mutual synchronization of oscillators. As the forcing amplitude
is increased, the oscillator which is subjected to external forcing
synchronizes with the forcing signal first, followed by the forced
synchronization of the entire system [20]. Similar observations were made in a
system of coupled Van der Pol and Duffing oscillators [21]. Such an approach
was utilized for modeling the effect of a pacemaker on the dynamics of the
human heart [22]. On the flip side, to the best of our knowledge, only one
experimental study has been conducted to understand the effect of forcing on
mutually coupled electronic circuits with the aim of verifying the phase
dynamics associated during regimes of desynchronized, mutually synchronized,
two-frequency quasiperiodic, three-frequency quasiperiodic, and forced
synchronized oscillations [18]. Further, all the aforementioned studies focus
on the phase dynamics of the system with particular attention afforded to the
dynamics of bifurcations among quasiperiodic, periodic, and chaotic regimes.
In light of the above discussed limitations of past studies, there is a need
to experimentally quantify the effect of mutual coupling and external forcing
on the amplitude of the limit cycle oscillations (LCO) observed in both the
oscillators. Thus, we analyze the phase as well as the amplitude response of
two coupled thermoacoustic oscillators under the condition of asymmetric
external forcing, and model the observations with a low-order model. This
makes up the key objective of the present study, and is of general interest to
the nonlinear dynamics community.

Biwa et al. [4] were the first to experimentally demonstrate the control of
LCO through amplitude death in coupled thermoacoustic engines using both the
dissipative and time-delay couplings. Thomas et al. [23] investigated
amplitude death in a model of coupled Rijke tubes. They found that the
simultaneous presence of dissipative and time-delay couplings was far more
effective in attaining AD in the Rijke tubes than either of the two types of
coupling applied separately. In a follow-up study with additive Gaussian white
noise, Thomas et al. [24] observed that the presence of noise affects the
coupled behavior of oscillators and mutual coupling only leads to about 80%
suppression in the amplitude of the uncoupled LCO as opposed to the state of
AD observed in the absence of noise in the system. Dange et al. [25] performed
detailed experimental characterization of coupled Rijke tube oscillators and
found that only time delay coupling is sufficient to achieve AD of low
amplitude LCO. In oscillators undergoing high amplitude oscillations,
frequency detuning is needed in addition to time delay coupling for attaining
AD and PAD. They also reported the phenomenon of phase-flip bifurcation in the
coupling of identical thermoacoustic oscillators. Recently, Hyodo et al. [26]
used double-tube coupling method to significantly reduce the tube diameter
necessary for demonstrating AD in a system of coupled identical Rijke tubes.
Jegal et al. [27] demonstrated the occurrence of AD in a practical turbulent
system consisting of two lean-premixed model swirl combustors. Moon et al.
[28] compared the characteristics of mutual synchronization of these systems
when the length and diameter of the coupling tube were changed. Jegal et al.
[27] also found that under different conditions, mutual synchronization can
also trigger the strong excitation of a new mode, even when the two combustors
are individually stable in the absence of coupling.

On the other hand, Balusamy et al. [29] experimentally studied the forced
response of LCO in a swirl-stabilized turbulent combustor using the framework
of forced synchronization and observed different states such as phase-locking,
phase drifting, and phase trapping in the system. Kashinath et al. [30]
highlighted the route to forced synchronization of limit cycle, quasiperiodic,
and chaotic oscillations in a numerical model of the premixed flame. These
findings were verified in experiments on Rijke tubes [16] and laminar
combustors [31, 32, 33]. Mondal et al. [16], Guan et al. [31] and Roy et al.
[32] further showed that forcing at frequencies away from the natural
frequency of LCO lead to greater than 80% decrease in their amplitude through
a phenomenon known as asynchronous quenching.

The aforementioned studies demonstrate the possibility of controlling
thermoacoustic instability based on mutual or forced synchronization. However,
the scope of these studies is still limited as external forcing can quench
thermoacoustic instability in a single system in a specific range of forcing
parameters, whereas practical engines generally have multiple combustors
working in tandem. Hence, the information on how the forcing of thermoacoustic
instability in one combustor affects the thermoacoustic instability developed
in another combustor is still unknown. Similarly, mutual coupling works for
two coupled oscillators; however, the parametric regime for which amplitude
death is observed is limited. Therefore, there is a need to combine both these
methodologies to overcome their individual limitations. Essentially, we aim to
implement a proof-of-concept capable of combining the best of both strategies
– asynchronous quenching and mutual synchronization – to control
thermoacoustic oscillations. Towards this purpose, we couple two Rijke tubes
during the state of thermoacoustic instability and subject one to external
harmonic forcing (asymmetric forcing). We then measure the phase and amplitude
response of the resultant acoustic pressure oscillations in the combustors at
different conditions of forcing and coupling parameters. We perform asymmetric
forcing experiments on identical and non-identical thermoacoustic oscillators
to comprehensively assess the response to forcing. We find that through
asymmetric forcing, we can expand the region of oscillation quenching of
thermoacoustic instability in the system of coupled identical Rijke tubes by
compounding the effect of asynchronous quenching and mutual synchronization.
Finally, we develop a model where two Rijke tube oscillators are coupled
through dissipative and time-delay coupling and are forced asymmetrically. We
show that the model compares favorably with the experimental results,
indicating the usefulness of reduced-order modeling of coupled oscillator
models under forcing.

The rest of the paper is organized as follows. In §II we describe the
experimental setup. In §III.1, we investigate the forced synchronization
characteristics of a single Rijke tube. We then demonstrate the presence of AD
and PAD in mutually coupled Rijke tubes in §III.2. In §III.3, we characterize
the response of coupled identical Rijke tubes to asymmetric forcing. We
consider the response of non-identical oscillators in §III.4. In §IV, we
describe the model and numerically investigate the dynamics of coupled Rijke
tube oscillators under forcing. Finally, we present the conclusions from the
study in §V.

##  II Experimental Setup

The experimental setup used in this study consists of a pair of horizontal
Rijke tubes (Fig. 1). Rijke tube A has a cross-section of $9.3\times 9.4$
$\text{cm}^{2}$ with a length of $102$ cm. Rijke tube B has a cross-section of
$9.3\times 9.5$ $\text{cm}^{2}$ and a length of $104$ cm. A decoupler of
dimensions $102\times 45\times 45$ $\text{cm}^{3}$ is attached to the inlet of
the respective Rijke tube to eliminate upstream disturbances and ensure a
steady flow in the system. Each Rijke tube consists of a separate electrically
heated wire mesh, powered by an external DC power supply, which acts as a
compact heat source. The heaters are located $27$ cm downstream of the inlet
in each duct. The air flow rate is maintained constant in each of the Rijke
tubes through separate mass flow controllers (MFC, Alicat Scientific) of
uncertainty $\pm$(0.8% of the measured reading + 0.2% of the full scale
reading).

Figure 1:  Schematic of the experimental setup having two horizontal Rijke
tube oscillators A and B, which are coupled using a connecting tube. Rijke
tube A is acoustically forced with 4 acoustic drivers attached to its sides.
An extension duct in Rijke tube A is used for implementing frequency detuning
in the system.

Both the Rijke tubes are coupled using a single vinyl tube of length $L$ and
internal diameter $d$ (see Fig. 1). The ports for the connecting tube are
located $57$ cm downstream of the inlet. Ball-type valves are manually opened
to establish coupling between the two oscillators. Four wall mounted acoustic
drivers (Minsound TD-200A) are attached on each side of Rijke tube A, $41$ cm
downstream of the inlet. The acoustic drivers are connected in parallel to a
power amplifier (Ahuja UBA-500M). Sinusoidal forcing signal with mean-to-peak
amplitude ($A_{f}$, in mV) and frequency ($f_{f}$, in Hz), generated using a
Tektronix function generator (Model No. AFG1022), is input to the power
amplifier to drive the coupled system.

