4.1 Before lock-in: $f_{f}$ close to $f_{n}$

First we force the system at a frequency close to its natural frequency: $f_{f}/f_{n}=0.98$ . The responses for states A and B are qualitatively similar to each other, so for brevity only state A is presented here: a period-1 oscillation arising from a supercritical Hopf bifurcation (Kashinath et al. Reference Kashinath, Waugh and Juniper2014, figure 3 at $x_{f}=0.166$ ). Figure 2 shows the time series and Poincaré maps for this state at different forcing amplitudes ( $\unicode[STIX]{x1D716}$ ).

Figure 2. Saddle-node bifurcation to lock-in: the forced response of the system during period-1 oscillations (state A) when the forcing frequency is close to the natural frequency, $f_{f}/f_{n}=0.98$ , where $f_{n}=2.304$ . Time series and Poincaré maps are shown for increasing forcing amplitudes: (a)  $\unicode[STIX]{x1D716}\equiv u_{f}^{\prime }/U_{0}=0.03$ , (b)  $0.06$ , (c)  $0.09$ , (d)  $0.11$ , (e)  $0.12$ , (f)  $0.125$ and (g)  $0.13$ .

When forced, the system responds at both its natural frequency and the forcing frequency but, as will be shown later, the former ( $f_{n}^{\prime }$ , where the prime indicates the presence of forcing) shifts towards the latter ( $f_{f}$ ) and is therefore no longer equal to the natural frequency of the unforced system ( $f_{n}$ ). The oscillations in figure 2(a–f) are quasi-periodic and arise from a torus-birth bifurcation (i.e. a Neimark–Sacker bifurcation) of the unforced period-1 oscillation. The power spectrum, which is not shown here but is similar to that reported for forced self-excited hydrodynamic systems (Li & Juniper Reference Li and Juniper2013a ,Reference Li and Juniper b ), contains peaks at linear combinations of $f_{n}^{\prime }$ and $f_{f}$ , indicating nonlinear triad interactions between the natural and forced modes. The Poincaré maps show two rings, indicating that the phase trajectory is not closed but spirals around the surface of a stable ergodic 2-torus. The time series shows evidence of beating, a low-frequency modulation of the pressure amplitude at the beating frequency, $\unicode[STIX]{x0394}f=f_{n}^{\prime }-f_{f}$ .

When forced above a critical amplitude (figure 2 g, $\unicode[STIX]{x1D716}=0.13$ ), the system locks in to the forcing and oscillates only at $f_{f}$ . The Poincaré map shows two discrete points, indicating a closed period-1 orbit in phase space. This transition from a quasi-periodic oscillation to a period-1 oscillation is abrupt, which, as will be confirmed later, reveals a saddle-node bifurcation to lock-in. The attractor at lock-in is a stable periodic orbit on the surface of the 2-torus that existed before the saddle-node bifurcation (Balanov et al. Reference Balanov, Janson, Postnov and Sosnovtseva2009).

Figure 3. Saddle-node bifurcation to lock-in: the phase difference (normalized by $2\unicode[STIX]{x03C0}$ ) between the system and the forcing at the conditions of figure 2.

Figure 3 shows the phase difference $\unicode[STIX]{x1D719}_{1,2}$ (normalized by $2\unicode[STIX]{x03C0}$ ) between the system and the forcing at the conditions of figure 2. The response pressure is measured at the same position in the duct as the forcing (i.e. the burner lip). Furthermore, we assume acoustic compactness of the flame. Different $x$ positions in the duct will have different phase, but focusing on one location is sufficient to understand the complete dynamics because other locations will behave similarly with a constant (time-invariant) phase difference. To explore the different states of synchronization, it is necessary to consider only the temporal evolution of $\unicode[STIX]{x1D719}_{1,2}$ and not its absolute value (Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2003). The average slope of each curve is the beating frequency, $\unicode[STIX]{x0394}f=f_{n}^{\prime }-f_{f}$ . In all cases except at $\unicode[STIX]{x1D716}=0.128$ (strong forcing), the oscillations are quasi-periodic, as indicated by the non-zero slope of $\unicode[STIX]{x1D719}_{1,2}$ , showing that $f_{n}^{\prime }\neq f_{f}$ . As $\unicode[STIX]{x1D716}$ increases, two trends emerge. First, the linear decrease in $\unicode[STIX]{x1D719}_{1,2}$ becomes increasingly wavy, with alternating periods of synchronicity (flat slope) and asynchronicity (steep slope), the latter of which are called phase slips (Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2003). Second, the magnitude of the average slope of each curve decreases, indicating that $f_{n}^{\prime }\rightarrow f_{f}$ . This behaviour, called frequency pulling, is characteristic of lock-in via a saddle-node bifurcation and can be modelled with universal low-dimensional oscillators (Balanov et al. Reference Balanov, Janson, Postnov and Sosnovtseva2009).

