3.1. Superharmonic secondary excitation

Figure 3 shows the Feigenbaum diagrams across a range of primary excitation amplitudes $A_p=0-1$ at $f_p=0.9$. In these panels, we consider superharmonic secondary excitation with frequency $f_s=2f_p$ and amplitude $A_s=0$, 0.05, 0.1 and $0.15$ (figure 3a–d). During single-frequency excitation (figure 3a), shedding occurs at the natural vortex shedding time period (period-1 state) when $A_p=0$. As the amplitude $A_p$ is gradually increased, torus birth bifurcation occurs, and shedding endures a quasi-periodic route, which is evident from the dense points near the low $A_p$ range. This quasi-periodic behaviour is found to extend to $A_p=0.119$.

Figure 3. Plots showing the last 1000 iterations (Feigenbaum diagrams) of $\Delta t_m$ across $A_p$ for superharmonic secondary excitation frequency $f_s=2f_p$. With $f_p=0.9$, each panel indicates Feigenbaum diagrams for a fixed secondary excitation $(a)$ $A_s=0$, $(b)$ $A_s=0.05$, $(c)$ $A_s=0.1$ and $(d)$ $A_s=0.15$. Panels (e–h) show the phase portraits between $\Delta t_{m}$ and $\Delta t_{m-1}$ for amplitudes associated with (a–d), respectively, at $A_p=0.02$ (shown with red dashed line in (a–d)).

A second bifurcation occurs (saddle-node bifurcation as reported in our previous study Britto & Mariappan Reference Britto and Mariappan2019a) when the amplitude $A_p$ is increased further. In this amplitude region, shedding exhibits a period-1 state, locked-in with the excitation frequency $f_p$. Finally, when $A_p$ is increased above $0.75$, a transition from period-1 state to a higher-periodic orbit is observed. It is observed in various dynamical systems that are susceptible to lock-in (Gollub & Benson Reference Gollub and Benson1980; Franceschini Reference Franceschini1983; Anishchenko, Safonova & Chua Reference Anishchenko, Safonova and Chua1993; Guan et al. Reference Guan, Murugesan and Li2018).

In figure 3(b), we have introduced non-zero secondary excitation ($A_s=0.05$). Since the total excitation amplitude of $u'$ starts from $0.05$ at $A_p=0$, vortex shedding never occurs at its natural time period. Quasi-periodic behaviour is observed right from $A_p=0$. Comparing figure 3(a–d), the extent of quasi-periodicity reduces with an increase in secondary excitation amplitude ($0< A_p<0.085$ for $A_s=0.05$ and $0< A_p<0.022$ for $A_s=0.1$). Finally, after a certain amplitude $A_s$, quasi-periodicity ceases to exist, and shedding begins to occur at the locked-in time period $1/2f_s$ (2 : 1 lock-in) as shown in figure 3(d).

Figure 3(e–h) shows phase portraits $\Delta t_m-\Delta t_{m-1}$ at $A_p=0.02$ (indicated in red dashed line in figure 3a–d). Referring to figure 4 of Britto & Mariappan (Reference Britto and Mariappan2019a), it was reported that during single-frequency excitation, the phase portrait in the quasi-periodic region exhibits a two-dimensional torus. With the increase in amplitude $A_p$, the size of the torus increases with its shape being retained. In the current scenario, we observe that the additional secondary excitation changes the shape of the torus as it evolves to a 2-quasi-periodic curve (f). Similar shape change was reported by Gardini et al. (Reference Gardini, Lupini, Mammana and Messia1987), Van Veen (Reference Van Veen2005) and Gyllenberg, Jiang & Niu (Reference Gyllenberg, Jiang and Niu2020). In the Lotka–Voltera model, Gardini et al. (Reference Gardini, Lupini, Mammana and Messia1987) first reported that the quasi-periodic curve (two-dimensional torus in the phase portrait) undergoes a Hopf bifurcation and the curve rounds twice to exhibit a 2-quasiperiodic curve (named recently by Gyllenberg et al. Reference Gyllenberg, Jiang and Niu2020). Subsequently, when the parameters are varied, the system undergoes a cascade of Hopf bifurcations and attains chaos (similar to classical period-doubling bifurcation to chaos). In the present model, as the secondary amplitude is increased gradually, the shape of the two-dimensional torus evolves and a bifurcation occurs therein, and the curve in the phase portrait rounds twice retaining its dense points, thereby exhibiting a 2-quasi-periodic curve. Unlike Lotka–Voltera model, we do not observe quasi-period-doubling. Whereas, when the amplitude $A_s$ is further increased, the size of the 2-quasi-periodic curve increases gradually and it breaks into a higher-periodic orbit (seen in figure 3(g,h)) before it encounters lock-in. Similar behaviour is observed when the secondary excitation frequency is switched to a subharmonic of the primary component ($f_s=f_p/2$), as described next.

3.2. Subharmonic secondary excitation

Figure 4(a–d) shows the Feigenbaum diagrams of $\Delta t_m$ across $A_p=0-1$ for $A_s=0$, 0.05, 0.1 and $0.2$. When a subharmonic ($f_s=f_p/2$) secondary excitation is introduced, the lock-in region becomes a period-2 orbit. For comparison, primary excitation frequency $f_p=0.9$ is kept the same as that of the superharmonic excitation. We observe the following similarities with the superharmonic case (§ 3.1): (1) before the onset of lock-in, a quasi-periodic behaviour is observed; (2) in the presence of (non-zero) secondary excitation (b–d), shedding does not occur at its natural frequency (even at $A_p=0$); (3) from the phase portraits ($\Delta t_{m-2}-\Delta t_m$ considering period-2 orbit) in the quasi-periodic regime, shown in figure 4(e–h), the two-dimensional torus undergoes a fold and evolves into a 2-quasi-periodic curve (f) as secondary excitation is introduced; and (4) when $A_s$ is increased further, the size of the 2-quasi-periodic curve increases and transitions to a higher-periodic orbit (g,h). However, unlike the superharmonic case, quasi-periodic shedding never disappears in the subharmonic secondary excitation (compare 3d and 4d).