A piezoelectric pressure transducer (PCB 103B02, sensitivity $217.5$ mV/kPa,
uncertainty $\pm 0.15$ Pa) is mounted close to the midpoint along the length
of the duct in each Rijke tube and is used for measuring the acoustic pressure
fluctuations in the system. Data are acquired simultaneously from both the
Rijke tubes at a sampling rate of $10$ kHz for a duration of $5$ s for each
parametric condition using a data acquisition system (NI USB 6343). The
resolution of frequency in the power spectrum of the signals is equal to 0.2
Hz. The rate of decay of an acoustic pulse generated from loudspeakers in the
absence of flow is used to measure the damping in the two Rijke tubes. The
acoustic decay rate values for Rijke tubes A and B are measured to be $16.5\pm
2$ $\text{s}^{-1}$ and $12.9\pm 1.8$ $\text{s}^{-1}$, respectively. To ensure
repeatability of the results and consistency of the experimental conditions,
the experiments are conducted only when the measured acoustic decay rates lie
in the range mentioned above. A change in the decay rate only changes the
critical parameter values at which the different dynamical states are
observed; however, the overall dynamics remain the same. During each
experiment, both the Rijke tubes are preheated for $10$ minutes by supplying
DC power at $1$ V to the wire mesh. The preheating ensures a steady
temperature profile inside the Rijke tubes.

Figure 2: Forced response of a single Rijke tube. (a,c) The phase response
shown in terms of $PLV$ and (b,d) amplitude response in terms of fractional
change in the amplitude of LCO in the Rijke tube for (a,b)
$p^{\prime}_{0}=120$ Pa and (c,d) $p^{\prime}_{0}=200$ Pa for different values
of $\bar{A_{f}}$ and $f_{f}$. The synchronization boundaries in (a-d) are
obtained through least-square-fit of points where $PLV=0.98$. The $R^{2}$
values for the least-square fitting are given in Section I of the Supplemental
Material. The region of forced synchronization decreases with increase in
$p^{\prime}_{0}$ (a,c), while quenching of LCO is observed only for
$f_{f}<f_{n0}$ (b,d). Green region around $f_{f}/f_{n0}=1$ in (b,d) signifies
the amplification of LCO above twice the value of unforced amplitude, such
that $\Delta p^{\prime}_{rms}/p^{\prime}_{0}$ varies in the range (-5.76, -1)
in (b), and (-4.47, -1) in (d).

A telescopic slide mechanism of 12 cm length is used to vary the length of
Rijke tube A. The natural frequency of Rijke tube A can be varied between
$f_{n0}=162$ Hz to $f_{n0}=148$ Hz with an uncertainty of $\pm 2$ Hz due to
the uncertainty in measuring the length of the Rijke tube. After preheating
both the Rijke tubes, the heater power is increased such that the system
undergoes subcritical Hopf bifurcation. The heater power is maintained away
from the bistable region in all experiments. LCO are maintained in each of the
Rijke tubes before coupling is induced. The amplitude of the LCO are
controlled by varying the heater power and the air flow rate. All the
experiments are reported for two different amplitudes (root-mean-square) of
the LCO in both the Rijke tubes in the uncoupled state: (a) lower amplitude
$p^{\prime}_{0}=120$ Pa maintained by supplying a constant air flow rate of
$2.4\pm 0.01$ g/s (Re = $1370$) in each Rijke tube, and (b) higher amplitude
$p^{\prime}_{0}=200$ Pa maintained by supplying a constant air flow rate of
$3.95\pm 0.01$ g/s (Re = $2284$) in each Rijke tube. After both the
oscillators exhibit LCO in their uncoupled state, the valves are opened to
couple these oscillators. The coupled system is then asymmetrically forced
through the loudspeakers connected to Rijke tube A.

##  III Results and discussions

###  III.1 Response of a thermoacoustic oscillator to external forcing

We begin by investigating the forced acoustic response of LCO in a single
Rijke tube. Figure 2 shows a two-parameter bifurcation plot on an
$\bar{A_{f}}-f_{f}$ plane illustrating the phase (Figs. 2a,c) and amplitude
(Figs. 2b,d) response of LCO under external forcing. The natural frequency of
LCO is $f_{n0}=160\pm 2$ Hz. The forced response is studied for two different
amplitudes of unforced LCO: $p^{\prime}_{0}=120$ Pa corresponding to
$p_{0}=26$ mV (Figs. 2a,b) and $p^{\prime}_{0}=200$ Pa corresponding to
$p_{0}=43$ mV (Figs. 2c,d). Here, $p^{\prime}_{0}$ refers to the amplitude of
the unforced LCO in pascal (Pa), and $p_{0}$ refers to the equivalent reading
obtained from the piezoelectric transducer in mV. The values of forcing
amplitude $A_{f}$ (in mV) are normalized with the unforced amplitude of LCO
measured in mV such that $\bar{A_{f}}=A_{f}/p_{0}$.

In Figs. 2a,c, we plot the distribution of phase locking value ($PLV$) between
the forced LCO and the external forcing signal on the $\bar{A_{f}}-f_{f}$
plane. $PLV$ quantifies the degree of synchronization between a pair of
oscillators at any given condition of forcing ($A_{f}$, $f_{f}$). It is
defined as [7],

$PLV=\dfrac{1}{N}\Bigg{\lvert}\sum\limits_{n=1}^{N}\text{exp}(i\Delta\phi)\Bigg{\lvert},$
(1)

where $N$ is the length of the $p^{\prime}$ signal, and $\Delta\phi$ is the
instantaneous phase difference between the $p^{\prime}$ and forcing signals.
Here, forcing is assumed to be sinusoidal of the form $F(t)=A_{f}\hskip
3.0pt\text{sin}(2\pi\hskip 1.0ptf_{f}\hskip 1.0ptt)$. The instantaneous phase
of the signals is determined using the analytic signal approach utilizing the
Hilbert transform [7]. A $PLV$ of $1$ indicates synchronization of two
oscillators, while a $PLV$ of $0$ indicates desynchronization of the
oscillators. In experimental situations, it is difficult to obtain $PLV=1$;
hence, we denote phase synchronization boundaries of LCO through a least-
square-fit of the points where $PLV=0.98$. The $R^{2}$ values of the least-
square-fit lines are given in Section I of the Supplemental Material. The
V-shaped region, called the Arnold tongue, separates the region of forced
synchronization from the region desynchronization. The critical amplitude of
forcing required for forced synchronization of LCO, $A_{f}=A_{f,c}$, increases
almost linearly with an increase in the frequency difference $\Delta
f=|f_{f}-f_{n0}|$.

From Figs. 2a,c, we notice that for $p^{\prime}_{0}=200$ Pa, the range of
$f_{f}$ for which forced synchronization occurs in the system is smaller than
that observed for $p^{\prime}_{0}=120$ Pa, indicating the dependence of forced
synchronization of LCO on their amplitude in the unforced state. Further, at
higher values of $\Delta f$, achieving forced synchronization of both LCO
becomes difficult. We also observe regions of desynchronized oscillations
(indicated in red) in the range of $f_{f}/f_{n0}=0.62-0.70$ at high values of
$\bar{A_{f}}$ ($\bar{A_{f}}>0.35$ in Figs. 2a,c). The reason for desynchrony
in this region is a result of period-3 behavior in $p^{\prime}$, which leads
to low values of PLV (see Section II of the Supplemental Material).

Figure 3: Coupled response of two Rijke tubes. Fractional change in the
amplitude of LCO for Rijke tube A and B as functions of the frequency detuning
($\Delta f_{n0}=|f_{n0}^{A}-f_{n0}^{B}|$) and length of the connecting tube
($L$) for (a,b) $p^{\prime}_{0}=120$ Pa and (c,d) $p^{\prime}_{0}=200$ Pa in
both the oscillators. The region of AD and PAD are indicated. At other regions
in the plots, LCO are observed at reduced amplitude due to coupling. The
parametric region exhibiting AD shrinks in size as $p^{\prime}_{0}$ is
increased from $120$ Pa to $200$ Pa.

