4.1.1. Network I: intermittent frequency locking and a breathing chimera

Figure 2 shows the collective dynamics of network I, whose oscillators are identical in equivalence ratio ($\phi _{1,2,3,4} = 0.61$) and cross-talk position ($\xi _{1,2,3,4} = 1600$ mm). We first consider the early stages of the experiment, $0 \leq t \leq 2$ s. In this interval (figure 2(a1), where the time span $0.98 \leq t \leq 1.02$ s is magnified), the time traces of $p^{\prime }$ for all four oscillators show temporally synchronized peaks (red bands) and troughs (blue bands), indicating that the entire network is in a state of global in-phase synchronization (Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2003; Balanov et al. Reference Balanov, Janson, Postnov and Sosnovtseva2008). This synchronicity extends to the HRR fluctuations as well (figure 2(a2), $0.98 \leq t \leq 1.02$ s): strong temporal alignment can be observed between the peaks of $p^{\prime }$ and $q^{\prime }$ (red and yellow bands) and between their troughs (blue and green bands). The fact that $p^{\prime }$ and $q^{\prime }$ are evolving in phase (i.e. with a phase difference of less than ${\rm \pi} /2$) implies a positive Rayleigh (Reference Rayleigh1945) integral ($\int p^{\prime } q^{\prime }\,\mathrm {d}t > 0$), providing a physical mechanism by which energy is transferred from the flames to the acoustic modes to sustain the observed self-excited thermoacoustic oscillations (Lieuwen & Yang Reference Lieuwen and Yang2005; Nicoud & Poinsot Reference Nicoud and Poinsot2005; Magri, Juniper & Moeck Reference Magri, Juniper and Moeck2020).

Figure 2. Collective dynamics of network I, which exhibits a transition from intermittent frequency locking on a $\mathbb {T}^{3}$ quasiperiodic attractor to a breathing chimera. Shown at the top are time traces of (a1) $p^{\prime }$ and (a2) $p^{\prime }$ and $q^{\prime }$ for each of the four oscillators in the network (C1, C2, C3 and C4); both $p^{\prime }$ and $q^{\prime }$ have been normalized by their respective maximum values from the entire network. Also shown are the spectrograms and PSDs of (b1–4) $p^{\prime }$ and (c1–4) $q^{\prime }$, with the PSDs computed via the algorithm of Welch (Reference Welch1967). Panels (b1, c1), (b2, c2), (b3, c3) and (b4, c4) correspond to oscillators C1, C2, C3 and C4, respectively. The figure also shows (d1, e1) the temporal variation of $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ and $\Delta \psi _{p^{\prime }_{\mathbb {X}}q^{\prime }_{\mathbb {X}}}$, alongside (d2, e2) their probability distributions, $\zeta _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ and $\zeta _{p^{\prime }_{\mathbb {X}}q^{\prime }_{\mathbb {X}}}$, where the values of $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ and $\Delta \psi _{p^{\prime }_{\mathbb {X}}q^{\prime }_{\mathbb {X}}}$ in (d2, e2) are wrapped around the interval $[-{\rm \pi}, {\rm \pi}]$. In the inset of (e1), each curve has been shifted by even integer multiples of ${\rm \pi}$ for clearer visualization. The ( f1) time trace, ( f2) spectrogram and PSD of the Kuramoto order parameter $R_K$ are shown in order to evaluate the phase coherence of the network. In (d1–2, e1–2), the dark- and light-grey regions denote in-phase and anti-phase dynamics, respectively. In (d1, f1–2), the yellow regions denote epochs of global in-phase synchronization.