Figure 4. Plots showing the last 1000 iterations (Feigenbaum diagrams) of $\Delta t_m$ across $A_p$ for subharmonic secondary excitation frequency $f_s=f_p/2$. With $f_p=0.9$, each panel shows the Feigenbaum diagram for a fixed secondary excitation amplitude $(a)$ $A_s=0$, $(b)$ $A_s=0.05$, $(c)$ $A_s=0.1$ and $(d)$ $A_s=0.2$. Panels (e–h) show the phase portraits for amplitudes associated with (a–d), respectively at $A_p=0.1$ (red dashed line).

Furthermore, we observe the following differences between single- and two-frequency excitation from Feigenbaum diagrams and phase portraits: (1) the reduction in the extent and subsequent disappearance of the quasi-periodic nature followed by a lock-in observed during superharmonic excitation (figure 3, (a) and (b–d)) highlights an important quantitative difference; and (2) the absence of a natural vortex shedding frequency and the appearance of folds in the phase portraits (observed both in subharmonic and superharmonic excitation) indicates a significant qualitative difference ((a) and (b–d) of figures 3 and 4).

In addition, we observe the following difference between subharmonic and superharmonic excitation. For $f_p=0.9$, the gradual increase in $A_s$ decreases the critical amplitude at which the onset of lock-in occurs and it leads to early lock-in (figure 3a–d). This behaviour is absent in the subharmonic excitation case. In contrast, the quasi-periodic nature with intermediate periodic windows never disappears during subharmonic excitation, whereas, when $A_s$ is gradually increased, the shedding occurs at a wider range of time period before the onset of lock-in (figure 4a–d).

These differences motivated further analyses that are presented in the rest of this paper. Since multiple frequencies are involved in excitation, lock-in is investigated from the perspective of phase lock-in. In the next section, we introduce the phase lock-in phenomenon often observed in interacting dynamical systems. The procedure to identify it in the current model is discussed in detail.

4.1. Hilbert transform for single- and two-frequency excitation

As mentioned in the previous paragraph, phase lock-in is identified by the difference between the phase of the system's response and excitation. In the present case, the system's response are the shedding time instances $t_1,t_2,\ldots$. Between two successive shedding instances, say $t_{m-1}$ and $t_m$, the circulation of the $m$th vortex begins to grow from zero and reaches $\varGamma _{sep}$, after which the vortex sheds. This forms a cycle, and the associated phase increases by $2{\rm \pi}$. Therefore, phase ($\psi _{R,m}$) of the response is defined only at vortex shedding instances ($t_m$s), given as $\psi _{R,m}=2m{\rm \pi}$ where $m=1,2,3,~\dots$ indicates vortex shedding events. Subscripts $R$ and $m$ represent response and $m$th vortex, respectively.

On the other hand, the phase of the excitation signal $u'$ is obtained using the concept of the analytic signal (Pikovsky & Rosenblum Reference Pikovsky and Rosenblum2007). An analytic signal $\zeta (t)$ is constructed from a signal or time series $u'(t)$ as below:

(4.1)\begin{equation} \zeta(t)=u'(t)+{i}H(u'(t)), \end{equation}

where $H(u'(t))$ is the Hilbert transform of $u'(t)$. As usual ${i}=\sqrt {-1}$. The instantaneous phase ($\psi _E(t$), subscript $E$ stands for excitation signal) of $u'$ is given by the following:

(4.2)\begin{equation} \psi_E(t)=\arctan \left(\frac{H(u'(t)}{u'(t)}\right)=\arctan \left(\frac{-A_p \cos(2{\rm \pi} f_p t)-A_s \cos (2{\rm \pi} f_s t+\gamma)}{A_p \sin (2{\rm \pi} f_p t)+A_s \sin(2{\rm \pi} f_s t+\gamma)}\right) . \end{equation}

For the functional form of $u'$ (2.3), $H(u'(t))=-A_p \cos (2 {\rm \pi}f_p t)-A_s \cos (2{\rm \pi} f_s t+\gamma )$, and $\psi _E(t)$ is, in general, a continuous function of time. We calculate it at discrete vortex shedding time instances $t_1,t_2,\ldots$ to obtain the phase of excitation signal, $\psi _{E,m}=\psi _E (t_m)$.

To identify 1:1 phase lock-in, the phase difference ($\Delta \psi _{m}$) between excitation signal and response at vortex shedding time instances is calculated as $\Delta \psi _{m}=\psi _{E,m}-\psi _{R,m}$. The dynamics of $\Delta \psi _{m}$ with $m$ allows one to identify phase lock-in. The same idea can be extended to identify a general p : q phase lock-in, where $\Delta \psi _{m}=q\psi _{E,m}-p\psi _{R,m}$ is a constant value.

In the subsequent sections, we first discuss the dynamics of $\Delta \psi _m$ in the vicinity of the 1:1 lock-in region in the $A_p - f_p$ plane. In § 4.3, we explore the nature of p : q lock-in during two-frequency excitation.

4.2. Phase lock-in during single-frequency excitation

Identification of phase lock-in through $\Delta \psi$ is first demonstrated in single-frequency excitation ($A_s=0$). The lock-in region (represented by green in figure 5a) is obtained from the analytical solution discussed in our previous paper (Britto & Mariappan Reference Britto and Mariappan2019a). Phase difference $\Delta \psi$ is evaluated at three significant regions in $A_p - f_p$ plane: (i) before the onset of lock-in; and (ii) inside the lock-in region and (iii) in the region after the transition from lock-in. In the following discussions, we highlight the behaviour of $\Delta \psi$ at these locations.

Figure 5. Phase lock-in during single-frequency excitation. Lock-in boundary is shown in (a). Phase difference $\Delta \psi _m$ for various forcing amplitudes and excitation frequency (b) $f=0.8$ and (c) $f=1.1$ are illustrated. Excitation amplitudes chosen in (b,c) are colour coded and are shown in (a) using cross symbols with the corresponding colours. Lock-in region is obtained from the analytical expression given in Britto & Mariappan (Reference Britto and Mariappan2019a).