We also plot the normalized change in the amplitude of LCO due to external
forcing, $\Delta
p^{\prime}_{rms}/p^{\prime}_{0}=(p^{\prime}_{0}-p^{\prime}_{rms})/p^{\prime}_{0}$
in Figs. 2b,d. Here, $\Delta p^{\prime}_{rms}/p^{\prime}_{0}\sim 1$
corresponds to complete suppression of LCO in the Rijke tube due to forcing,
whereas $\Delta p^{\prime}_{rms}/p^{\prime}_{0}<0$ indicates increase in the
amplitude of LCO above the unforced value due to forcing. The green region in
Figs. 2b,d corresponds to an increase in the amplitude of LCO above twice of
its unforced value (i.e., $\Delta p^{\prime}_{rms}/p^{\prime}_{0}<-1$) due to
resonant amplification of the acoustic pressure signal as $f_{f}$ is very
close to $f_{n0}$. $\Delta p^{\prime}_{rms}/p^{\prime}_{0}<-1$ implies that
the resultant amplitude is more than twice the amplitude of LCO in unforced
conditions. The simultaneous occurrence of forced synchronization for
$f_{f}\approx f_{n0}$ along with the resonant amplification of the amplitude
of LCO is referred to as synchronance [16]. We further observe a large region
with a reduction in amplitude greater than $80\%$ of $p^{\prime}_{0}$ only for
$f_{f}<f_{n0}$. A significant decrease in the amplitude of LCO at non-resonant
conditions ($f_{f}\neq f_{n0}$) of forcing is a result of asynchronous
quenching [16, 31]. Asynchronous quenching of LCO is achieved through forced
synchronization, where the response $p^{\prime}$ signal oscillates at $f_{f}$,
which can be seen from the coincidence of the boundaries of the Arnold tongue
with region corresponding to $\Delta p^{\prime}_{rms}/p^{\prime}_{0}\approx 1$
(see Figs. 2b,d). The asymmetry in the characteristics of the Arnold tongue
and asynchronous quenching of LCO arises due to the inherent nonlinearity of
the thermoacoustic system [30].

###  III.2 Response of coupled thermoacoustic oscillators

We now examine the response of two mutually coupled Rijke tubes when the
length of the coupling tube ($L$) and the frequency detuning $\Delta f_{n0}$
$(=f^{B}_{n0}-f^{A}_{n0})$ between them are varied independently, where
$f^{a}_{n0}$ and $f^{B}_{n0}$ are natural frequencies of oscillators A and B
in the uncoupled state, respectively. In this study, the diameter of the
connecting tube is kept constant at 1 cm for all the experiments (see Section
III of Supplemental Material).

We note that the use of the connecting tube induces an acoustic time delay in
the coupling between the two Rijke tubes, as a finite time is required for the
acoustic waves in the two oscillators to interact with one another [25]. A
change in the length of the coupling tube changes the acoustic time-delay
between the two mutually interacting Rijke tubes. A change in the diameter of
the coupling tube changes the coupling strength between the two interacting
Rijke tubes. The coupling strength comprises of two parts: time-delay coupling
and dissipative coupling. Dissipative coupling is related to the interaction
that arises due to mass transfer between the two ducts [34]. We observe the
presence of both time delay and dissipative coupling in our system.

In Fig. 3, we plot the fractional change in the amplitude of LCO in these
oscillators, $\Delta p^{\prime}_{rms}/p^{\prime}_{0}$, as a function of $L$
and $\Delta f_{n0}$ for two different values of $p^{\prime}_{0}=120$ Pa (Figs.
3a,b) and $200$ Pa (Figs. 3c,d). The colorbar illustrating $\Delta
p^{\prime}_{rms}/p^{\prime}_{0}$ values range from 0 to 1. During mutual
coupling of oscillators, we seldom observe any amplification of LCO.As a
result, $\Delta p^{\prime}_{rms}/p^{\prime}_{0}$ is never negative. In
contrast, during forcing experiments in Fig. 2b,d, we observe resonant
amplification of LCO.

Figure 4:  (a)-(c) Time series of $p^{\prime}$ in Rijke tubes A (blue) and B
(brown) sequentially illustrating the effect of coupling and forcing on the
amplitude of LCO in both the Rijke tubes for different coupling and forcing
parameters. In (a,c), the Rijke tubes are non-identical and have a frequency
difference of 10 Hz. (i,ii) The enlarged portions of the desynchronized LCO
and the state of PAD in (c), respectively. The coupling of oscillators always
leads reduction in the amplitude of LCO, while forcing of oscillators can have
both reduction or amplification effects, depending on the values of coupling
and forcing parameters. The common parameters in all plots are: $d=10$ mm,
$f_{f}=140$ Hz, and $A_{f}=30$ mV.

In Fig. 3, for identical oscillators ($\Delta f_{n0}=0$ Hz), we do not observe
any perceivable reduction in the amplitude of LCO due to coupling for either
values of $p^{\prime}_{0}$. However, a significant reduction in the amplitude
of LCO is observed when non-identical oscillators are coupled. We observe two
states of oscillation quenching, i.e., amplitude death (AD) and partial
amplitude death (PAD), in coupled Rijke tubes for a particular range of $L$.
For example, when $\Delta f_{n0}=5-10$ Hz and $L=85$ cm, we witness the
presence of PAD in the system (see green box in Fig. 3), while for $\Delta
f_{n0}=7-10$ Hz and $L=95$ cm, we notice the occurrence of AD in the system
(see yellow box Fig. 3). During the state of PAD, LCO in Rijke tube B
(unforced) undergoes suppression and that in Rijke tube A (forced) remains at
the reduced amplitude. For the state of AD, we observe greater than $95\%$
decrease in the amplitude of LCO in both the oscillators due to coupling. The
range of $\Delta f_{n0}$ for which AD is observed in the system is smaller for
$p^{\prime}_{0}=200$ Pa (Fig. 3c,d) when compared to that for
$p^{\prime}_{0}=120$ Pa (Fig. 3a,b). Thus, the occurrence of AD in Rijke tube
oscillators has a dependence on the amplitude of uncoupled LCOs. It is quite
clear from Fig. 3 that the mutual coupling induced through the coupling tube
is not capable of inducing AD in identical oscillators, and the occurrence of
AD in high amplitude LCO is restricted to a small range of $L$ and requires a
finite value of $\Delta f_{n0}$ in the system [25].

###  III.3 Response of coupled identical thermoacoustic oscillators under
asymmetric forcing

As observed in Fig. 3, in the absence of frequency detuning, mutual coupling
is ineffective in achieving oscillation quenching (i.e., AD or PAD) in the
coupled system. Hence, we asymmetrically force the coupled thermoacoustic
oscillators to enhance the quenching of LCOs. Figure 4 shows the
representative time series of the acoustic pressure fluctuations in Rijke tube
A (in blue) and B (in brown) under the effect of mutual coupling and external
forcing when introduced sequentially. Both the oscillators are initially
coupled through a single tube and then oscillator A is forced through
loudspeakers. The time instant when coupling and forcing are switched on/off
are marked as well. In some situations, we notice that coupling and forcing
can marginally reduce the amplitude of LCO in the coupled system (Fig. 4a).
While in others, forcing can completely quench LCO in one and reduce the
amplitude of LCO in the other oscillator (Fig. 4b). This situation is akin to
the state of PAD. Thus, forcing aids in attaining PAD. On the flip side, in
Fig. 4c, we observe that forcing can also lead to an amplification of the
acoustic pressure oscillations, as can be seen from the increased amplitude of
these oscillations in Rijke tube A above the uncoupled value. In contrast, the
acoustic pressure oscillations remain quenched in Rijke tube B. Thus, the
effectiveness of coupling and forcing of thermoacoustic oscillators is
restricted to a particular range of coupling and forcing parameters and hence,
we focus on the identification of such regions in the subsequent discussion.

In Fig. 5, we plot the amplitude (Fig. 5a,b) and the phase (Fig. 5c,d)
response of identical Rijke tubes as a function of the forcing amplitude
($A_{f}$) and the length of coupling tube ($L$). The phase response is
determined from the $PLV$ calculated between the external forcing signal and
$p^{\prime}$ for either of the Rijke tubes. Only Rijke tube A is subjected to
external forcing. The forcing frequency is chosen as $f_{f}=100$ Hz
($f_{f}/f_{n0}\approx 0.6$) for which we observe asynchronous quenching in a
single forced Rijke tube oscillator (Fig. 2d).

For $A_{f}=0$, we observe that a change in $L$ does not lead to any
suppression of LCO in the oscillators (Fig. 5a,b). With an increase in
$A_{f}$, we observe a gradual decrease in the amplitude of LCO in both the
oscillators with the effect on Rijke tube B (unforced) being more pronounced
than Rijke tube A (forced). Complete suppression of LCO ($\Delta
p^{\prime}_{rms}/p^{\prime}_{0}\approx 1$) is observed in a particular range
of $L$ in Rijke tube B (Fig. 5b), while a significant decrease in the
amplitude of LCO (i.e., $\Delta p^{\prime}_{rms}/p^{\prime}_{0}\approx 0.8$)
is observed over a wider range of $L$ in Rijke tube A (Fig. 5a). For the
coupled systems without forcing in Fig. 3, we did not observe AD or PAD when
identical Rijke tubes were coupled for any $L$. However, the same coupled
identical oscillators when asymmetrically forced exhibit the state of PAD for
a range of $L$ (see point e in Fig. 5a). Thus, we notice that a larger region
over which suppression of high amplitude LCO can be achieved in coupled
identical thermoacoustic oscillators when the oscillators are coupled in a
particular range of L and asymmetric forcing is applied at non-resonant
frequencies. We reiterate that we do not observe complete suppression of
amplitude of LCO in Rijke tube A.