With strong forcing ( $\unicode[STIX]{x1D716}\geqslant 0.12$ ), there are times when $\unicode[STIX]{x1D719}_{1,2}$ is nearly constant, indicating synchronicity, with phase slips occurring relatively abruptly. The phase slips are equal to $-2\unicode[STIX]{x03C0}$ because here $f_{f}/f_{n}<1$ , which means that the system overtakes the forcing by one full cycle during a phase slip. As $\unicode[STIX]{x1D716}$ increases further, both the abruptness of the phase slips and the time interval between them increase, ultimately leading to infinitely long intervals of constant $\unicode[STIX]{x1D719}_{1,2}$ and thus lock-in.

The characteristics observed in figures 2 and 3, which correspond to $f_{f}<f_{n}$ , are also observed when $f_{f}>f_{n}$ as long as $f_{f}$ is close to $f_{n}$ . Crucially, the sequence of bifurcations remains unchanged: it begins with a torus-birth bifurcation to quasi-periodicity from a period-1 oscillation, followed by a saddle-node bifurcation to lock-in at a critical forcing amplitude. However, when $f_{f}>f_{n}$ , the phase slips occur in the opposite direction ( $+2\unicode[STIX]{x03C0}$ ) because the system loses one full cycle with respect to the forcing when it phase slips.

4.2 Before lock-in: $f_{f}$ far from $f_{n}$

Next we force the system at a frequency far from its natural frequency: $f_{f}/f_{n}=0.89$ . For completeness, here we present state B, which responds qualitatively similarly to state A and is likewise a period-1 oscillation. However, state B arises from a subcritical, rather than a supercritical, Hopf bifurcation of the steady base state (Kashinath et al. Reference Kashinath, Waugh and Juniper2014, figure 3). Figure 4 shows the time series, power spectra and phase portraits for this state at increasing $\unicode[STIX]{x1D716}$ , starting with the unforced case (figure 4 a).

Figure 4. Torus-death bifurcation to lock-in: the forced response of the system during period-1 oscillations (state B) when the forcing frequency is far from the natural frequency, $f_{f}/f_{n}=0.89$ , where $f_{n}=1.125$ . Time series, power spectra and phase portraits are shown for increasing forcing amplitudes: (a)  $\unicode[STIX]{x1D716}\equiv u_{f}^{\prime }/U_{0}=0.00$ or unforced, (b)  $0.02$ , (c)  $0.08$ , (d)  $0.14$ , (e)  $0.16$ , (f)  $0.18$ , (g)  $0.20$ and (h)  $0.23$ .

Compared to figure 2 ( $f_{f}$ close to $f_{n}$ ), figure 4 ( $f_{f}$ far from $f_{n}$ ) shows many similarities but also some key differences. In both cases, at intermediate forcing amplitudes, the system responds at both $f_{n}^{\prime }$ and $f_{f}$ , becoming quasi-periodic via a torus-birth bifurcation of the unforced period-1 oscillation. Furthermore, the time series show low-frequency beating and the phase portraits show a stable ergodic 2-torus.

For $f_{f}$ far from $f_{n}$ (figure 4), $f_{n}^{\prime }$ remains unchanged from its unforced value ( $f_{n}$ ) as $\unicode[STIX]{x1D716}$ increases; but for $f_{f}$ close to $f_{n}$ (§ 4.1), $f_{n}^{\prime }$ shifts towards $f_{f}$ , leading to frequency pulling. Moreover, for $f_{f}$ far from $f_{n}$ , the power spectra show a steady decrease in the amplitude of the natural mode (at $f_{n}^{\prime }$ ) and a corresponding increase in the amplitude of the forced mode (at $f_{f}$ ). At lock-in (figure 4 h), the system oscillates only at $f_{f}$ , with the natural mode becoming suppressed and the phase trajectory in a closed period-1 orbit, similar to the case for $f_{f}$ close to $f_{n}$ (figure 2 g). However, unlike for $f_{f}$ close to $f_{n}$ , here the transition from quasi-periodicity to lock-in is gradual rather than abrupt, revealing an inverse Neimark–Sacker bifurcation (i.e. a torus-death bifurcation) rather than a saddle-node bifurcation. This is consistent with predictions from generic models of self-excited oscillators, such as the forced van der Pol oscillator (Balanov et al. Reference Balanov, Janson, Postnov and Sosnovtseva2009). Finally, the response amplitude at lock-in is significantly lower than that of the unforced case; at this particular value of $f_{f}/f_{n}$ , it is approximately 70 % lower (compare figure 4 a with figure 4 h). As will be discussed in § 4.4, this decrease tends to occur when $f_{f}$ is far from $f_{n}$ . When $f_{f}$ is close to $f_{n}$ , however, the response amplitude at lock-in tends to be higher than that of the unforced case. Nevertheless, the decrease shows that weakening of self-excited thermoacoustic oscillations is possible via open-loop acoustic forcing at far-off-resonance frequencies, confirming the results of Bellows et al. (Reference Bellows, Hreiz and Lieuwen2008).