For two primary excitation frequencies on either side of $f_p=1$ ($f_p=0.8$ and $f_p=1.1$), the evolution of $\Delta \psi _m/2{\rm \pi}$ with $m$ at the selected locations in the $A_p - f_p$ plane is illustrated in figure 5(b,c). The colour of the curves corresponds to the cross symbols shown in the $A_p - f_p$ plane of (a). For $f_p=0.8$, we begin with a low amplitude, $A_p=0.1$ (illustrated with a blue marker). Clearly, there is no lock-in, and $\Delta \psi _m$ increases unboundedly, exhibiting a large positive slope (blue curve in (b)). This phenomenon is called phase drifting (Pikovsky & Rosenblum Reference Pikovsky and Rosenblum2007; Li & Juniper Reference Li and Juniper2013c). As the amplitude is increased gradually, the slope of $\Delta \psi _m$ approaches zero. This phenomenon is called frequency pulling (Pikovsky & Rosenblum Reference Pikovsky and Rosenblum2007; Li & Juniper Reference Li and Juniper2013c), where the average frequency of the shedding is pulled towards the forcing frequency. Finally, the instantaneous phase difference, $\Delta \psi _{m}$, attains a constant value leading to phase lock-in (green curve). Near the lock-in boundary (in the unlocked state), $\Delta \psi _m$ is almost constant with phase slips occurring at discrete instances (red and yellow curves). During a phase slip, $\Delta \psi _m$ jumps by $2{\rm \pi}$. Phase slips occur less frequently as one approaches the lock-in boundary (compare red and yellow curves). This indicates that shedding begins to exhibit intermittent, long epochs of lock-in between phase slips as $A_p$ approaches the lock-in boundary. In the phase lock-in region (green shaded portion), $\Delta \psi _m$ remains constant without any phase slips. When the amplitude reaches higher values, a transition out of lock-in occurs. The black marker in panel $a$ is outside the lock-in boundary, and the corresponding $\Delta \psi _m$ shows phase slips. It must be noted that for all $A_p$ in $f_p<1$, $\Delta \psi _m$ remains positive. Since $f_p$ is less than the natural vortex shedding frequency, the phase of the excitation signal $\psi _E (t)$ is greater than $\psi _R (t)$. Hence the positive slope is consistent in no lock-in amplitudes. The three salient features of phase that are observed in the current model, namely phase drifting, phase slipping and frequency pulling, were reported in experiments (Li & Juniper Reference Li and Juniper2013c).

In (c), the above exercise is repeated for $f_p=1.1$ (a case where $f_p>1$), we see that $\Delta \psi _m$ exhibits similar characteristics as that for $f_p=0.8$, but with the slope of $\Delta \psi _m$ negative before the onset of lock-in (since $\psi _E (t) < \psi _R (t)$). However, during the transition out of lock-in, the phase difference exhibits a positive slope similar to the $f_p<1$ case. Overall, we have shown that the identification of a lock-in region through $\Delta \psi _m$ is possible for single-frequency excitation. We now proceed to extend the same idea during two-frequency excitation.

4.3. Phase lock-in during two-frequency excitation

Figure 6(a) shows the boundary of lock-in in the $A_p - f_p$ plane, during a two-frequency excitation. In this case, the excitation signal, apart from the primary frequency $(f_p)$ contains a secondary superharmonic component with amplitude $A_s=0.05$ and frequency $f_s=2f_p$. Similar to the previous single-frequency excitation, we consider two cases for the frequency of primary excitation: $f_p<1$ and $f_p>1$. The corresponding evolution of phase difference $\Delta \psi _m$ is shown in (b) and (c), respectively. Considering first the case at $f_p=0.85$ in the no lock-in, $\Delta \psi _{m}$ (blue curve in (b)) increases with time, and exhibits phase slipping with an intermediate $2{\rm \pi}$ jump. As $A_p$ is increased, the interval between the jump increases (refer to red and yellow curves). Finally during lock-in (green curve), $\Delta \psi _{m}$ attains a constant value. Overall, both for $f_p<1$ and $f_p>1$ (see (b,c)), $\Delta \psi _{m}$ follows qualitatively similar characteristics as that of their single-frequency excitation counterpart. It is important to mention the following observation. For the case, $f_p=1.4$ which is relatively far (in comparison with the case, $f_p=0.85$) from the natural vortex shedding frequency, $\Delta \psi _{m}$ becomes constant during lock-in ($A_p=0.25$, green curve in (c)) and decreases unboundedly near the lock-in boundary ($A_p=0.24$, pink curve in (c)). This shows that the current model does not exhibit phase trapping, a behaviour in which $\Delta \psi _{m}$ remains bounded but fluctuates near the lock-in boundary. The same observation is found in our previous investigation (Britto & Mariappan Reference Britto and Mariappan2019a) which involved single-frequency excitation.

Figure 6. Phase lock-in during two-frequency excitation. Lock-in boundary is shown in (a). Phase difference $\Delta \psi _{m}$ obtained for various forcing amplitudes and excitation frequency $f_s=2f_p$ (b) $f_p=0.85$ and (c) $f_p=1.4$. Lock-in region can be identified when the phase difference is constant. The chosen excitation amplitudes in (b,c) are colour-coded and are marked in (a) using cross symbols with the corresponding colours.

Unlike the case of single-frequency excitation, where the lock-in boundary is obtained analytically (Britto & Mariappan Reference Britto and Mariappan2019a), the boundary for the case of two (or multiple)-frequency excitation is identified by tracking the evolution of $\Delta \psi _{m}$.