Figure 5:  Response of coupled, identical oscillators to asymmetric forcing.
(a,b) Amplitude and (c,d) phase response as functions of the forcing amplitude
($A_{f}$) and length of the coupling tube ($L$). (e-g) Representative time
series of $p^{\prime}$ for the points indicated in (a) depicting difference in
the response of Rijke tube A and B. Complementary forcing and coupling
enhances the region of $L$ over which the suppression of LCO is observed in
both the oscillators. Forcing is ineffective in synchronization of LCO in
Rijke tube B (the black region in (d) denotes the complete desynchrony of LCO
in Rijke tube B with the forcing signal), while the region of quenching of LCO
in Rijke tube A nearly coincides with the region of forced synchronization
($PLV\approx 1$). The common parameters in all the plots are:
$p^{\prime}_{0}=200$ Pa, $f_{f}=100$ Hz ($f_{f}/f_{n0}\approx 0.6$), $\Delta
f_{n}=0$ Hz, and $d=1$ cm.

Further, we notice that the region of quenching of LCO in Rijke tube A (i.e.
$p^{\prime}_{rms}/p^{\prime}_{0}>0.8$ in Fig. 5a) nearly coincides with the
region of forced synchronization (i.e., $PLV\approx 1$ in Fig. 5c) while the
other regions remain desynchronized with the forcing signal. However, the LCO
in Rijke tube B always remains desynchronized with the forcing signal, which
is observed from the value of $PLV\sim 0$ in Fig. 5d. For the regions where we
do not observe suppression of LCO in Rijke tube B, difference between $f_{f}$
and $f_{n0}$ is too large such that there is no phase locking between
oscillations in Rijke tube B and the forcing signal, resulting in low $PLV$.
For the regions where we notice complete suppression in Rijke tube B (Fig.
5e), the oscillations are noisy with very low amplitude, leading to low $PLV$.

Figure 6:  Amplitude and phase response (black lines) of Rijke tubes A and B
when coupled identical oscillators are subjected to asymmetric forcing for
(a,b) $p^{\prime}_{0}=120$ Pa and (c,d) $p^{\prime}_{0}=200$ Pa. The LCO in
Rijke tube A is externally forced through the acoustic drivers. The forced
synchronization region is wider for Rijke tube A than that observed for Rijke
tube B, whereas a much larger magnitude of suppression of LCO is observed in
Rijke tube B as compared to that in Rijke tube A. Inside the green region, the
oscillations are amplified to values above twice the value of LCO amplitude in
the uncoupled-unforced Rijke tubes, such that $\Delta
p^{\prime}_{rms}/p^{\prime}_{0}$ is observed in the range (-6.47, -1) in (a),
(-3.54, -1) in (b), (-6.67, -1) in (c), and (-2.66, -1) in (d). Experimental
conditions: $f^{A}_{n0}=f^{B}_{n0}\approx 160$ Hz, $L=105$ cm, $d=1$ cm.

The blue and red time series shown in Fig. 5g depict the acoustic pressure
fluctuations in Rijke tubes A and B, respectively and the acoustic pressure
time series correspond to data point (g) in Figs. 5a,b. Although these signals
appear mutual phase synchronized with each other in Fig. 5g, they are
desynchronized with the forcing signal due to a difference in the frequencies
of their oscillations (i.e., $f_{n0}=160$ Hz and $f_{f}=100$ Hz). Therefore,
the PLV distribution quantifying forced synchronization in Fig. 5c and Fig 5d
show zero values for data point (g).

Next, we measure the effect of forcing on the Arnold tongue and the amplitude
quenching characteristics of the coupled identical Rijke tubes. The length and
internal diameter of the coupling tube are fixed at $L=105$ cm and $d=1$ cm,
respectively. Only Rijke tube A is forced externally. The natural frequency of
both the oscillators is maintained at $f^{A}_{n0}=f^{B}_{n0}\approx 160$ Hz.
In Fig. 6, we depict the fractional change in the amplitude of LCO for each
oscillator overlapped with the Arnold tongue on the $\bar{A_{f}}-f_{f}$ plane
for $p^{\prime}_{0}=120$ Pa (Figs. 6a,b) and $p^{\prime}_{0}=200$ Pa (Figs.
6c,d).

Figure 7: The amplitude and phase response (black lines) of Rijke tubes A and
B when coupled non-identical oscillators are subjected to asymmetric forcing
for (a,b) $p^{\prime}_{0}=120$ Pa and (c,d) $p^{\prime}_{0}=200$ Pa. The
effect of forcing is less effective in synchronizing and quenching of LCO in
Rijke tube B, while it shows regions of forced synchronization and quenching
of LCO in Rijke tube A. Inside the region marked in green colour, $\Delta
p^{\prime}_{rms}/p^{\prime}_{0}$ varies in the range (-6.03, -1) in (a),
(-1.47, -1) in (b), and (-3.04, -1) in (c). The common parameters in all the
plots are: $\Delta f_{n0}=10$ Hz, $L=105$ cm, $d=1$ cm.

When the coupled Rijke tubes are asymmetrically forced, we notice forced
synchronization at a lower value of $\bar{A_{f}}$ for Rijke tube A than for
Rijke tube B at any value of $f_{f}$. Therefore, the boundaries of the Arnold
tongue are observed to be longer and the forced synchronization region is
wider for Rijke tube A (Figs. 6a,c) than that for Rijke tube B (Figs. 6b,d).
Furthermore, we notice that the boundaries of the Arnold tongue for
$p^{\prime}_{0}=120$ Pa are less steeper than that for $p^{\prime}_{0}=200$
Pa. This means that a significantly larger value of forcing amplitude is
required to synchronize and quench the large amplitude LCO in the coupled
Rijke tube oscillators. As the external forcing is applied to Rijke tube A,
the region of resonant amplification is larger than that for Rijke tube B.
Moreover, as the forcing amplitude is increased, the oscillator which is
subjected to external forcing (i.e., Rijke tube A) synchronizes with the
forcing signal first, followed by the forced synchronization of the entire
system, i.e., the coupled system of Rijke tubes A & B. Further, the transition
from desynchronized to forced synchronization for Rijke tubes A & B takes
place through a sequence of bifurcations: (i) from desynchronized limit cycle
at $f_{n0}^{A,B}$ to two-frequency quasiperiodicity through torus-birth
bifurcation ($f_{n0}^{A,B},f_{f}$); and (ii) from two-frequency
quasiperiodicity to synchronized limit cycle ($f_{f}$) either through saddle-
node bifurcation if $f_{n0}^{A,B}$ is close to $f_{f}$ or through torus-death
bifurcation if $f_{n0}^{A,B}$ is far from $f_{f}$. These two routes are quite
well-known [14] and is observed for the identically coupled Rijke tubes under
forcing. $f_{n0}$ denotes the natural frequency of the Rijke tubes, and
$f_{n}$ denotes the frequency of the acoustic pressure oscillations exhibited
by the Rijke tubes after the ducts are coupled and asymmetrically forced.

Rijke tube A (Figs. 6a,c) shows a similar trend of quenching of different
amplitude LCO as observed for the single oscillator (Figs. 2b,d). The
similarities include an approximate coincidence of maximum amplitude
suppression with the synchronization boundary, significant amplitude
suppression only for forcing frequencies of $f_{f}<f_{n0}$, and the presence
of resonant amplification region around $f_{f}/f_{n0}\sim 1$ (shown in green).
However, the $\bar{A_{f}}$ required for forced synchronization at any $f_{f}$
for the coupled identical oscillators (Figs. 6a,c) is higher than that for a
single oscillator (Figs. 2b,d).