Although $p^{\prime }$ and $q^{\prime }$ evolve in phase, their dynamics are not simply periodic: the spectrogram and power spectral density (PSD) of both $p^{\prime }$ (figures 2b(1–4)) and $q^{\prime }$ (figures 2c(1–4)) reveal the coexistence of three discrete modes, whose frequencies are incommensurable ($f_1 = 167$, $f_2 = 186$, $f_3 = 201$ Hz). This implies the presence of quasiperiodicity on an ergodic three-dimensional torus attractor (3-torus), otherwise known as $\mathbb {T}^{3}$ quasiperiodicity (Hilborn Reference Hilborn2000). Similar $\mathbb {T}^{3}$ quasiperiodic states have been observed by Guan et al. (Reference Guan, Gupta, Wan and Li2019a,Reference Guan, He, Murugesan, Li, Liu and Lib) in a laminar thermoacoustic system consisting of a single Bunsen-flame combustor subjected to external periodic forcing. Other systems in which $\mathbb {T}^{3}$ quasiperiodicity has been observed include Rayleigh–Bénard convection cells (Gollub & Benson Reference Gollub and Benson1980), barium–sodium niobate crystals (Martin, Leber & Martienssen Reference Martin, Leber and Martienssen1984) and electrical circuits (Borkowski et al. Reference Borkowski, Perlikowski, Kapitaniak and Stefanski2015). In nonlinear dynamical systems, quasiperiodic states on $\mathbb {T}^{n}$ with $n \geq 3$ are known to be unstable to arbitrarily small disturbances (Newhouse, Ruelle & Takens Reference Newhouse, Ruelle and Takens1978). As a result, they tend to collapse into disorganized states, such as chaos, via a sequence of folding and stretching operations (Hilborn Reference Hilborn2000). In our system, we find that the $\mathbb {T}^{3}$ quasiperiodic state is indeed unstable, existing only transiently between asynchronous epochs, before eventually giving way to a breathing chimera in the late stages of the experiment ($t \geq 2$ s), as will be discussed below.

To verify that all four oscillators in the network are in the same $\mathbb {T}^{3}$ quasiperiodic state, we use the Hilbert transform to compute the instantaneous phase difference, $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$, between the pressure signals in each pair of directly/indirectly coupled oscillators ($\mathbb {X}$ and $\mathbb {Y}$). Focusing still on the early stages ($0 \leq t \leq 2$ s), we find that all six oscillator-pair combinations exhibit long epochs in which $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ oscillates within $\pm {\rm \pi}/2$ of even integer multiples of ${\rm \pi}$, indicating in-phase synchronization (figure 2(d1), yellow regions and, in the left inset, dark-grey regions). During these long synchronous epochs, the time-averaged slope of $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$, or $\langle \dot {\Delta \psi }_{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}\rangle$, is zero for all combinations, indicating that all four oscillators are evolving at the same time-averaged frequency, a phenomenon known as frequency locking (Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2003). The fact that the instantaneous values of $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ do not remain steady in time, however, implies the absence of phase locking (Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2003). The presence of frequency locking without phase locking is a characteristic state in synchronization theory, known as phase trapping. Phase trapping is not unique to thermoacoustic systems (Balusamy et al. Reference Balusamy, Li, Han, Juniper and Hochgreb2015; Kashinath, Li & Juniper Reference Kashinath, Li and Juniper2018; Guan et al. Reference Guan, Gupta, Wan and Li2019a) or even to fluid mechanical systems (Li & Juniper Reference Li and Juniper2013b,Reference Li and Junipera), but can be found in various physical systems, such as solid-state lasers (Thévenin et al. Reference Thévenin, Romanelli, Vallet, Brunel and Erneux2011). Its detection here in a ring-coupled thermoacoustic system thus provides further evidence of its universality (Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2003; Balanov et al. Reference Balanov, Janson, Postnov and Sosnovtseva2008).