To find lock-in regions in multiple-frequency excitation, we use the criteria $\Delta \psi _m=C$, where $C$ is a constant. Throughout this study, we restrict ourselves to only rational relations between primary and secondary excitation frequencies, $f_s=p_f f_p/q_f$, where $p_f$ and $q_f$ are integers. A discussion on such a choice of secondary excitation is provided in § 4.4. During phase lock-in, the iterations of $\Delta \psi _m$ are period-$q_f$ and bounded. Therefore, a p : q lock-in region is identified using the following relations:

(4.3)\begin{equation} \Delta \psi_m-\Delta \psi_{m-q_f}=0,\quad \textrm{with} \ \Delta \psi_m=q \psi_{E,m}-p \psi_{R,m}. \end{equation}

To identify a given p : q lock-in, $\Delta \psi _m-\Delta \psi _{m-q_f}=K$ is plotted in the $A_p - f_p$ plane. For illustration, we choose $f_s=2f_p$ for superharmonic and $f_s=f_p/2$ for subharmonic excitations. Consequently, we expect 1:1, 2 : 1 and 1:1, 1 : 2 lock-in for super- and subharmonic excitations, respectively. Figures 7(a,b) and (d,e) show the contour plots of $K$ corresponding to superharmonic and subharmonic secondary excitation, respectively, with $A_s=0.05$. The dark blue region ($K=0$) in the contour represents phase lock-in regions. The corresponding boundary of lock-in is shown in figure 7(c,f). This boundary is obtained by increasing $A_p$ gradually till $K=0$ is attained for the first time, at each $f_p$, blue and red curves represent 1:1 and 2 : 1/1 : 2 lock-in, respectively, in figures 7(c) and 7(f).

Figure 7. Illustration of boundary of lock-in obtained using the condition (4.3). Panels (a–c) and (d–f) correspond to the superharmonic and subharmonic secondary excitation, respectively, with $A_s=0.05$. In (a,d), the dark blue region ($K=0$) represents 1:1 phase lock-in. Similarly, in (b,e), it represents 2 : 1 and 1 : 2 phase lock-in, respectively. The boundary of blue regions in these panels are mapped in (c,f).

The following observations are common to both variants of secondary excitations: (i) when $A_p< A_s$, the system exhibits 2 : 1 or 1 : 2 lock-in for super- and subharmonic excitation, respectively; and (ii) when $A_p>A_s$, vortex shedding encounters 1:1 lock-in. For superharmonic excitation (a,b), both 1:1 and 2 : 1 lock-in regions show a plateau boundary (upper and lower plateau) at their onsets (c). On the other hand, a prominent plateau boundary is observed only at the onset of the 1:1 lock-in boundary (upper plateau) for subharmonic excitation (f). Our simulations indicate that the lower plateau is so small that it appears to be absent.

Furthermore, in contrast to the lock-in boundary of single-frequency excitation (figure 5a), we observe the following difference in its shape during two-frequency excitation (the differences are common in both super- and subharmonic secondary excitation). The 1:1 lock-in forms a plateau boundary at $A_p=A_s$ which is in contrast to the classical $V$-shaped boundary at $f_p=1$ during single-frequency excitation (figure 5a). Although the contribution of secondary excitation is present throughout the $A_p - f_p$ plane, it is dominant in the region where $A_p< A_s$. This dominance restricts the 1:1 lock-in region to the $A_p>A_s$ region. This behaviour is observed for other superharmonic ($f_s=3f_p$) and subharmonic ($f_s=f_p/3$) secondary excitation (figure 8). The same figure shows that the 1:1 lock-in boundary stemmed at $f_p=1$ for single-frequency excitation, detaches from $(A_p,f_p)=(0,1)$ and shifts to a value of $A_p=A_s$, forming a plateau.

Figure 8. The 1:1 phase lock-in regions for (a) $f_s=2f_p$, (b) $f_s=3f_p$, (c) $f_s=f_p/2$ and (d) $f_s=f_p/3$. Three different secondary excitation amplitudes are chosen to illustrate the variation of lock-in boundary.

At this point, it is worth to mention the parallels with the experimental study by Guan et al. (Reference Guan, Gupta, Wan and Li2019a). They periodically forced a ducted laminar premixed flame, exhibiting quasi-periodic unforced (natural) oscillations. The parallel is that the shape of the lock-in boundary is altered by the presence of two natural frequencies. This is similar to the present altered lock-in boundary (figure 8), where there is a change in the order of lock-in. In our investigations, two forcing frequencies are used, while in Guan et al. (Reference Guan, Gupta, Wan and Li2019a), two natural frequencies are present. In both studies, three modes are involved.

These results show the possibility of controlling and tailoring the lock-in region by introducing a secondary excitation component in the flow stream. Since some of the previous experimental results do indicate the occurrence of 1:1 lock-in during combustion instability (Poinsot et al. Reference Poinsot, Trouve, Veynante, Candel and Esposito1987; Singh & Mariappan Reference Singh and Mariappan2021), the above method can be considered as active control for its mitigation.

In order to understand the formation of a plateau in the lock-in boundary, analytical investigation through first return maps is performed in § 5. This requires a geometrical interpretation of phase difference $\Delta \psi _m$, discussed in the next section.

4.4. Geometric interpretation of $\Delta \psi _m$

We begin by studying the geometrical interpretation of the phase of the excitation signal $\psi _{E,m}$. As explained in § 2, vortex shedding time instances $(t_m)$ are identified by the condition $\varGamma _m (t_m)= \varGamma _{sep}(t_m)$, $\varGamma _{sep}(t_m)$ is linearly related to excitation $u'$. Since the excitation is sinusoidal, for a particular set of $A_p$, $f_p$, $A_s$ and $f_s$, $\varGamma _{sep}(t_m)$ can be located as phase points on a geometry consisting of two circles: primary and secondary (figure 9). The primary excitation along with the steady base flow is represented by the primary circle of radius $A_p/2$, centred (point $O$) at 1/2 from the origin (point $S$) in the vertical direction. The shift 1/2 is to accommodate the steady flow velocity in $\varGamma _{sep}(t_m)$. The vertical distance between points $P$ and $S$ equals $1/2+A_p\sin (2{\rm \pi} f_p)/2$, which represents the contribution of steady and primary frequency components of $u'$ excitation in $\varGamma _{sep}(t_m)$ (refer to (a)). Over time $t$, the point $P$ traces over the primary circle and forms an angle $2 {\rm \pi}f_p t$. Similarly, the presence of the secondary component ($A_s \sin (2{\rm \pi} f_s t$)) can be represented by another point $Q$ lying on the secondary circle having a radius $A_s/2$. This circle is centred at the primary phase point $P$, allowing the secondary circle to move along the primary circle. Geometrically, the vertical distance between points $Q$ and $S$ equals $\varGamma _{sep}(t_m)=1/2+A_p\sin {(2{\rm \pi} f_p t_m)}/2+A_s\sin {(2{\rm \pi} f_s t_m+\gamma )}/2$ (refer to (b)). Therefore, the evolution of $\varGamma _{sep}$ is the movement of point $Q$ with time. Crucially, the angle ($\phi _{m}$) subtended between the line joining points $Q$ and $O$ with the horizontal ($OR$) equals