In contrast to the response of Rijke tube A, simultaneous effect of coupling
and external forcing lead to a much greater suppression of LCO along the
boundaries of forced synchronization in Rijke tube B (Figs. 6b,d). Suppression
of LCO in Rijke tube B is observed for both $f_{f}<f_{n0}$ and $f_{f}>f_{n0}$,
unlike Rijke tube A where we notice a reduction only for $f_{f}<f_{n0}$.
Further, the range of $f_{f}$ and $\bar{A_{f}}$ over which the suppression of
LCO occurs in Rijke tube B is larger than that observed for Rijke tube A.
Thus, we reassert that asymmetrically forced coupled system exhibits
suppression of higher amplitude LCO for a larger range of forcing parameters
than that observed when the oscillators are forced or coupled individually.

###  III.4 Forced response of coupled non-identical limit cycle oscillators

Now, we study the response of coupled non-identical Rijke tubes under
asymmetric forcing. Only Rijke tube A is externally forced. In Fig. 7, we
depict the Arnold tongue and the fractional change in the amplitude of LCO for
$p^{\prime}_{0}=120$ Pa (Figs. 7a,b) and $p^{\prime}_{0}=200$ Pa (Figs. 7c,d).
The oscillators are coupled through a tube of $L=105$ cm and $d=1$ cm. A
frequency detuning between the uncoupled oscillators is fixed at 10 Hz (i.e.,
$\Delta f_{n0}=f^{B}_{n0}-f^{A}_{n0}=10$ Hz). We notice that the effect of
forcing is quite insignificant for synchronization of LCO in Rijke tube B. We
observe a small range of $f_{f}$ (i.e., $f_{f}<f^{B}_{n0}$) over which Rijke
tube B is synchronized to forcing for $p^{\prime}_{0}=120$ Pa (Fig. 7b). The
region of forced synchronization is completely absent when $p^{\prime}_{0}$ is
increased to 200 Pa (Fig. 7d). In contrast, as the external forcing is applied
directly to Rijke tube A, it is easily synchronized with the forcing, as seen
from the longer boundaries of the Arnold tongue for both $p^{\prime}_{0}=120$
and 200 Pa in Figs. 7a,c, respectively.

Finally, the mechanism through which forced sychronization is attained for
$p^{\prime}_{0}=120$ Pa in the non-identical system is quite different from
that of the coupled system. As the forcing amplitude is increased, the
oscillator which is subjected to external forcing (i.e., Rijke tube A)
synchronizes with the forcing signal first, followed by the forced
synchronization of the entire system, i.e., the coupled system of Rijke tubes
A & B. The entire system undergoes forced synchronization only in
$f_{f}/f_{n0}^{A,B}<1$ range. The transition from desynchronized to forced
synchronization for Rijke tube A takes place through the following sequence of
bifurcations: (i) from desynchronized two-frequency quasiperiodicity
($f_{n0}^{A},f_{n0}^{B}$) to three-frequency quasiperiodicity
($f_{n0}^{A},f_{n0}^{B},f_{f}$) as $A_{f}$ is increased; and (ii) from three-
frequency quasiperiodicity to synchronized limit cycle ($f_{f}$). This route
was reported for coupled Van der Pol oscillators under asymmetric forcing [20,
19]. However, Rijke tube B goes from two-frequency quasiperiodicity
($f_{n0}^{A},f_{n0}^{B}$) to synchronized limit cycle ($f_{f}$) when
$f_{f}/f_{n0}<1$. For $p^{\prime}_{0}=200$ Pa, the acoustic pressure
fluctuations in Rijke tube B remain desynchronized with the forcing signal
throughout the $\bar{A_{f}}-f_{f}$ parameter plane.

The amplitude response of Rijke tube A and B shows that the significant
suppression of LCO can still be achieved at non-resonant conditions of
forcing. The region of resonant amplification observed around
$f_{f}/f_{n0}\sim 1$ is very small for Rijke tube B as compared to Rijke tube
A. Furthermore, the comparison of the forced response of the coupled
oscillators with $p^{\prime}_{0}=120$ Pa (Figs. 7a,b) and $p^{\prime}_{0}=200$
Pa (Figs. 7c,d) shows that for the higher amplitude LCO, we need significantly
larger values of $\bar{A_{f}}$ for synchronization and quenching of
oscillations in both the Rijke tubes. This is similar to the observations made
for identical oscillators in Fig. 6. Note that the ordinate in Figs. 7c,d is
much larger than that in Figs. 7a,b. Thus, we re-emphasize that the effect of
forcing is more significant in suppressing LCO in both coupled thermoacoustic
oscillators if their natural frequencies are nearly the same as compared to
that seen in the case of non-identical oscillators.

##  IV Mathematical model

In this section, we will discuss a reduced-order model developed for the
system of coupled horizontal Rijke tubes subjected to asymmetric forcing. The
model of the uncoupled oscillator is based on Balasubramanian and Sujith [35].
We neglect the effects of mean flow and mean temperature gradient in the duct.
The temporal evolution of a single Rijke tube is described by the following
set of ODEs

$\dfrac{d\eta_{j}}{dt}=\dot{\eta_{j}},$ (2)
$\begin{split}\dfrac{d\dot{\eta_{j}}}{dt}&+2\xi_{j}\omega_{j}\dot{\eta_{j}}+k_{j}^{2}\eta_{j}\\\
&=-j\pi
K\left[\sqrt{\left|\dfrac{1}{3}+u_{f}^{\prime}(t-\tau)\right|}-\sqrt{\dfrac{1}{3}}\right]\hskip
2.0pt\text{sin}(j\pi x_{f}),\end{split}$ (3)

where $k_{j}=j\pi$ refers to the non-dimensional wave number and
$\omega_{j}=j\pi$ refers to the non-dimensional angular frequency of the
$j^{\text{th}}$ mode. Other parameters are the non-dimensional heater power
$K$, and non-dimensional velocity, $u^{\prime}_{f}$ at the non-dimensional
heater location, $xf$. The thermal inertia of the heat transfer in the medium
is captured by a parameter time lag $\tau$.The coefficient $\xi_{j}$ appearing
in the second term of Eq. (3) represents the frequency-dependent damping [36],
and is given by the following ansatz [37]:

$\xi_{j}=\dfrac{c_{1}\dfrac{\omega_{j}}{\omega_{1}}+c_{2}\sqrt{\dfrac{\omega_{1}}{\omega_{j}}}}{2\pi}$
(4)

Here, $c_{1}$ and $c_{2}$ are the damping coefficients which determine the
amount of damping. We choose the values $c_{1}=0.1$ and $c_{2}=0.06$ based on
[37] for all simulations. We also set $x_{f}=0.25$ ($=L/4$), as it is the most
favorable location for the onset of thermoacoustic instability [36]. The non-
dimensional velocity $u^{\prime}$ and non-dimensional pressure $p^{\prime}$
fluctuations in the model are written in terms of the Galerkin modes:

$p^{\prime}(x,t)=\sum_{j=1}^{N}-\dfrac{\gamma M}{j\pi}\dot{\eta_{j}}(t)\hskip
2.0pt\text{sin}(j\pi x),$ (5)

$u^{\prime}(x,t)=\sum_{j=1}^{N}\eta_{j}(t)\hskip 2.0pt\text{cos}(j\pi x).$ (6)

For Rijke tubes, Matveev [36] and Sayadi  _et al._ [38] have shown that the
first mode is the most unstable of all the other modes. Consequently, we
consider only the first mode ($N=1$) in our numerical analysis, to keep the
model simple.

Figure 8: (a) The phase response and (b) the amplitude response obtained from
the model of a single Rijke tube under external forcing. The synchronization
boundaries are obtained through a least-square-fit of points where $PLV=0.98$.
The Arnold tongue and extent of asynchronous quenching from the model show a
qualitative match with the experimental results in Fig. 2. In (b), the green
region around $f_{f}/f_{n0}=1$ indicates doubling of LCO amplitude from its
unforced value, where $\Delta p^{\prime}_{rms}/p^{\prime}_{0}$ varies in the
range (-6.35, -1) in (b).