Interspersed between those long epochs of in-phase frequency locking (figure 2(d1), yellow regions) are short epochs of asynchrony in which $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ jumps by even integer multiples of ${\rm \pi}$. These jumps, known as phase slips, are a classic feature of forced or coupled self-excited systems exposed to strong or unbounded noise (Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2003). In effect, such noise drives the system from one stable equilibrium point (potential well) to an adjacent one. In our thermoacoustic system, the source of such noise is thought to be turbulence in the underlying reactive flow field. Put together, these observations indicate that during $0 \leq t \leq 2$ s, the system switches intermittently between two distinct regimes: (i) in-phase frequency locking on a $\mathbb {T}^{3}$ quasiperiodic attractor; and (ii) desynchronization due to noise-induced phase slipping. In this study, we refer to this alternating state as intermittent frequency locking. Similar states have been observed previously in other forced or coupled self-excited systems, such as the human brain exposed to external visual stimuli (Pisarchik, Chholak & Hramov Reference Pisarchik, Chholak and Hramov2019). However, it is important to recognize that the intermittent frequency locking observed here differs from the intermittent phase synchronization observed by Pawar et al. (Reference Pawar, Seshadri, Unni and Sujith2017): the former involves switching between frequency-locked $\mathbb {T}^{3}$ quasiperiodicity and desynchronization in a small network of four ring-coupled combustors represented by their pressure signals, whereas the latter involves switching between phase-locked periodicity and desynchronization in a single isolated combustor represented by its pressure and HRR signals. Crucially, in our ring-coupled network, all four oscillators are either globally synchronized or globally desynchronized at any given instant (within $0 \leq t \leq 2$ s). In other words, the inter-combustor $p^{\prime }_{\mathbb {X}}$–$p^{\prime }_{\mathbb {Y}}$ interactions of the entire network enter the frequency-locked epochs at the same time, and then they switch to the asynchronous epochs also at the same time (figure 2(d1)). Later, however, we will see that such simultaneous global switching does not continue indefinitely, with the network eventually transitioning to a breathing chimera.

Next we examine the probability distribution of $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$, denoted as $\zeta _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$, where $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ is wrapped around the interval $[-{\rm \pi}, {\rm \pi}]$ (figure 2(d2)). In a continuously desynchronized system (i.e. one without any synchronous epochs), $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ would drift unboundedly in time, causing $\zeta _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ to be distributed uniformly across all possible values of $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$. By contrast, in a continuously synchronized system (i.e. one without any asynchronous epochs), $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ would be locked to certain values, causing $\zeta _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ to exhibit dominant peaks. In the special case of intermittent frequency locking, the tail properties of those $\zeta _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ peaks would depend on the amplitude of phase trapping and on the frequency and duration of phase slipping. In figure 2(d2) (top sub-panel, $0 \leq t \leq 2$ s), we find that nearly all the $p^{\prime }_{\mathbb {X}}$–$p^{\prime }_{\mathbb {Y}}$ interactions lead to a unimodal $\zeta _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ distribution with a peak at $-{\rm \pi} /2 < \Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}} < {\rm \pi}/2$. This is consistent with the observed intermittent switching between long epochs of in-phase frequency locking and short epochs of desynchronization associated with phase slipping (figure 2(d1)). However, one specific pair of indirectly coupled oscillators (C1–C3) exhibits a bimodal $\zeta _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ distribution centred on $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}} \approx 0$. This is caused by the square-like waveform of $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ for C1–C3 (figure 2(d1), see left inset), which shows that C1 leads C3 roughly half of the time, and vice versa the other half. Meanwhile, the other pair of indirectly coupled oscillators (C2–C4) exhibits a unimodal $\zeta _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ distribution rather than a bimodal one. This indicates an asymmetry in the evolution of $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$, despite the overall system being composed of identical oscillators. We attribute this asymmetry to subtle differences in the operating conditions (e.g. the temperature, velocity and $\phi$ of the reactants) and in the dimensions of the combustors and cross-talk tubes. Indeed, Moon et al. (Reference Moon, Jegal, Gu and Kim2019) have shown that two nominally identical uncoupled combustors operated at nominally identical conditions can exhibit different limit-cycle amplitudes in their pressure oscillations. Moreover, Ghirardo et al. (Reference Ghirardo, Di Giovine, Moeck and Bothien2019) have shown that even slight asymmetries in can-annular combustors can have a noticeable effect on their modal dynamics. Nevertheless, for all six oscillator-pair combinations, every peak in $\zeta _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ resides well within the in-phase limits of $-{\rm \pi} /2 < \Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}} < {\rm \pi}/2$ (figure 2(d2), top sub-panel, dark-grey region), indicating that the intermittent epochs of phase slipping are neither frequent nor long enough to prevent the four oscillators from evolving in phase with one another on a time-averaged basis.