Figure 5. Torus-death bifurcation to lock-in: the phase difference (normalized by $2\unicode[STIX]{x03C0}$ ) between the system and the forcing at the conditions of figure 4.

Figure 5 shows the phase difference $\unicode[STIX]{x1D719}_{1,2}$ (normalized by $2\unicode[STIX]{x03C0}$ ) at the conditions of figure 4. For weak forcing ( $0<\unicode[STIX]{x1D716}\leqslant 0.14$ ), $\unicode[STIX]{x1D719}_{1,2}$ decreases linearly with an average slope equal to the beating frequency, $\unicode[STIX]{x0394}f=f_{f}-f_{n}^{\prime }$ . For moderate forcing ( $0.14<\unicode[STIX]{x1D716}\leqslant 0.18$ ), the magnitude of the average slope increases via large phase slips (many of them ${>}2\unicode[STIX]{x03C0}$ ), indicating frequency pushing: $f_{n}^{\prime }$ shifts away from $f_{f}$ as $\unicode[STIX]{x1D716}$ increases, which will be discussed below. For stronger forcing, the magnitude of the average slope decreases slightly ( $0.18<\unicode[STIX]{x1D716}\leqslant 0.218$ ), indicating frequency pulling, before abruptly snapping to zero at $\unicode[STIX]{x1D716}=0.218\rightarrow 0.22$ . After this, the phase slips continue to occur, keeping $\unicode[STIX]{x1D719}_{1,2}$ bounded within a $\pm \unicode[STIX]{x03C0}$ band around $\unicode[STIX]{x1D719}_{1,2}=0$ . The result is a partially synchronous state known as phase trapping (Aronson et al. Reference Aronson, Ermentrout and Kopell1990), which was only recently discovered in hydrodynamics (Li & Juniper Reference Li and Juniper2013c ) and thermoacoustics (Balusamy et al. Reference Balusamy, Li, Han, Juniper and Hochgreb2015). During phase trapping, the oscillations are still quasi-periodic and frequency-locked but are not phase-locked. They become phase-locked only at lock-in, when the amplitude at $f_{n}^{\prime }$ is completely quenched ( $\unicode[STIX]{x1D716}=0.23$ ). In figure 4, this last sequence of events (frequency pulling $\rightarrow$ phase trapping $\rightarrow$ lock-in) coincides with the forced mode overtaking the natural mode in amplitude (figure 4 e–h) and was recently reported for a forced hydrodynamically self-excited low-density jet (Li & Juniper Reference Li and Juniper2013c ). It is worth mentioning that, although the results shown in figures 4 and 5 are for $f_{f}<f_{n}$ , the same qualitative behaviour is seen for $f_{f}>f_{n}$ as long as $f_{f}$ is far from $f_{n}$ .

The phenomenon of frequency pushing has been observed by Bellows et al. (Reference Bellows, Hreiz and Lieuwen2008) and Balusamy et al. (Reference Balusamy, Li, Han, Juniper and Hochgreb2015) in experiments on lean-premixed turbulent combustors at similar forcing conditions, i.e. for $f_{f}$ far from $f_{n}$ . Those researchers mentioned that this shift in the natural frequency could not be explained simply. Frequency pushing is well known, however, in magnetrons (Chen Reference Chen1990) and has been modelled successfully by adding a Duffing (cubic restoring force) term to the van der Pol equation (Walsh et al. Reference Walsh, Johnston, Davidson and Sullivan1989). In magnetrons, frequency pushing arises from highly nonlinear electron–wave interactions that change the mean field (Chen Reference Chen1990). Given the similarities between this thermoacoustic system and universal model oscillators, we speculate that frequency pushing in thermoacoustics could also be modelled by adding a Duffing term.