(4.4)\begin{equation} \phi_{m}=\arctan{\frac{Y}{X}} , \end{equation}

where $X=A_p \cos (2{\rm \pi} f_p t_m)+A_s \cos ( 2{\rm \pi} f_s t_m +\gamma )$ and $Y=A_p \sin (2{\rm \pi} f_p t_m)+A_s \sin (2{\rm \pi} f_s t_m +\gamma )$ (b). Now, $\phi _m$ at time instances $t_m$, where $m=1,2,3,\ldots$ are the phase points of the vortex shedding events on the $\varGamma _{sep}(t)$. The angle $\phi _m$ can be related to the phase $\psi _{E,m}$ of the excitation signal $u'$ using (4.2):

(4.5)\begin{equation} \tan \psi_{E,m} ={-}\frac{1}{\tan \phi_m},\quad \implies \quad \phi_m=\psi_{E,m}-{\rm \pi}/2. \end{equation}

Figure 9. Time instances of vortex shedding are illustrated as phase points on the circle. (a) The circle centred at $O$ represents the primary excitation signal with the parameters $A_p$ and $f_p$. The inclusion of a secondary component results in an additional circle of radius $A_s$ with frequency $f_s$ centred at the point $P$. Over time $t$ points $P$ and $Q$ trace over the primary and secondary circle, respectively. (b) At any time instant $t$, the phase point $Q$ lies at a vertical distance $\varGamma _{sep}(t)$ from $S$. The angle subtended by $QO$ with $OR$ is $\phi _m$.

As discussed in § 4.1, a p : q lock-in is identified when $\Delta \psi _m$ is constant. For a secondary excitation given by $f_s=p_f/q_f f_p$, where $p_f$ and $q_f$ are integers, the constant phase criteria translates to $\Delta \psi _m-\Delta \psi _{m-q_f}=0$, from which we obtain the following relation between $\phi _m$ and $\phi _{m-q_f}$:

(4.6)\begin{equation} \left.\begin{gathered} q \left(\psi_{E,m}-\psi_{E,m-q_f}\right)-p\left(\psi_{R,m}-\psi_{R,m-q_f}\right)=0,\\ q \left(\phi_{m}+\frac{\rm \pi}{2}-\phi_{m-q_f}-\frac{\rm \pi}{2}\right)-p\left(q_f 2{\rm \pi}\right)=0,\\ \implies \phi_m-\phi_{m-q_f}=\frac{2{\rm \pi} p q_f}{q}. \end{gathered}\right\} \end{equation}

Equation (4.6) is similar to that of the phase relation reported for single-frequency excitation during p : q lock-in (Britto & Mariappan Reference Britto and Mariappan2019a).

Specifically, for the superharmonic excitations considered in this paper, $q_f=1$ (refer to figure 8a,b). The phase relation becomes $\phi _m-\phi _{m-1}={2{\rm \pi} p/q}$. Furthermore, in the present vortex shedding model, we always observed lock-in with $q=1$. Therefore, in a p : 1 lock-in, the consecutive phases $\phi _m$s are $2{\rm \pi} p$ radians apart. On the other hand, for the present subharmonic excitations, $p_f=1$ (refer to figure 8c,d). Therefore, the condition for lock-in becomes $\phi _m-\phi _{m-q_f}=2{\rm \pi} q_f/q$. Specifically, during 1 : 2 lock-in, owing to subharmonic ($f_s=f_p/2$) secondary excitation, the phase follows $\phi _{m}-\phi _{m-2}={2{\rm \pi} }$. Therefore, phases $\phi _m$s are $2{\rm \pi}$ radians apart for every two iterations. Depending on the lock-in order and frequency ratio, the phase relation exhibits different periodicity. Overall, the period of $\phi _m$ is equivalent to the lowest frequency among the primary and secondary excitation signals. This behaviour was recently reported by Kashinath, Li & Juniper (Reference Kashinath, Li and Juniper2018) in a model oscillator subjected to large amplitude forcing. Hence, an appropriate modulo operation (modulo $2{\rm \pi} , ~4{\rm \pi} ,~6{\rm \pi} ,\ldots$) on $\phi _m$ leads to a fixed point during lock-in. We utilise this fact to study bifurcations across the lock-in boundary in § 5.

Since the condition of lock-in can be expressed elegantly using (4.6), we prefer to study a map relating $\phi _{m-q_f}$ and $\phi _{m}$, instead of investigating (3.1) in time $t$. Furthermore, the transformation allows us to conveniently study dynamical features of lock-in, such as bifurcations, through first return maps (discussed in § 5).

At this point, it is necessary to highlight lock-in in the presence of quasi-periodic excitations. Quasi-periodic behaviour of oscillations were reported during combustion instability in multiple previous investigations Kabiraj & Sujith (Reference Kabiraj and Sujith2012). During lock-in, under a non-zero secondary excitation ($f_s=p_f f_p/q_f$), (4.6) implies that the oscillations ($\phi _m$) are $q_f$ periodic. As $q_f$ is infinite for the quasi-periodic excitation, it is not possible to study phase lock-in in the current framework (refer to (4.6)). In fact, only frequency lock-in is viable during the quasi-periodic excitation, which is shown in figure 10.

Figure 10. Response caused by quasi-periodic excitation with $f_p=0.9$, $f_s=\sqrt {2} f_p$ and $A_s=0.05$. Panel (a) shows the evolution of $\Delta \psi _m$ with $A_p=0.08 - 0.12$. Panels (b–d) show the corresponding phase portrait between $\Delta t_m$ and $\Delta t_{m-1}$ for amplitudes associated with (a) (curves are colour coded in (a)).