Moving on to the intra-combustor flame–acoustic interactions, we again use the Hilbert transform to compute the instantaneous phase difference, $\Delta \psi _{p^{\prime }_{\mathbb {X}}q^{\prime }_{\mathbb {X}}}$, but this time between the pressure and HRR signals in each individual oscillator ($\mathbb {X}$). We find that $\Delta \psi _{p^{\prime }_{\mathbb {X}}q^{\prime }_{\mathbb {X}}}$ evolves differently across all four oscillators (figure 2(e1), $0 \leq t \leq 2$ s), despite them being nominally identical in geometry and operating conditions. The cause of this asymmetry is believed to be related to that observed in $\zeta _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ between C1–C3 and C2–C4. All four oscillators exhibit phase slipping between $p^{\prime }_{\mathbb {X}}$ and $q^{\prime }_{\mathbb {X}}$ but to different degrees, with C3 showing the largest and most frequent phase slips. This may be attributed to the relatively strong low-frequency HRR components in C3 (figure 2(c3), see inset). These components are at linear combinations of $f_1$, $f_2$ and $f_3$, implying that they may be due to beating, possibly promoted by the slow temporal variations observed in the mode amplitudes of the HRR signal. These low-frequency components are more prominent in $q^{\prime }_{\mathbb {X}}$ than in $p^{\prime }_{\mathbb {X}}$ (figure 2(c3) versus figure 2(b3)), which would suggest that they are caused by vortical interactions rather than by acoustic interactions. Similar low-frequency components in the HRR signal, arising from an intrinsic hydrodynamic mode in the flame, have been identified by Guan et al. (Reference Guan, Li, Ahn and Kim2019c) as a potential cause of $p^{\prime }_{\mathbb {X}}$–$q^{\prime }_{\mathbb {X}}$ decoupling in a liquid-fuelled combustor with a turbulent diffusion flame. The fact that the oscillator with the lowest degree of phase slipping between $p^{\prime }_{\mathbb {X}}$ and $q^{\prime }_{\mathbb {X}}$ (C1) exhibits the weakest low-frequency HRR components lends support to this hypothesis.

Between the epochs of phase slipping, phase locking occurs, as evidenced by $\Delta \psi _{p^{\prime }_{\mathbb {X}}q^{\prime }_{\mathbb {X}}}$ remaining relatively constant in time (figure 2(e1), see inset, where the data have been shifted by even integer multiples of ${\rm \pi}$ for clearer visualization). This indicates the presence of intermittent phase locking between $p^{\prime }_{\mathbb {X}}$ and $q^{\prime }_{\mathbb {X}}$ in each oscillator. The absence of continuous phase locking between $p^{\prime }_{\mathbb {X}}$ and $q^{\prime }_{\mathbb {X}}$ is not a violation of the Rayleigh criterion. This is because although the instantaneous values of $\Delta \psi _{p^{\prime }_{\mathbb {X}}q^{\prime }_{\mathbb {X}}}$ do not always remain at a fixed value, the probability distribution of $\Delta \psi _{p^{\prime }_{\mathbb {X}}q^{\prime }_{\mathbb {X}}}$, denoted as $\zeta _{p^{\prime }_{\mathbb {X}}q^{\prime }_{\mathbb {X}}}$, still shows a statistical preference for in-phase $p^{\prime }_{\mathbb {X}}$–$q^{\prime }_{\mathbb {X}}$ dynamics ($-{\rm \pi} /2 < \Delta \psi _{p^{\prime }_{\mathbb {X}}q^{\prime }_{\mathbb {X}}} < {\rm \pi}/2$) in each oscillator (figure 2(e2), top sub-panel). Finally, it is worth recalling that the evolution of $\Delta \psi _{p^{\prime }_{\mathbb {X}}q^{\prime }_{\mathbb {X}}}$ differs among the four oscillators, with phase slipping in some oscillators often coinciding with phase locking in other oscillators (figure 2(e1)). This stands in stark contrast to the evolution of $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ (figure 2(d1)), demonstrating that global synchronization of $p^{\prime }_{\mathbb {X}}$–$p^{\prime }_{\mathbb {Y}}$ across the entire network does not necessarily imply simultaneous phase locking of $p^{\prime }_{\mathbb {X}}$–$q^{\prime }_{\mathbb {X}}$ in each oscillator.