4.3 Before lock-in: beating frequency and summary

For a closer examination of the beating frequency, figure 6 shows the dependence of $\unicode[STIX]{x0394}f=f_{f}-f_{n}^{\prime }$ on $f_{f}$ , both normalized by $f_{n}$ at two different values of $\unicode[STIX]{x1D716}$ . The data shown are for state B but are also representative of state A. In figure 6(a), where the forcing is weak ( $\unicode[STIX]{x1D716}=0.04$ ), $\unicode[STIX]{x0394}f/f_{n}$ around $f_{f}/f_{n}=1$ is zero, indicating lock-in. For higher or lower values of $f_{f}/f_{n}$ , the magnitude of $\unicode[STIX]{x0394}f/f_{n}$ increases (i) nonlinearly close to the lock-in boundary, indicating frequency pulling, but (ii) linearly away from it, indicating no frequency pulling. This behaviour of $\unicode[STIX]{x0394}f/f_{n}$ has been derived analytically for low-order model oscillators and is well known in the literature of nonlinear dynamics and synchronization (Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2003): $\unicode[STIX]{x0394}f/f_{n}$ has a square-root dependence on frequency detuning ( $f_{f}-f_{n}$ ) close to the saddle-node bifurcation, where $f_{f}$ is relatively close to $f_{n}$ . In figure 6(a), an approximately square-root dependence is seen on both sides of $f_{f}/f_{n}=1$ .

Figure 6. Dependence of the beating frequency, $\unicode[STIX]{x0394}f=f_{f}-f_{n}^{\prime }$ , on the forcing frequency, $f_{f}$ , both normalized by the unforced natural frequency, $f_{n}$ , at two forcing amplitudes: (a)  $\unicode[STIX]{x1D716}=0.04$ , where lock-in occurs via a saddle-node bifurcation for both $f_{f}<f_{n}$ and $f_{f}>f_{n}$ , and (b)  $\unicode[STIX]{x1D716}=0.06$ , where lock-in occurs via a saddle-node bifurcation for $f_{f}<f_{n}$ but via a torus-death bifurcation for $f_{f}>f_{n}$ .

Figure 6(b) is with stronger forcing ( $\unicode[STIX]{x1D716}=0.06$ ). As in figure 6(a), $\unicode[STIX]{x0394}f/f_{n}$ is zero near $f_{f}/f_{n}=1$ (indicating lock-in) and has a similar square-root-like dependence when $f_{f}/f_{n}<1$ (indicating frequency pulling). However, when $f_{f}/f_{n}>1$ , $\unicode[STIX]{x0394}f/f_{n}$ increases (i) abruptly at the lock-in boundary and (ii) linearly away from it, indicating an absence of frequency pulling. This suggests that, at this particular forcing amplitude ( $\unicode[STIX]{x1D716}=0.06$ ), lock-in occurs via a torus-death bifurcation when $f_{f}/f_{n}>1$ but via a saddle-node bifurcation when $f_{f}/f_{n}<1$ . This will be confirmed in § 4.4.

In summary, we have shown that, even when oscillating periodically, this self-excited thermoacoustic system responds to harmonic forcing in many different ways, depending on the proximity of the forcing frequency ( $f_{f}$ ) to the natural frequency ( $f_{n}$ ) and whether $f_{f}$ is above or below $f_{n}$ . When the forcing amplitude ( $\unicode[STIX]{x1D716}$ ) increases from zero, the system first undergoes a torus-birth bifurcation to quasi-periodicity from a period-1 oscillation. When $\unicode[STIX]{x1D716}$ exceeds a critical value, the system locks in to the forcing. If $f_{f}$ is close to $f_{n}$ , lock-in occurs via a saddle-node bifurcation with frequency pulling. If $f_{f}$ is far from $f_{n}$ , lock-in occurs via a torus-death bifurcation, with frequency pushing if $f_{f}$ and $f_{n}$ are sufficiently far apart. The lock-in process has two subtle features: (i) it is asymmetric about $f_{f}/f_{n}=1$ and (ii) the response amplitude at lock-in may be larger or smaller than that of the unforced system. These and other lock-in features will be discussed next.

4.4 At lock-in: the 1 : 1 Arnold tongue, response amplitudes and jump phenomena

In this section, we continue to examine the system during period-1 oscillations (state B), but we focus on the forced response at lock-in. Figure 7 shows the $1:1$ Arnold tongue: the minimum forcing amplitude required for lock-in, $\unicode[STIX]{x1D716}_{lock}$ , as a function of the frequency detuning around $f_{f}/f_{n}=1$ . The diamond markers denote saddle-node bifurcations, and the circular markers denote torus-death bifurcations. The dashed lines are linear fits to the saddle-node data on either side of $f_{f}/f_{n}=1$ .

Figure 7. The $1:1$ Arnold tongue for period-1 oscillations (state B) when forced across (a) a small range of frequency detuning ( $0.98<f_{f}/f_{n}<1.02$ ) and (b) a large range of frequency detuning ( $0.85<f_{f}/f_{n}<1.15$ ). The diamond markers denote saddle-node bifurcations, and the circular markers denote torus-death bifurcations. The dashed lines are linear fits to the saddle-node data on either side of $f_{f}/f_{n}=1$ .