Figure 10(a) represents the evolution of the instantaneous phase difference $\Delta \psi _m$ for $f_p=0.9$, $A_s=0.05$ and $f_s=\sqrt {2} f_p$. At the lowest amplitude $A_p=0.08$, $\Delta \psi _m$ decreases unboundedly indicating phase drifting (unlocked state). When the amplitude is increased gradually, the slope of $\Delta \psi _m$ moves towards zero. At $A_p=0.12$, $\Delta \psi _m$ becomes bounded, although fluctuating about a mean. The bounded nature indicates frequency lock-in, while the fluctuations show the absence of phase lock-in. Panels (b–d) show the corresponding phase portraits between $\Delta t_m - \Delta t_{m-1}$, which reveal the occurrence of frequency lock-in. At the lowest amplitude ($A_p=0.08$), we observe that the iterations occupy a dense region in the phase portrait, exhibiting three frequency quasi-periodicity (3-tori shown in (b,c)). During frequency lock-in (d), the phase portrait collapses to a 2-torus quasi-periodic orbit. Since the present framework allows us to study only phase lock-in, we restrict ourselves to super- and subharmonic secondary excitations, which are rational multiples of the primary.

It is found that a dynamical map between $\phi _{m-q_f}$ and $\phi _m$ for the two-frequency excitation cannot be obtained explicitly. However, the map can be indirectly determined. For this purpose, we introduce $\alpha _m=2{\rm \pi} f_p t_m$ and $\beta _m=2 {\rm \pi}f_s t_m+\gamma$ as notations for phase angles of points P and Q, respectively, in their corresponding circles (refer to figure 9). The following relation between $\alpha _m$ and $\beta _m$ exists:

(4.7)\begin{equation} \beta_m=\frac{p_f}{q_f} \alpha_m +\gamma, \end{equation}

where $\gamma$ is the initial ($t=0$) relative phase between the primary and secondary excitation. Using the above notations in (3.1), the map equation connecting successive vortex shedding events become

(4.8)\begin{equation} F(\alpha_m,\alpha_{m-1},A_p,A_s,f_p,f_s,\gamma)=0 . \end{equation}

The full implicit equation (4.8) is given in Appendix A. The parameters $A_p$, $A_s$, $f_p$ and $f_s$, along with the initial condition $\gamma$ govern the successive instantaneous phase $\alpha _m$ of vortex shedding. In the next section, the influence of these parameters on the dynamics of $\alpha _m$, and hence $\phi _m$, is studied in detail using return maps and cobweb diagrams.

5.1. Bifurcations in the $A_p - f_p$ plane

In previous work (Britto & Mariappan Reference Britto and Mariappan2019a), for a single-frequency excitation case, we observed the birth of stable and unstable fixed points in the return maps at the onset of lock-in. This occurred through a saddle-node bifurcation. An analytical stability analysis was performed to confirm the same. In this study, we follow a similar procedure and analyse our return maps. Since a functional relation between $\phi _{m}$ and $\phi _{m-1}$ (or $\phi _{m}$ and $\phi _{m-2}$) is not available, we rely entirely on the geometry of return maps and cobweb diagrams to study bifurcations. In the present study, we observed two orders of lock-in in three regions of the $A_p - f_p$ plane (refer to figure 7). Therefore, we explore the nature of bifurcations across these regions: $A_p< A_s$, $A_p=A_s$ and $A_p>A_s$. First, we study the geometry of return maps in the $A_p< A_s$ region where the secondary excitation is dominant over the primary (discussed in figure 7). Further, the bifurcation associated when both primary and secondary excitation amplitudes are equal across $A_p=A_s$. In the end, the bifurcation beyond the $A_p>A_s$ region is studied.

5.1.1. Phase dynamics in the $A_p< A_s$ region

Figure 13(a) shows the lock-in region in focus. Using criteria (4.6), we find the occurrence of 2 : 1 lock-in for $A_p< A_s$. To illustrate the occurrence of bifurcations, we choose $A_p=0.01< A_s=0.05$ with $f_s=2f_p$ (superharmonic excitation). As we increase the primary excitation frequency, from $f_p=0.935 - 1.07$ (refer to (b–h)), the first return map exhibits a series of saddle-node bifurcations ($SN_1-SN_4$) (see supplementary movie 1, available at https://doi.org/10.1017/jfm.2021.624). Using (4.6), we have set modulo $4{\rm \pi}$ on $\phi _m$ to observe fixed points on return maps during 2 : 1 lock-in. Initially, in (b), $f_p=0.935$ lies outside the lock-in region and, hence, there are no fixed points in the first return map. The flow of cobwebs occupy a dense region in the map representing quasi-periodic behaviour. The parameter $f_p$ now enters the 2 : 1 phase lock-in region illustrated in (c). The return map, in turn, intersects with $\phi _m=\phi _{m-1}$ at two locations (solid and hollow green circles), marking the formation of two new fixed points through saddle-node bifurcation ($SN_1$). At one of the fixed points (hollow circle), the return map has a slope greater than one, identifying it as linearly unstable. Therefore, the cobwebs flow towards the stable fixed point (solid circle), where the slope is less than one (solid green circle). The dynamics governed by the geometry of the first return map alters once $f_p$ enters the blue shaded region shown in (a). Although $f_p$ lies inside the 2 : 1 phase lock-in region, a new pair of fixed points (solid and hollow orange circles) are formed through a second saddle-node bifurcation ($SN_2$), in addition to the existing set (d). This causes the cobweb to choose between the two stable fixed points (between solid green and orange circles). The flow of cobwebs now depends on the initial $\phi _0$, which defines a bistable region that exists throughout the blue shaded region shown in (a). When the cobweb switches from one fixed point to another, it changes the vortex shedding phase instances as illustrated in (i), through threshold curve $\varGamma _{sep} (t)$. Although $\varGamma _{m}$ remains almost same, both $\phi _m$s and, hence, $t_m$s change with the initial condition ($\phi _0$). This behaviour is critical from the perspective of combustion instability. The phase of heat release in comparison to the acoustic pressure fluctuation in the combustion chamber largely alters the Rayleigh criterion (Lieuwen Reference Lieuwen2012), which governs the growth/decay of pressure oscillations. If the conditions are right, one of the fixed points can have an appropriate $\phi _m$ for the occurrence of instability. So the bistable region has the potential to exhibit pulsed combustion instability (Culick, Burnley & Swenson Reference Culick, Burnley and Swenson1995).