We now consider the late stages of the experiment, $2 \leq t \leq 4$ s. We find that a key difference relative to the early stages ($0 \leq t \leq 2$ s) is that synchronization of $p^{\prime }_{\mathbb {X}}$–$p^{\prime }_{\mathbb {Y}}$ no longer occurs simultaneously in every pair of oscillators. Although intermittent frequency locking continues to occur across the entire network, its degree and timing vary locally. For example, in a sample time window $2.3 \leq t \leq 2.34$ s (figure 2(d1), green region), three oscillator-pair combinations (C1–C2, C2–C3, C1–C3) exhibit frequency locking, as evidenced by their $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ oscillating boundedly with $\langle \dot {\Delta \psi }_{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}\rangle = 0$. In the same time window, however, the other three combinations (C3–C4, C4–C1, C2–C4) exhibit desynchronization, as evidenced by their $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ slipping in time. Crucially, in the epochs of frequency locking, $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ shows none of the statistical preference for even integer multiples of ${\rm \pi}$ (in-phase synchronization) seen in the early stages. Taking C2–C3 as an example, we find that some of its frequency-locked epochs show temporal switching between anti-phase and in-phase synchronization (figure 2(d1), top middle inset), some epochs show only in-phase synchronization (figure 2(d1), bottom right inset), and other epochs show only anti-phase synchronization (figure 2(d1), top right inset). Such switching between anti-phase and in-phase synchronization can be seen directly in the pressure traces (figure 2(a1)), where the peaks of C2 are initially aligned with the troughs of C3 ($t \approx 2.3$ s) but then become aligned with the peaks of C3 shortly afterwards ($t \approx 2.34$ s). As expected, this mix of in-phase and anti-phase synchronization causes $\zeta _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ to be uniformly distributed with no clear peak (figure 2(d2), bottom sub-panel, $2 \leq t \leq 4$ s). From these observations, we can conclude that the system has transitioned from a state in which intermittent frequency locking occurs simultaneously across the entire network ($0 \leq t \leq 2$ s) to one in which intermittent frequency locking occurs at different times and to different degrees in different parts of the network ($2 \leq t \leq 4$ s). As noted in § 1.1, such a hybrid state containing both synchronous and asynchronous spatial domains is called a chimera (Abrams & Strogatz Reference Abrams and Strogatz2004).

To identify the specific type of chimera present in this system, we compute $R_K$ using the instantaneous phase of $p^{\prime }$ in all four oscillators (Kuramoto Reference Kuramoto2003). On examining the time trace, spectrogram and PSD of $R_K$ (figures 2( f1–2)), we find two different types of behaviour. Early on ($0 \leq t \leq 2$ s), we observe long epochs in which $R_K$ is relatively steady and close to 1 (figures 2( f1–2), yellow regions), indicating that the phasors of all four oscillators are well aligned with one another, consistent with frequency locking occurring in the entire network. Interspersed between those long epochs of $R_K \approx 1$ are short epochs in which $R_K$ fluctuates between 0.5 and 1; these $R_K$ fluctuations arise from rotating phasors associated with phase slipping. Later on ($2 \leq t \leq 4$ s), we find that $R_K$ oscillates continuously with a large amplitude, indicating that the system has transitioned to a non-stationary state in which the positions of the coherent (synchronous) and incoherent (asynchronous) spatial domains vary in time. Such a state is known as a breathing chimera and was first discovered by Abrams et al. (Reference Abrams, Mirollo, Strogatz and Wiley2008) in a network of phase oscillators. An examination of the $R_K$ spectra shows that the breathing frequencies are $f_3-f_1$, $f_3-f_2$ and $f_2-f_1$ (figure 2( f2)), which are linear combinations of the three incommensurable modes identified earlier in the $\mathbb {T}^{3}$ quasiperiodic attractor. It is worth mentioning that a breathing chimera was observed previously by Mondal et al. (Reference Mondal, Unni and Sujith2017) in a turbulent combustor during its transition to intermittency. In that study, however, the instantaneous phase was extracted from the HRR field in a single combustor, without any inter-combustor coupling. In the present study, we show that a breathing chimera can also emerge in a small network of four ring-coupled thermoacoustic oscillators, strengthening the universality of this state.