In figure 7(a), $\unicode[STIX]{x1D716}_{lock}$ increases linearly with $|f_{f}-f_{n}|$ when $f_{f}$ is close to $f_{n}$ , producing a characteristic V-shaped lock-in curve for saddle-node bifurcations. Similar V-shaped curves have been reported for other self-excited, but physically disparate, systems such as turbulent lean-premixed combustors (Bellows et al. Reference Bellows, Hreiz and Lieuwen2008), momentum-dominated low-density jets (Li & Juniper Reference Li and Juniper2013a ), laminar jet diffusion flames (Li & Juniper Reference Li and Juniper2013b ), low-density and equidensity cross-flowing jets (Davitian et al. Reference Davitian, Getsinger, Hendrickson and Karagozian2010; Getsinger, Hendrickson & Karagozian Reference Getsinger, Hendrickson and Karagozian2012), capillary jets (Olinger Reference Olinger1992) and cylinder wakes (Provansal, Mathis & Boyer Reference Provansal, Mathis and Boyer1987). The slope of the V is asymmetric about $f_{f}/f_{n}=1$ : lock-in occurs more readily for $f_{f}/f_{n}>1$ than for $f_{f}/f_{n}<1$ . A similar asymmetry has been observed in laminar jet diffusion flames (Li & Juniper Reference Li and Juniper2013b ) and equidensity cross-flowing jets (Davitian et al. Reference Davitian, Getsinger, Hendrickson and Karagozian2010), but an opposite asymmetry has been observed in cylinder wakes (Blevins Reference Blevins1985), low-density cross-flowing jets (Getsinger et al. Reference Getsinger, Hendrickson and Karagozian2012) and momentum-dominated low-density jets (Li & Juniper Reference Li and Juniper2013a ). The exact cause of the asymmetry is unknown.

The boundary between saddle-node and torus-death bifurcations lies closer to $f_{f}/f_{n}=1$ when $f_{f}/f_{n}>1$ . Beyond this boundary, the loci of the torus-death bifurcations deviate from the linear fits to the saddle-node data, which is a trend that has been proved analytically for the forced van der Pol oscillator (Holmes & Rand Reference Holmes and Rand1978). Furthermore, the loci of the torus-death bifurcations are different on either side of $f_{f}/f_{n}=1$ (i.e. the left and right branches of the $1:1$ Arnold tongue). Figure 7(b) shows the same Arnold tongue as figure 7(a) but over a larger range of $f_{f}/f_{n}$ . The data for $f_{f}/f_{n}>1$ (right branch) saturate at $\unicode[STIX]{x1D716}_{lock}\approx 0.12$ , but the data for $f_{f}/f_{n}<1$ (left branch) peak at $\unicode[STIX]{x1D716}_{lock}\approx 0.32$ (where $f_{f}/f_{n}\approx 0.97$ ) before decreasing with decreasing $f_{f}/f_{n}$ . This behaviour of the left branch is identical to that seen in experiments on hydrodynamically self-excited jet diffusion flames (Li & Juniper Reference Li and Juniper2013b ). It is attributed to subharmonic lock-in arising from the overlap of the adjacent $3:4$ Arnold tongue.

Figure 8. (a) The system response at lock-in, i.e. at the critical forcing amplitude, $\unicode[STIX]{x1D716}_{lock}$ . The response is defined as the non-dimensional amplitude of the pressure oscillations, $A_{lock}=p^{\prime }/\unicode[STIX]{x1D6FE}Mp_{0}$ . As in figure 7(b), the diamond markers denote saddle-node bifurcations, and the circular markers denote torus-death bifurcations, with the different colours indicating different regimes. (b) The frequency-response curve for a forced Duffing oscillator with a soft cubic spring. The dashed branch (between points 3 and 6) is unstable and cannot be reached in experiments or simulations, leaving a discontinuous jump that resembles that seen in panel (a).

Figure 8(a) shows the response amplitude of the system at lock-in (i.e. at $\unicode[STIX]{x1D716}_{lock}$ ) as a function of $f_{f}/f_{n}$ . Here the response amplitude is defined as the non-dimensional amplitude of the pressure oscillations, $A_{lock}=p^{\prime }/\unicode[STIX]{x1D6FE}Mp_{0}$ . The peak in $A_{lock}$ at a frequency below $f_{f}/f_{n}=1$ and the sharp decrease on either side of it give rise to a characteristic ‘shark-fin’ curve that has been observed in other nonlinear systems, such as hydrodynamically self-excited jet diffusion flames (Li Reference Li2012) and thermoacoustically self-excited turbulent premixed flames (Bellows et al. Reference Bellows, Hreiz and Lieuwen2008). Crucially, this behaviour is also observed in the forced response of low-dimensional model oscillators, such as the Duffing oscillator with a soft cubic spring (Thompson & Stewart Reference Thompson and Stewart2002). This similarity in the forced response of physically disparate systems is further evidence that, with more analysis, some aspects of thermoacoustically self-excited systems can be represented by simple model oscillators.

