  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.