Let superscripts “$a$” and “$b$” denote Rijke tube A and B, respectively. We
assume that the two Rijke tubes are coupled through time-delay and dissipative
couplings. The governing equations for coupled non-identical Rijke tubes with
asymmetric sinusoidal forcing can then be written as:

$\dfrac{d\eta_{j}^{a,b}}{dt}=\dot{\eta_{j}}^{a,b},$ (7)

$\dfrac{d\dot{\eta_{j}}^{a,b}}{dt}+2\xi_{j}\left(\dfrac{\omega_{j}}{r^{a,b}}\right)\dot{\eta_{j}}^{a,b}+\left(\dfrac{k_{j}}{r^{a,b}}\right)^{2}\eta_{j}^{a,b}\\\
=-\dfrac{j\pi}{{r^{a,b}}^{2}}K\left[\sqrt{\left|\dfrac{1}{3}+u_{f}^{\prime
a,b}(t-\tau)\right|}-\sqrt{\dfrac{1}{3}}\right]\hskip
2.0pt\text{sin}\left(\dfrac{j\pi x_{f}}{r^{a,b}}\right)\\\
+\underbrace{K_{d}(\dot{\eta_{j}}^{b}-\dot{\eta_{j}}^{a})}_{\text{Dissipative
coupling}}+\underbrace{K_{\tau}(\dot{\eta_{j}}^{b}(t-\tau_{tube})-\dot{\eta_{j}}^{a}(t))}_{\text{Time-
delay coupling}}\\\ +\underbrace{[A_{f}\hskip 2.0pt\text{sin}(2\pi
f_{f}t)]^{a}}_{\text{Forcing term}},$ (8)

where, $r^{a,b}$ is defined as the ratio of the length of the duct to a
reference length, $L_{a,b}/L_{ref}$. We consider $L_{a}$ to be the reference
length. For identical oscillators, $r^{a}=r^{b}=1$. For non-identical
oscillators, $r_{a}=1$ and $r_{b}=L_{b}/L_{a}=\omega_{a}/\omega_{b}$. The
detailed derivation of Eq. (8) is provided in Section IV of the Supplemental
Material.

The second and third terms on the right-hand side of Eq. (8) are the
dissipative and time-delay coupling terms, respectively. Dissipative coupling
encapsulates the interaction that arises from the mass transfer between the
two ducts [34]. Time-delay coupling quantifies the time taken by acoustic
waves to propagate through the coupling tube connecting the two Rijke tubes
[4, 23]. Thus, $\tau_{tube}$ denotes the time-delay in the response induced by
one Rijke tube on the other, and is proportional to the length of the coupling
tube, i.e., $\tau_{tube}\propto L/c$, where $c$ is the speed of sound inside
the coupling tube. The fourth term is the sinusoidal forcing term with
amplitude $A_{f}$ and frequency $f_{f}$. The external forcing is applied only
to Rijke tube A.

The effect of the source term is characterized by the time-delay $\tau$, which
captures the thermal inertia of the heat transfer in the medium. The ODEs
given in Eqs. (7) and (8) are solved numerically using dde23, an inbuilt
function for solving delay differential equations in MATLAB [39], and
$p^{\prime}$ is calculated using Eq. (5). The parameters which are kept
constant during the simulations are indicated in Table 1. These parameters
have been chosen such that the both the oscillators exhibit limit cycle
oscillations at a parametric location away from the bistable regime and also
to quantitatively match the experimental results.

Parameter | Value | Parameter | Value | Parameter | Value
---|---|---|---|---|---
$N$ | 1 | $\tau$ | 0.25 | $K$ | 5
$\gamma$ | 1.4 | $M$ | 0.005 | $x_{f}$ | 0.25
$c_{1}$ | 0.1 | $c_{2}$ | 0.06 | $\omega_{f}$ | $j\pi$
Table 1: Model parameters kept constant throughout the numerical analysis of
the reduced-order model.

###  IV.1 Model results for a single oscillator and mutually coupled
oscillators

Figure 9: Two-parameter bifurcation plots between frequency detuning between
the two Rijke tube model oscillators and $\tau_{tube}$. $K_{d}$ and $K_{\tau}$
are kept constant at $1.0$ and $0.2$, respectively. The dark red region in
both the plots represent regions of AD observed in the model oscillators. At
other places, LCO are observed at reduced amplitude due to coupling.  Figure
10:  (a,b) Amplitude and (c,d) phase response of coupled, identical
oscillators under asymmetric forcing for different values of $\tau_{tube}$ and
$A_{f}$. Amplitude response is indicated by the fractional change in the
amplitude of LCO ($\Delta p^{\prime}_{rms}/p^{\prime}_{0}$) while phase
response is quantified by the $PLV$ between $p^{\prime}$ and forcing signal.
For $\tau_{tube}>0.4$ and $A_{f}>1.5$, large suppression in the amplitude of
LCO is observed in the two Rijke tubes.

We begin by analyzing the forced response of a single Rijke tube by adding the
forcing term $A_{f}\sin(2\pi f_{f}t)$ to the right hand side Eq. (3).
Considering only the first mode, we get the following set of ODEs is:

$\dfrac{d\eta_{1}}{dt}=\dot{\eta_{1}},$ (9)
$\begin{split}\dfrac{d\dot{\eta_{1}}}{dt}&+2\xi_{1}\omega_{1}\dot{\eta_{1}}+k_{1}^{2}\eta_{1}=-\pi
K\left[\sqrt{\left|\dfrac{1}{3}+u_{f}^{\prime}(t-\tau)\right|}-\sqrt{\dfrac{1}{3}}\right]\\\
&\times\hskip 2.0pt\text{sin}(\pi x_{f})+\underbrace{A_{f}\hskip
2.0pt\text{sin}(2\pi f_{f}t)}_{\text{Forcing term}}.\end{split}$ (10)

Figure 8 shows the two-parameter bifurcation plot on the $A_{f}-f_{f}$ plane
illustrating the phase and amplitude dynamics of the limit cycle oscillations
in a single Rijke tube, obtained from the mathematical model. We observe the
existence of Arnold tongue in Fig. 8a along with the region of asynchronous
quenching ($\Delta p^{\prime}_{rms}/p^{\prime}_{0}>0.5$) in Fig. 8b. We
observe the qualitative match between experimental results shown in Fig. 2 and
those obtained from the model in Fig. 8. However, there are a number of
differences. First, the region of asynchronous quenching observed in the model
is smaller than that observed in the experiments. Second, the resonant
amplification region is spread over a larger extent of parameter values in the
model. Finally, the Arnold tongue obtained from the model is symmetrical as
opposed to left-skewed in the experiments. The skewness of the Arnold tongue
increases as the model heater power $K$ is increased to higher-values,
indicating the nonlinear behavior of the overall systems. Figure 8 has been
shown for a model heater power $K$ for which we get the best match with the
experimental results in Fig. 2.

In Fig. 3, we showed that the state of AD was attained in coupled Rijke tubes
only when there was a frequency detuning present in the two oscillators, in
addition to the dissipative and time-delay coupling due to the connecting
tube. In Fig. 9, we show the effect of coupling and frequency detuning on the
occurrence of the state of AD in the model. We fixed the value of $K_{d}=1.0$
and $K_{\tau}=0.2$, and varied $\tau_{tube}$ and frequency detuning $\Delta
f_{n0}$. For identical oscillators ($\Delta f_{n0}=0$), we did not observe any
appreciable reduction in the amplitude of LCO. However, for finite value of
detuning, we observe the state of AD in the model, consistent with our
experimental observations in Fig. 3. Thus, the effect of change of the length
of coupling tube is captured quite well by the time-delay $\tau_{tube}$.

The results summarized in Figs. 8 and 9 validate our model against
experimental observations for a single and coupled Rijke tubes reported in the
present study (Figs. 2 and 3), as well as those made in past studies [23, 40,
25]. We now turn our attention towards modeling the effect of asymmetric
forcing on the coupled Rijke tubes.

###  IV.2 Model results for coupled behavior of thermoacoustic oscillators
under asymmetric forcing

Figure 11: The amplitude response and overlapped Arnold tongue for identical,
coupled Rijke tubes from the numerical model. External forcing is applied to
Rijke tube A. The synchronization boundaries are obtained through least-
square-fit of points where $PLV=0.98$. For the green region around
$f_{f}/f_{n0}=1$, values of $\Delta p^{\prime}_{rms}/p^{\prime}_{0}$ vary in
the range (-3.20, -1) in (a), and (-2.08, -1) in (b). The coupling parameters
are: $K_{d}=1.0$, $K_{\tau}=0.2$ and $\tau_{tube}=0.4$.

In Fig. 10, we plot the amplitude and the phase response of the coupled
identical Rijke tubes under asymmetric forcing, i.e., only Rijke tube A is
subjected to external forcing according to Eq. (8). The effect of coupling is
parameterized by $\tau_{tube}$, while that of external forcing by $A_{f}$. The
forcing frequency is fixed at $f_{f}/f_{n0}=0.6$. $K_{d}$, $K_{\tau}$ & $K$
values are fixed at $1.0$, $0.2$ and $5.0$, respectively. Note that the heater
power $K=5.0$ leads to high amplitude LCO in the two Rijke tubes
($p^{{\prime}{A,B}}_{0}=648$ Pa).