A discontinuous jump in $A_{lock}$ occurs at a critical value of $f_{f}/f_{n}$ (0.97), suggesting a region of bistability and the possibility of hysteresis. This jump phenomenon is a classical feature of a cusp catastrophe (Thompson & Stewart Reference Thompson and Stewart2002). It arises when variations in one or two of the control parameters (here $f_{f}$ and $\unicode[STIX]{x1D716}$ ) cause the system to cross the fold curve on the catastrophe surface, jumping from a point on the upper (lower) stable manifold to a point on the lower (upper) stable manifold (Nayfeh & Mook Reference Nayfeh and Mook1995). To produce figure 8(a), we started the simulations at $f_{f}/f_{n}=1$ and worked outwards, increasing $f_{f}$ to get to $f_{f}/f_{n}>1$ and decreasing $f_{f}$ to get to $f_{f}/f_{n}<1$ . Hence, the bifurcation at $f_{f}/f_{n}=0.97$ is reached by decreasing $f_{f}$ . This corresponds to path $1\rightarrow 2\rightarrow 3\rightarrow 4\rightarrow 5$ in figure 8(b), which is a sketch of the frequency-response curve for a forced Duffing oscillator with a soft cubic spring (Nayfeh & Mook Reference Nayfeh and Mook1995). The reverse path $5\rightarrow 4\rightarrow 6\rightarrow 2\rightarrow 1$ is not explored in this study, but is expected to extend the lower branch of figure 8(a) (green circles) further to the right.

Like the V-shaped lock-in curve and the ‘shark-fin’ frequency-response curve, the jump phenomenon has been observed in various nonlinear systems, such as electronic circuits (Giannakopoulos & Deliyannis Reference Giannakopoulos and Deliyannis2001), hypoid gears (Wang & Lim Reference Wang and Lim2011), ecosystems (Scheffer et al. Reference Scheffer, Carpenter, Foley, Folke and Walker2001), shape-memory alloys (Xia & Sun Reference Xia and Sun2015) and turbulent premixed combustors (Bellows et al. Reference Bellows, Hreiz and Lieuwen2008). Crucially, it can be modelled accurately with a forced Duffing oscillator, a second-order nonlinear damped oscillator with cubic elasticity subjected to periodic forcing (Nayfeh & Balachandran Reference Nayfeh and Balachandran2004):

Here $x$ is the position at time $t$ , $\unicode[STIX]{x1D701}$ is the damping constant, $\unicode[STIX]{x1D714}_{0}$ is the undamped natural frequency and $\unicode[STIX]{x1D6FD}$ controls the degree of nonlinearity in the restoring force. On the right-hand side, $\unicode[STIX]{x1D716}$ is the forcing amplitude and $\unicode[STIX]{x1D714}_{f}$ is the angular frequency of the forcing. Figure 8(b) shows the typical frequency-response curve for a forced Duffing oscillator with a soft cubic spring ( $\unicode[STIX]{x1D6FD}<0$ ). The dashed branch (between points 3 and 6) is unstable and cannot be reached in experiments or simulations, leaving a discontinuous jump that resembles that seen in figure 8(a). The forced Duffing oscillator is also able to reproduce the peak in the response amplitude at $f_{f}/f_{n}<1$ , as well as predicting hysteresis. Moreover, when $f_{f}$ is far from $f_{n}$ , both the Duffing oscillator and the thermoacoustic system oscillate at amplitudes that are far below those of their unforced states. This implies that it may be possible (i) to use open-loop harmonic forcing to weaken self-excited thermoacoustic oscillations and (ii) to understand and predict how this occurs through analysis of low-order model oscillators. Quantitatively relating the coefficients of such model oscillators to the system characteristics is possible but beyond the scope of this study.

In summary, lock-in occurs most readily when $f_{f}$ is close to $f_{n}$ , but the details depend on whether $f_{f}>f_{n}$ or $f_{f}<f_{n}$ . When $f_{f}<f_{n}$ , stronger forcing is required for lock-in than when $f_{f}>f_{n}$ . However, when $f_{f}$ is gradually decreased from $f_{n}$ , the response amplitude at lock-in first increases, reaches a maximum and then drops abruptly in turn. This jump phenomenon is a well-known hysteretic feature of nonlinear oscillators undergoing a cusp catastrophe. When $f_{f}$ is far from $f_{n}$ , the response amplitude at lock-in drops to as low as 10 % of that of the unforced system. This shows that lock-in can be an effective means of weakening self-excited thermoacoustic oscillations, provided that $f_{f}$ is chosen carefully with respect to $f_{n}$ . Finally, the similarities in the forced response of this thermoacoustic system and that of universal model oscillators suggest that the behaviour seen here is not limited to this particular system, but is representative of an entire class of self-excited oscillators with a single dominant oscillatory mode.