As for the intra-combustor flame–acoustic interactions in the late stages ($2 \leq t \leq 4$ s), we find that the trends identified in the early stages ($0 \leq t \leq 2$ s) are still present. In particular, intermittent phase locking between $p^{\prime }_{\mathbb {X}}$ and $q^{\prime }_{\mathbb {X}}$ continues to occur to varying degrees in all four oscillators (figure 2(e1)): C1 exhibits almost continuous phase locking, but C2, C3 and C4 exhibit increasingly pronounced epochs of phase slipping amidst intermittent phase locking. Nevertheless, as in the early stages, all four oscillators spend considerable time with their $\Delta \psi _{p^{\prime }_{\mathbb {X}}q^{\prime }_{\mathbb {X}}}$ at even integer multiples of ${\rm \pi}$, causing $\zeta _{p^{\prime }_{\mathbb {X}}q^{\prime }_{\mathbb {X}}}$ to exhibit a dominant peak within the in-phase limits of $-{\rm \pi} /2 < \Delta \psi _{p^{\prime }_{\mathbb {X}}q^{\prime }_{\mathbb {X}}} < {\rm \pi}/2$ (figure 2(e2), bottom sub-panel, $2 \leq t \leq 4$ s). This implies that across the entire network, $p^{\prime }_{\mathbb {X}}$ and $q^{\prime }_{\mathbb {X}}$ are in phase on a time-averaged basis, even though they are not always so on an instantaneous basis. The in-phase relationship between $p^{\prime }_{\mathbb {X}}$ and $q^{\prime }_{\mathbb {X}}$ in each oscillator (figure 2(e2), bottom sub-panel) stands in stark contrast to the absence of any statistically preferred phase relationship between $p^{\prime }_{\mathbb {X}}$ and $p^{\prime }_{\mathbb {Y}}$ in the entire network (figure 2(d2), bottom sub-panel). This contrasting behaviour provides further evidence that the phase dynamics of the intra-combustor flame–acoustic interactions does not necessarily dictate that of the inter-combustor acoustic–acoustic interactions.

In summary, we have shown that under certain conditions, a small network of four ring-coupled thermoacoustic oscillators can transition from (i) intermittent frequency locking on a $\mathbb {T}^{3}$ quasiperiodic attractor to (ii) a breathing chimera in which the positions of the synchronous and asynchronous spatial domains vary in time. Furthermore, we have provided evidence showing that the phase dynamics of the $p^{\prime }_{\mathbb {X}}$–$p^{\prime }_{\mathbb {Y}}$ interactions between coupled oscillators does not necessarily match the phase dynamics of the $p^{\prime }_{\mathbb {X}}$–$q^{\prime }_{\mathbb {X}}$ interactions within those oscillators.