For $A_{f}=0$, we notice that the change in $\tau_{tube}$ leads to very low
suppression in either of the oscillators ($\Delta
p^{\prime}_{rms}/p^{\prime}_{0}<0.5$). This is again a reflection of the fact
that the region of AD is quite limited when identical oscillators are coupled.
Our aim is to illustrate that external forcing can lead to significant
quenching of LCO in the identical oscillators. When $A_{f}$ is increased, we
observe a decrease in the amplitude of LCO in both the oscillators after a
critical value of $A_{f}$. The effect of forcing is more pronounced in Rijke
tube B as compared to that in Rijke tube A, which is evident from the larger
“red” region in Fig. 10b compared to that in Fig. 10a.

The results obtained from the model (Fig. 10) can be compared with the
experimental results shown in Fig. 5. The ratio of forcing frequency to
natural frequency is kept same for ease of comparison. Although the magnitude
of amplitude suppression is not captured quantitatively by the model, the
qualitative trends such as the extent of amplitude suppression in the two
model oscillators along with the forced synchronization characteristics are
consistent between the experimental and numerical results. Compared to the
pressure oscillations in Rijke tube B exhibiting complete suppression in the
experimental study (for e.g., point (e) in Fig. 5a), we observe around $70\%$
suppression in the amplitude of LCO in Rijke tube B.

Similar to experimental results, we also observe that the LCO in Rijke tube B
remains desynchronized with the forcing signal through a large extent of the
parameter plane. The region of quenching of LCO in Rijke tube A ($\Delta
p^{\prime}_{rms}/p^{\prime}_{0}\sim 1$) nearly coincides with the onset of
forced synchronization ($PLV\sim 1$). The results shown in Fig. 10 are in
qualitative agreement with the experimental results indicated in Figs. 5a-d.

In Fig. 11, we depict the fractional change in the amplitude of LCO for each
oscillator overlapped with the Arnold tongue on the $A_{f}-f_{f}$ plane,
obtained from the numerical simulations for the case where identical
oscillators are asymmetrically forced. The values of the coupling constants
are fixed at $K_{d}=1.0$, $K_{\tau}=0.20$ and $\tau_{tube}=0.4$. The
normalized natural frequency of both the oscillators is $0.5$, and the
normalized heater power are maintained at $K=5.0$.

We see that the amplitude dynamics observed during the experiments, as
depicted in Fig. 6, is well captured by the numerical model in Fig. 11. From
the numerical results, we observe larger magnitude of suppression of LCO in
Rijke tube B, as compared to that in Rijke tube A. Also, the high magnitude of
suppression ($\Delta p^{\prime}_{rms}/p^{\prime}_{0}>0.5$) in Rijke tube B
nearly coincides with the boundary of Arnold tongue in Rijke tube A. The
resonant synchronization region in Rijke tube B is smaller in size compared to
that in Rijke tube A. The model does not adequately capture the phase dynamics
as observed in the experiments. Although the Arnold tongue is similar for both
the Rijke tubes, the $PLV$ values observed prior to the onset of forced
synchronization is lower in Rijke tube B than that in Rijke tube A. Also, the
resonant amplification region (shown in green) is skewed towards the left side
of $f_{f}/f_{n0}=1.00$ value. This is different from the experimental results
(Fig. 6), where the resonant amplification region was almost symmetrically
located around the $f_{f}/f_{n0}=1.00$ value.

Figure 12: The amplitude response and overlapped Arnold tongue for non-
identical, coupled Rijke tubes from the numerical model. External forcing is
applied to Rijke tube A. The synchronization boundaries in are obtained
through least-square-fit of points where $PLV=0.98$. Inside the green region
in (a), $\Delta p^{\prime}_{rms}/p^{\prime}_{0}$ varies in (-2.66, -1) range.
The coupling parameters are: $K_{d}=1.0$, $K_{\tau}=0.2$ and
$\tau_{tube}=0.4$.

Finally, we use our model to characterize the response of asymmetric forcing
on the Arnold tongue and amplitude characteristics of coupled non-identical
Rijke tubes. The coupling constants and non-dimensional heater power values
are kept same as those used to get the results shown in Fig. 11,i.e.,
$K_{d}=1.0$, $K_{\tau}=0.2$, $\tau_{tube}=0.4$, and $K=5$. This ensures the
consistency of initial and boundary conditions, as was ensured in the
experiments. In §III.4, to study the forced response of coupled non-identical
Rijke tube oscillators, the natural frequencies of the LCO in Rijke tubes A
and B were maintained at $150$ Hz and $160$ Hz, respectively, which equals a
frequency ratio of $160/150=1.067$. Consistent with the experiments, the value
of $r$ is kept as $1.066$ to analyze the reduced-order model given in Eqs. (7)
and (8).

We observe that the effect of forcing is quite insignificant for forced
synchronization of LCO in Rijke tube B, consistent with the experimental
results (Fig. 7). A significant suppression of the amplitude of LCO is
observed at non-resonant conditions of forcing in Rijke tube A. The region of
resonant amplification is absent for Rijke tube B, although we do observe
amplification of the amplitude of LCO at higher values of $A_{f}$ in Fig. 12b.
Some differences are present between the experimental (Fig. 7) and numerical
results (Fig. 12). First, the forced synchronization of Rijke tube B is
observed for $f_{f}/f_{n0}^{B}<1$ values during experiments, whereas, the
pressure oscillations in Rijke tube B model oscillator undergo forced
synchronization for $f_{f}/f_{n0}^{B}>1$ values. Second, the Arnold tongue is
spread over a larger extent of parameter values for a coupled non-identical
oscillators system as opposed to that in a coupled identical oscillator system
(Fig. 11). During experiments, an opposite behavior was observed in Fig. 7c,
where the size of the Arnold tongue of Rijke tube A for a coupled non-
identical system is less when compared to that in the coupled identical system
(Fig. 6c). The qualitative differences between the results can be attributed
to the various assumptions used while deriving the reduced-order model, such
as absence of mean flow, zero temperature gradient, non-heat conducting gas,
etc. A more detailed study is needed to examine the effects of such
assumptions on the numerical results.

##  V Conclusions

In this proof-of-concept study, we investigated the phase and the amplitude
dynamics of coupled thermoacoustic oscillators under asymmetric forcing, and
presented a model which satisfactorily captured the experimental results
qualitatively. In particular, we discuss the viability of simultaneous
coupling and forcing as a method for controlling thermoacoustic instability in
a system of multiple combustors.

The forced response of limit cycle oscillations (LCO) in a single oscillator
(Rijke tube) shows the presence of Arnold tongue along with the region of
asynchronous quenching for the parametric range of $f_{f}<f_{n0}$. The region
of asynchronous quenching coincides with the boundary of forced
synchronization of the acoustic pressure fluctuations in the system [16]. The
presence of Arnold tongue and asynchronous quenching are consistent with the
results observed in previous studies conducted on Rijke tubes [16] and laminar
combustors [31, 32, 33]. We notice that the characteristics of forced
synchronization of LCO are dependent on the amplitude of LCO in the unforced
state. In particular, the region of forced synchronization (or Arnold tongue)
of the oscillator gets narrower as the amplitude of LCO is increased. In
addition, the coupled response of two Rijke tube oscillators (A and B) show
the occurrence of two different states of oscillation quenching, i.e.,
amplitude death and partial amplitude death. These states occur only when a
finite frequency detuning is present between the oscillators, and the length
of the coupling tube lies within a specific range.