4.5 An alternative route to lock-in: intermittency

In §§ 4.1 and 4.2, we showed that lock-in can occur via a torus-birth bifurcation followed by either (i) a saddle-node bifurcation with frequency pulling or (ii) a torus-death bifurcation without frequency pulling. Although this is true for many types of periodic oscillations, the transition to lock-in may sometimes involve other bifurcations, including transition to chaos. This may arise from large forcing amplitudes, forcing frequencies far away from the natural frequency or self-excited oscillations that are extremely resistant to external forcing (Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2003).

Figure 9. Intermittency route to lock-in: the forced response of the system during period-1 oscillations (state A) when $f_{f}/f_{n}=0.97$ , where $f_{n}=2.304$ . Time series and Poincaré maps are shown for increasing forcing amplitudes: (a)  $\unicode[STIX]{x1D716}\equiv u_{f}^{\prime }/U_{0}=0.06$ , (b)  $0.12$ , (c)  $0.16$ , (d)  $0.18$ , (e)  $0.20$ , (f)  $0.21$ , (g)  $0.22$ , (h)  $0.23$ and (i)  $0.24$ .

Figure 9 shows the transition to lock-in when the system, undergoing period-1 oscillations (state A), is forced at $f_{f}/f_{n}=0.97$ . As $\unicode[STIX]{x1D716}$ increases from zero, the first transition is a torus-birth bifurcation to quasi-periodicity, as seen previously in §§ 4.1 and 4.2. However, this is followed by intermittent instability in the torus attractor: the phase trajectories in a neighbourhood around the surface of the 2-torus start to diverge. In the time series, this appears as mildly chaotic oscillations separated by ‘quiet’ quasi-periodic intervals, but complete breakdown of the 2-torus to fully chaotic oscillations does not occur. (It is worth noting that, at a different operating point, complete breakdown to chaos does occur, resembling the Ruelle–Takens–Newhouse route to chaos presented by Kashinath et al. (Reference Kashinath, Waugh and Juniper2014).) When $\unicode[STIX]{x1D716}$ increases further, the mildly chaotic oscillations eventually lock in to the forcing, resulting in period-1 oscillations again but this time at $f_{f}$ (figure 9 i).

Figure 10. Intermittency route to lock-in: the phase difference (normalized by $2\unicode[STIX]{x03C0}$ ) between the system and the forcing at the conditions of figure 9. The arrows show that phase slips can be greater than $2\unicode[STIX]{x03C0}$ .

Figure 10 shows $\unicode[STIX]{x1D719}_{1,2}$ at the conditions of figure 9. As $\unicode[STIX]{x1D716}$ increases from zero, the magnitude of the slope of $\unicode[STIX]{x1D719}_{1,2}$ decreases, indicating frequency pulling, similar to that seen in figure 3. For $\unicode[STIX]{x1D716}>0.15$ (solid lines), phase slips occur between periods of quasi-periodicity. The arrows show that the phase slips can be greater than $2\unicode[STIX]{x03C0}$ and cause the slope of $\unicode[STIX]{x1D719}_{1,2}$ to become positive, producing an overshoot in the frequency pulling that causes the system to oscillate at a frequency slightly higher than that of the forcing, even though $f_{f}/f_{n}=0.97$ . Inspection of figures 9 and 10 shows that the phase slips coincide with (i) intervals of intermittent ‘spikes’ in the time series and (ii) intermittent instability of the phase trajectories around the 2-torus. With stronger forcing ( $0.21\leqslant \unicode[STIX]{x1D716}\leqslant 0.23$ ), the phase slips become more infrequent and, at $\unicode[STIX]{x1D716}=0.24$ , the slope of $\unicode[STIX]{x1D719}_{1,2}$ is zero, indicating lock-in.

Intermittency is well-known in synchronization and has been studied extensively in many systems, from simple one-dimensional maps, such as the circle map, to more complex systems of coupled chaotic oscillators (Belair & Glass Reference Belair and Glass1983; Glass et al. Reference Glass, Guevara, Belair and Shrier1984). The reasons behind the various transitions and their bifurcations have been studied for low-dimensional dynamical systems using periodic orbit theory (Venkatesan & Lakshmanan Reference Venkatesan and Lakshmanan1997). A detailed investigation of these transitions and bifurcations in our thermoacoustic system is beyond the scope of this study.