Figure 13. Illustration of bifurcations occurring in region $A_p< A_s$. (a) The bistability region is shown using blue shade. It lies within the 2 : 1 lock-in boundary. (b–h) Return maps and cobweb diagrams for superharmonic secondary excitation $f_s=2f_p$. With the primary excitation amplitude set to $A_p=0.01$, $f_p$ is swept across the lock-in boundary (the chosen $f_p$s are marked using red cross symbols in (a)). Appearances of stable and unstable fixed points are marked using solid and hollow circles, respectively. Saddle-node bifurcations marked $SN_1-SN_4$ are shown in (c–g). Vortex shedding time instances for two initial conditions $\phi _0=0$, $0.5$ corresponding to (d) are shown using green and orange circles in (i). Refer to supplementary movie 1 showing the continuous variation of the return map with $f_p$, along with the occurrence of saddle-node bifurcations.

Now, as the primary excitation frequency crosses $f_p=1$, the shape of the first return map begins to exhibit another transformation (see (e–h)). The curve of the first return map near the first fixed point (green circle) begins to detach from the $\phi _m=\phi _{m-1}$ line. This causes another saddle-node bifurcation ($SN_3$), and the first fixed point disappears (g), marking the end of bistability. Hence, the cobweb flows towards the second fixed point (orange circle). Finally, when $f_p$ is increased beyond the lock-in boundary, the second fixed point disappears through the final saddle-node bifurcation ($SN_4$) and causes the cobweb to occupy a dense region in the map (h). It should be noted that bistability is only observed during superharmonic secondary excitation when $A_p< A_s$. In subharmonic secondary excitation, we observe only $SN_1$ and $SN_4$ bifurcations.

The origin of the bistability region is comprehended using the return maps. In figure 14(a–d) represent return maps for super- ($f_s=2f_p$) and subharmonic ($f_s=f_p/2$) secondary excitations, respectively. According to (4.6), modulo $4{\rm \pi}$ and $2{\rm \pi}$ operations must be performed on the corresponding return maps. However, we use modulo $2{\rm \pi}$ operation throughout (a–d) to illustrate better the origin of bistability. Consequently, the curves that span from $0 - 2{\rm \pi}$ and $2{\rm \pi} - 4{\rm \pi}$ are shown in blue and red colours, respectively. We begin with $A_p=0$, which corresponds to the single-frequency excitation. In this scenario, we observe that the return map (a,c) that spans between $\phi _m=2{\rm \pi} - 4{\rm \pi}$ (red curve) overlaps with the curve that lies within $0 - 2{\rm \pi}$ (blue curve). Hence, the two sets (both stable and unstable) of fixed points formed between $\phi _m=0 - 4{\rm \pi}$ are the same (solid and hollow red circles). This observation is common for both sub- and superharmonic secondary excitations.

Figure 14. Origin of bistability owing to the inclusion of primary excitation. Panels (a–d) represent the return maps for super- and subharmonic secondary excitations, respectively. In both the cases, the primary excitation frequency is set at $f_p=1.01$ and $A_p$ is increased from $0$ to $0.02$. Panels (e,g) and (f,h) represent the threshold circulation strength $\varGamma _{sep}$ for super- ($f_s=2 f_p$) and subharmonic ($f_s=f_p/2$) secondary excitations, respectively. Panels in the rows (e,f) and (g,h) correspond to primary excitation amplitudes $A_p=0.01$ and $0.045$, respectively. The evolution of $\varGamma _{sep}$ is shown as a blue solid curve, while black and red dashed curves correspond to the contributions from primary and secondary excitations, respectively.

Bistability occurs when a non-zero primary excitation is introduced into the flow field ($A_p=0.02$, refer to (b,d)). In the superharmonic excitation, the $2{\rm \pi}$ periodicity is broken as the first return map begins to exhibit $4{\rm \pi}$ periodicity (blue and red curves do not overlap). This behaviour is in line with the lock-in criterion given in (4.6). In (b), it can be observed that the return map and fixed points (blue and red circles) exhibit a slight difference between them. Therefore, it creates a bistable region during superharmonic secondary excitation. The absence of this behaviour during subharmonic secondary excitation (refer to (d)) is discussed below.

Panels (e–h) represents the time evolution of threshold circulation strength $\varGamma _{sep}$ (blue curve) for super- (e,g) and subharmonic (f,h) secondary excitations, respectively. In both cases, the primary excitation is increased from $A_p=0.01$ to $0.045$. In the considered parametric regime ($A_p< A_s$), secondary excitation is dominant, which dictates the vortex shedding. The presence of the primary excitation modulates $\varGamma _{sep}$. This can be observed by the evolution of the contributions of primary (black dashed curve) and secondary (red dashed curve) excitations to $\varGamma _{sep}$, shown in (e–h). In our current model, we observe that the period of oscillation of $\phi _m$ (also in $\varGamma _{sep}$) corresponds to that of the lowest frequency of excitation (also reported in Kashinath et al. Reference Kashinath, Li and Juniper2018). In the superharmonic case ($f_s=2f_p$), the lowest frequency corresponds to the primary component $A_p$ (black dashed curve). Hence, the non-zero primary excitation creates a change in the periodicity of the return map, leading to two stable solutions (bistability). Whereas, in the subharmonic case, the lowest frequency is caused by the dominant secondary excitation (red dashed curve). Hence, the introduction of the primary (having a larger frequency) does not break the $2{\rm \pi}$ periodicity in the corresponding return maps (c,d). Therefore, the presence of two-frequency excitation causes bistability only in the superharmonic case.