To expand the parametric range of oscillation quenching in two mutually
coupled Rijke tube oscillators, we acoustically force Rijke tube A. We observe
that the suppression of LCO in coupled identical Rijke tube oscillators is
possible through the combined effect of mutual coupling and asynchronous
quenching. A significantly larger value of forcing amplitude is required to
synchronize and quench the large amplitude LCO in the coupled thermoacoustic
oscillators. This behavior is depicted through the increase in the steepness
of the boundaries of Arnold tongue as the amplitude of LCO is increased.
Suppression of LCO is observed predominantly for $f_{f}<f_{n0}$ in Rijke tube
A, while it is observed on both sides of $f_{n0}$ in Rijke tube B. We notice
that external forcing widens the region of coupling and forcing parameters
over which the oscillation quenching states are observed in both the
oscillators than when only one of the two mechanisms of control are applied on
its own. Most importantly, although Rijke tube A is forced, the suppression of
LCO is more significant in Rijke tube B in comparison with that in Rijke tube
A. We also studied the coupled behavior of two non-identical thermoacoustic
oscillators under asymmetric forcing. We witness that because of direct
influence of forcing, Rijke tube A exhibits the features of forced
synchronization, while Rijke tube B (not directly forced) remains
desynchronized with the forcing signal. As a result, we observe a significant
suppression of LCO in Rijke tube A and not in Rijke tube B, which is opposite
to the behavior of forced response of coupled identical thermoacoustic
oscillators. Finally, we qualitatively capture the experimental results
through a reduced-order model of two coupled Rijke tubes. A good agreement is
obtained between the experimental and numerical results.

Thus, periodic forcing aids the mitigation of thermoacoustic instability
observed in coupled identical oscillators. We believe that the present
investigation on the asymmetrically forced prototypical coupled thermoacoustic
oscillators would prove to be a benchmark specifically for the control of
thermoacoustic oscillations observed in can combustors with multiple cans, and
for coupled nonlinear oscillators subjected to external forcing in general
nonlinear dynamics literature.

## Acknowledgments

A.S. and A.R. gratefully acknowledge the Ministry of Human Resource
Development (MHRD) for funding PhD through Half-Time Research Assistantship
(HTRA). The authors are grateful to Ms. Srikanth and Ms. Manoj for helping
with the MATLAB implementation of the numerical model. This work was supported
by the Office of Naval Research Global (Contract Monitor: Dr R. Kolar) Grant
no. N62909-18-1-2061.

## References

  * Jenkins [2013] A. Jenkins, Physics Reports 525, 167 (2013).
  * Strogatz [2004] S. Strogatz,  _Sync: The emerging science of spontaneous order_ (Penguin UK, 2004).
  * Strogatz  _et al._ [2005] S. H. Strogatz, D. M. Abrams, A. McRobie, B. Eckhardt, and E. Ott, Nature 438, 43 (2005).
  * Biwa  _et al._ [2015] T. Biwa, S. Tozuka, and T. Yazaki, Physical Review Applied 3, 034006 (2015).
  * Lieuwen and Yang [2005] T. C. Lieuwen and V. Yang,  _Combustion instabilities in gas turbine engines: operational experience, fundamental mechanisms, and modeling_ (American Institute of Aeronautics and Astronautics, 2005).
  * Sujith and Unni [2020] R. I. Sujith and V. R. Unni, Physics of Fluids 32, 061401 (2020).
  * Pikovsky  _et al._ [2003] A. Pikovsky, J. Kurths, M. Rosenblum, and J. Kurths,  _Synchronization: a universal concept in nonlinear sciences_ , Vol. 12 (Cambridge university press, 2003).
  * Juniper and Sujith [2018] M. P. Juniper and R. I. Sujith, Annual Review of Fluid Mechanics 50, 661 (2018).
  * Pyragas  _et al._ [2000] K. Pyragas, F. Lange, T. Letz, J. Parisi, and A. Kittel, Physical Review E 61, 3721 (2000).
  * Kumar  _et al._ [2011] A. Kumar, S. Cardanobile, S. Rotter, and A. Aertsen, Frontiers in Systems Neuroscience 5, 86 (2011).
  * Huddy and Skufca [2012] S. R. Huddy and J. D. Skufca, IEEE Transactions on Power Electronics 28, 247 (2012).
  * Mirollo and Strogatz [1990] R. E. Mirollo and S. H. Strogatz, Journal of Statistical Physics 60, 245 (1990).
  * Atay [2003] F. M. Atay, Physica D: Nonlinear Phenomena 183, 1 (2003).
  * Balanov  _et al._ [2008] A. Balanov, N. Janson, D. Postnov, and O. Sosnovtseva,  _Synchronization: from simple to complex_ (Springer Science & Business Media, 2008).
  * Keen and Fletcher [1970] B. Keen and W. Fletcher, Physical Review Letters 24, 130 (1970).
  * Mondal  _et al._ [2019] S. Mondal, S. A. Pawar, and R. I. Sujith, Journal of Fluid Mechanics 864, 73 (2019).
  * Battelino [1988] P. M. Battelino, Physical Review A 38, 1495 (1988).
  * Anishchenko  _et al._ [2009a] V. S. Anishchenko, S. V. Astakhov, T. E. Vadivasova, and A. V. Feoktistov, Nelineinaya Dinamika [Russian Journal of Nonlinear Dynamics] 5, 237 (2009a).
  * Anishchenko  _et al._ [2009b] V. Anishchenko, S. Astakhov, and T. Vadivasova, EPL (Europhysics Letters) 86, 30003 (2009b).
  * Anishchenko  _et al._ [2008] V. Anishchenko, S. Nikolaev, and J. Kurths, Chaos: An Interdisciplinary Journal of Nonlinear Science 18, 037123 (2008).
  * Wei  _et al._ [2011] X. Wei, M. Randrianandrasana, M. Ward, and D. Lowe, Mathematical Problems in Engineering 2011 (2011).
  * Honerkamp [1983] J. Honerkamp, Journal of Mathematical Biology 18, 69 (1983).
  * Thomas  _et al._ [2018a] N. Thomas, S. Mondal, S. A. Pawar, and R. I. Sujith, Chaos: An Interdisciplinary Journal of Nonlinear Science 28, 033119 (2018a).
  * Thomas  _et al._ [2018b] N. Thomas, S. Mondal, S. A. Pawar, and R. I. Sujith, Chaos: An Interdisciplinary Journal of Nonlinear Science 28, 093116 (2018b).
  * Dange  _et al._ [2019] S. Dange, K. Manoj, S. Banerjee, S. A. Pawar, S. Mondal, and R. I. Sujith, Chaos: An Interdisciplinary Journal of Nonlinear Science 29, 093135 (2019).
  * Hyodo  _et al._ [2020] H. Hyodo, M. Iwasaki, and T. Biwa, Journal of Applied Physics 128, 094902 (2020).
  * Jegal  _et al._ [2019] H. Jegal, K. Moon, J. Gu, L. K. Li, and K. T. Kim, Combustion and Flame 206, 424 (2019).
  * Moon  _et al._ [2020] K. Moon, Y. Guan, L. K. Li, and K. T. Kim, Chaos: An Interdisciplinary Journal of Nonlinear Science 30, 023110 (2020).
  * Balusamy  _et al._ [2015] S. Balusamy, L. K. Li, Z. Han, M. P. Juniper, and S. Hochgreb, Proceedings of the Combustion Institute 35, 3229 (2015).
  * Kashinath  _et al._ [2018] K. Kashinath, L. K. Li, and M. P. Juniper, Journal of Fluid Mechanics 838, 690 (2018).
  * Guan  _et al._ [2019a] Y. Guan, V. Gupta, K. Kashinath, and L. K. Li, Proceedings of the Combustion Institute 37, 5315 (2019a).
  * Roy  _et al._ [2020] A. Roy, S. Mondal, S. A. Pawar, and R. I. Sujith, Journal of Fluid Mechanics 884 (2020).
  * Guan  _et al._ [2019b] Y. Guan, V. Gupta, M. Wan, and L. K. Li, Journal of Fluid Mechanics 879, 390 (2019b).
  * Bar-Eli [1985] K. Bar-Eli, Physica D: Nonlinear Phenomena 14, 242 (1985).
  * Balasubramanian and Sujith [2008] K. Balasubramanian and R. Sujith, Physics of Fluids 20, 044103 (2008).
  * Matveev [2003] K. I. Matveev,  _Thermoacoustic instabilities in the Rijke tube: experiments and modeling_ , Ph.D. thesis, California Institute of Technology (2003).
  * Sterling and Zukoski [1991] J. Sterling and E. Zukoski, Combustion Science and Technology 77, 225 (1991).
  * Sayadi  _et al._ [2013] T. Sayadi, V. L. Chenadec, P. Schmid, F. Richecoeur, and M. Massot, arXiv preprint arXiv:1312.3101 (2013).
  * Shampine and Thompson [2001] L. F. Shampine and S. Thompson, Applied Numerical Mathematics 37, 441 (2001).
  * Hyodo and Biwa [2018] H. Hyodo and T. Biwa, Physical Review E 98, 052223 (2018).