4.6 Beyond lock-in: the stability of synchronized oscillations

As $\unicode[STIX]{x1D716}$ increases above $\unicode[STIX]{x1D716}_{lock}$ , the amplitude of the phase-locked oscillations at $f_{f}$ also increases. However, for large values of $\unicode[STIX]{x1D716}$ , the periodic orbit at $f_{f}$ can become unstable and transition to chaos. Pikovsky et al. (Reference Pikovsky, Rosenblum and Kurths2003) explain that there are three main routes to chaos when $\unicode[STIX]{x1D716}$ increases within an Arnold tongue. Route I, which is typically found near the centre of the Arnold tongue where the stable and unstable periodic orbits are far apart, involves a series of period-doubling bifurcations of the stable periodic orbit. And routes II and III, which are typically found near the outer edges of the Arnold tongue where the frequency detuning is large, involve intermittency, which manifests as long ‘laminar’ synchronized periods separated by phase slips at chaotic intervals (route II usually occurs at smaller values of $\unicode[STIX]{x1D716}$ (Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2003)). All three routes to chaos have been analysed by Afraimovich & Shilnikov (Reference Afraimovich and Shilnikov1983) and Aronson et al. (Reference Aronson, Ermentrout and Kopell1990).

Figure 11. Transition to chaos from lock-in via period-doubling bifurcations (route I): the forced response of the system during period-1 oscillations (state B) when $f_{f}/f_{n}=1$ , where $f_{n}=1.125$ . Time series, power spectra, phase portraits and Poincaré maps are shown for increasing forcing amplitudes: (a)  $\unicode[STIX]{x1D716}\equiv u_{f}^{\prime }/U_{0}=0.16$ , (b)  $0.24$ , (c)  $0.30$ , (d)  $0.34$ and (e)  $0.36$ .

Figure 11 shows an example of route I: transition to chaos via period-doubling bifurcations of the locked-in periodic orbit at $f_{f}$ . In our previous study, we observed this route to chaos when the flame position was varied (Kashinath et al. Reference Kashinath, Waugh and Juniper2014, figure 14). In figure 11, the same route to chaos is observed when $\unicode[STIX]{x1D716}$ is increased. The period-1 oscillations (arising from lock-in) undergo a period-doubling bifurcation to period-2 oscillations, followed by another period-doubling bifurcation to period-4 oscillations and so on. The power spectra show the emergence of a new subharmonic at each period-doubling bifurcation: $f_{f}/2$ in figure 11(b), $f_{f}/4$ in (c), $f_{f}/8$ in (d), ultimately leading to chaos in (e).

Figure 12. Switching between different stable attractors after lock-in: the forced response of the system during period-1 oscillations (state A) when $f_{f}/f_{n}=1.06$ , where $f_{n}=2.304$ . Time series, power spectra, phase portraits and Poincaré maps are shown for increasing forcing amplitudes: (a)  $\unicode[STIX]{x1D716}\equiv u_{f}^{\prime }/U_{0}=0.03$ , (b)  $0.16$ , (c)  $0.18$ , (d)  $0.21$ and (e)  $0.24$ .

Apart from chaos, there are other features of synchronization at large $\unicode[STIX]{x1D716}$ that are not observed at small $\unicode[STIX]{x1D716}$ . For example, different synchronization regions (Arnold tongues) can overlap, leading to multi-stability. This means that, for certain combinations of $f_{f}$ and $\unicode[STIX]{x1D716}$ , periodic oscillations with different rational ratios between the observed and forcing frequencies can coexist. This phenomenon has been experimentally observed by van der Pol & van der Mark (Reference van der Pol and van der Mark1927) in a low-dimensional oscillator circuit and, as figure 12 shows, is also present in our thermoacoustic system when $f_{f}/f_{n}=1.06$ . Figure 12(a) shows the familiar quasi-periodic oscillation that arises from a torus-birth bifurcation, followed by lock-in at $\unicode[STIX]{x1D716}=0.12$ via a torus-death bifurcation (not shown). At larger $\unicode[STIX]{x1D716}$ , the system switches to a periodic oscillation at $f_{f}/2$ (figure 12 b). At still larger $\unicode[STIX]{x1D716}$ , it switches back to the primary synchronization orbit at $f_{f}$ (figure 12 e). This occurs because the external forcing modifies the stability of the different attractors, altering their basins of attraction, thus making one state more stable than another depending on $\unicode[STIX]{x1D716}$ (Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2003). These results show that besides choosing $f_{f}$ carefully, in order to maximize the weakening of the self-excited mode, it is also important to examine the stability of the system at lock-in.