5.1.2. Phase dynamics across $A_p=A_s$

In this section, we discuss the dynamics of $\phi _m$ across the $A_p=A_s$ line that separates the relative dominance of the two excitation signals. As discussed in § 4.3, there are two orders of lock-in regions within the lock-in boundary in the $A_p - f_p$ plane (discussed in figure 7). This indicates a jump in the order of lock-in across $A_p=A_s$. Using the analytical criterion given in (4.6), for a superharmonic (or subharmonic) excitation $f_s=2f_p$ ($f_s=f_p/2$), the iterations of $\phi _m$ exhibit a phase difference of $4{\rm \pi}$ ($2{\rm \pi}$) and $2{\rm \pi}$ ($4{\rm \pi}$) in 2 : 1 (1 : 2) and 1:1 (1:1) lock-in regions. Hence, we use the appropriate modulo operation on their return maps to study the behaviour of fixed points and the flow of cobwebs. In figure 15, we have set $A_s=0.05$ and varied $A_p$ in the range 0.048–0.051. In the superharmonic excitation (a,b), the first return map spans from $0 - 4{\rm \pi}$ and $0 - 2{\rm \pi}$ when $A_p< A_s$ and $A_p>A_s$, respectively (refer to (4.6)). First, during 2 : 1 lock-in, a stable fixed point (slope less than one) exists at $\phi _{m-1}=10.15$ (green solid circle in figure 15a). When $A_p$ is increased gradually, the shape of the first return map changes slightly (not shown). As $A_p$ crosses $A_s$, the geometry of the first return map changes drastically and becomes $2{\rm \pi}$ periodic as shown in (b). A new pair of fixed points are formed at $\phi _{m-1}=1.14{\rm \pi}$ (solid green circle) and $1.85{\rm \pi}$ (hollow green circle) (refer to figure 15b). It should be noted that previously, there was no fixed point below $\phi _{m-1}=2{\rm \pi}$ (illustrated using vertical dashed line in (a)). This indicates that the map undergoes a saddle-node bifurcation when $A_p$ is increased across the $A_p=A_s$ line. The aforementioned can be observed in supplementary movie 2, where the return maps between $\phi _{m-1}-\phi _{m}$ (superharmonic, left panel) and $\phi _{m-2}-\phi _{m}$ (subharmonic, right panel) are animated in blue. Note that both the axes are between $0 - 4{\rm \pi}$. To highlight the occurrence of $2{\rm \pi}$ periodicity, the map between $2{\rm \pi} - 4{\rm \pi}$ is re-plotted between $0 - 2{\rm \pi}$ in red. From the movie, one can observe at $A_p=A_s$, the first return map (left panel) changes its periodicity from $4{\rm \pi}$ to $2{\rm \pi}$ (shown by the overlap between blue and red curves in $0 - 4{\rm \pi}$). A contrasting behaviour is observed during subharmonic secondary excitation shown in figure 15(c,d). Since $q_f=2$, we now study the second return map and its corresponding cobweb diagram in $\phi _m - \phi _{m-2}$ plane. When $A_p< A_s$ (c), the return map lies within $0 - 2{\rm \pi}$ with a fold near $\phi _{m-2}={\rm \pi} /2 - 3{\rm \pi} /2$ (between red vertical dashed lines in (c)). There are two stable fixed points (orange and green solid circles) that lie within $0 - 2{\rm \pi}$. When $A_p$ is increased above $0.05$, the return map unfolds and stretches between $0 - 4{\rm \pi}$ (d). One of the stable fixed points (solid green circle), along with its unstable counterpart, disappears and a new pair of fixed points is formed at $\phi _{m-2}=11.14$ and $11.61$ (hollow and solid pink circles). Therefore, a saddle-node bifurcation is observed when the primary excitation amplitude is increased across the $A_p=A_s$ line (refer to the right panel of supplementary movie 2). In the next section, we study the bifurcation observed beyond $A_p=A_s$.

Figure 15. Return maps and cobweb diagrams for (a,b) superharmonic $f_s=2f_p$ and (c,d) subharmonic secondary excitation. With the primary excitation frequency set to $f_p=1.05$, $A_p$ is swept across $A_s=0.05$. The transformation in the periodicity of the first return map and the flow of cobwebs are illustrated in supplementary movie 2.

5.1.3. Bifurcations in the $A_p>A_s$ region

We set the excitation amplitudes at $A_p=0.07$ and $A_s=0.05$. For the superharmonic secondary excitation, the iterations $\phi _m$ are $2{\rm \pi}$ periodic (refer to (4.6)). First, we start from $f_p=0.88$ and increase $f_p$ across the lock-in boundary. figure 16(a) shows the two frequencies considered across the 1:1 lock-in boundary (green and pink cross symbols). Initially, at $f_p=0.88$, cobwebs densely fill a region (as observed in § 5.1.1) of the return map indicating quasi-periodic nature of vortex shedding (b). Subsequently, when $f_p$ is increased to $f_p=0.92$ (c), the map intersects $\phi _{m-1}=\phi _{m}$ (dashed line) at two locations, indicated as solid and hollow green circles. They represent the stable and unstable fixed points of the map. The cobwebs converge to the solid green circle. In (b), there are no fixed points, while in (c) two fixed points (stable and unstable) occur. Therefore, 1:1 lock-in occurs through a saddle-node bifurcation.

Figure 16. Occurrence of saddle-node bifurcation through the appearance of a stable and unstable fixed point in the $A_p>A_s$ region. Panel (a) shows the lock-in boundary for $f_s=2f_p$ and $A_s=0.05$. The chosen parameters in (b,c) are represented using green and pink cross symbols. First return maps for (b) $f_p=0.88$ and (c) $f_p=0.92$ with cobweb diagrams show the presence of saddle-node bifurcation.

A similar behaviour is observed during subharmonic secondary excitation $f_s=f_p/2$ (not shown). Since $q_f=2$, the iterations are of period-2 and second return maps in the $\phi _m - \phi _{m-2}$ plane are used to identify saddle-node bifurcation.

Throughout, we have dealt with two-frequency excitation using the phase of the excitation signal. Using the geometric interpretation of phase $\phi _m$, we observed various dynamics of return map and cobweb diagrams across the phase lock-in boundary. We observed plateaus at (i) $A_p=0$ during superharmonic secondary excitation (the extent of the plateau is comparably small during subharmonic secondary excitation) and (ii) $A_p=A_s$ during both super- and subharmonic secondary excitation (refer to figure 8). In the next section, we obtain analytical relations for lock-in boundaries.