4.1.2. Network II: a two-cluster state of anti-phase synchronization

Figure 3 shows the collective dynamics of network II, whose oscillators are identical in equivalence ratio ($\phi _{1,2,3,4} = 0.65$) and cross-talk position ($\xi _{1,2,3,4} = 1600$ mm). In terms of operating conditions, network II differs from network I (§ 4.1.1) in that it is at a slightly higher equivalence ratio (0.65 versus 0.61). The layout of figure 3 (network II) is identical to that of figure 2 (network I). On examining the time traces of $p^{\prime }$ (figure 3(a1)), we find that unlike network I, network II exhibits no dynamical transitions during the entire sampling interval. Instead, it remains in a stationary state where anti-phase synchronization occurs between any two adjacent oscillators, i.e. between any two directly coupled oscillators (C1–C2, C2–C3, C3–C4, C4–C1). In the literature on can-annular combustors, this is known as a push–pull mode, which Ghirardo, Moeck & Bothien (Reference Ghirardo, Moeck and Bothien2020) and von Saldern et al. (Reference von Saldern, Orchini and Moeck2021b), among others, have recently studied via low-order modelling. Owing to the ring-coupled architecture of our system, the fact that anti-phase synchronization occurs between any two directly coupled oscillators (C1–C2, C2–C3, C3–C4, C4–C1) implies that in-phase synchronization occurs between any two indirectly coupled oscillators (C1–C3, C2–C4). The result is a globally synchronous state in which all four oscillators evolve at the same limit-cycle frequency ($f_1 = 210$ Hz, as per the PSDs of both $p^{\prime }$ and $q^{\prime }$; figures 3(b1–4) and 3(c1–4)) but are split into two clusters based on their instantaneous phases: cluster {C1,C3} is in anti-phase synchronization with cluster {C2, C4}. Similar states of clustering have been observed previously in networks of Stuart–Landau oscillators (Manrubia, Mikhailov & Zanette Reference Manrubia, Mikhailov and Zanette2004; Premalatha et al. Reference Premalatha, Chandrasekar, Senthilvelan and Lakshmanan2018), chemical oscillators (Wang, Kiss & Hudson Reference Wang, Kiss and Hudson2000; Kiss, Zhai & Hudson Reference Kiss, Zhai and Hudson2005) and candle-flame oscillators (Manoj et al. Reference Manoj, Pawar, Dange, Mondal, Sujith, Surovyatkina and Kurths2019; Manoj, Pawar & Sujith Reference Manoj, Pawar and Sujith2021). The phase difference $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ for each of the six oscillator-pair combinations remains largely constant in time (figure 3(d1)), indicating that the pressure oscillations in the entire network are not only frequency locked but also phase locked (Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2003), with no sign of the phase trapping or slipping seen in network I (figure 2(d1)). As noted earlier, the directly coupled oscillators (C1–C2, C2–C3, C3–C4, C4–C1) are anti-phase synchronized, implying that their $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ is locked to odd integer multiples of ${\rm \pi}$ (figure 3(d1), light-grey regions). By contrast, the indirectly coupled oscillators (C1–C3, C2–C4) are in-phase synchronized, implying that their $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ is locked to even integer multiples of ${\rm \pi}$ (figure 3(d1), dark-grey regions). This mix of anti-phase and in-phase synchronization leads to a clear separation of peaks in the $\zeta _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ distribution (figure 3(d2)), which is a characteristic feature of clustering based on the instantaneous phase (Manrubia et al. Reference Manrubia, Mikhailov and Zanette2004).

Figure 3. The same as in figure 2 but for network II, which exhibits a two-cluster state of anti-phase synchronization on a periodic limit cycle.

In contrast to the mix of anti-phase and in-phase dynamics observed in the inter-combustor $p^{\prime }_{\mathbb {X}}$–$p^{\prime }_{\mathbb {Y}}$ interactions, the intra-combustor $p^{\prime }_{\mathbb {X}}$–$q^{\prime }_{\mathbb {X}}$ interactions show only in-phase dynamics. This is apparent in the time traces of $p^{\prime }_{\mathbb {X}}$ and $q^{\prime }_{\mathbb {X}}$ (figure 3(a2)) and in the evolution of $\Delta \psi _{p^{\prime }_{\mathbb {X}}q^{\prime }_{\mathbb {X}}}$ (figures 3(e1–2)). Although all the $p^{\prime }_{\mathbb {X}}$–$p^{\prime }_{\mathbb {Y}}$ interactions are continuously phase locked (figure 3(d1)), the $p^{\prime }_{\mathbb {X}}$–$q^{\prime }_{\mathbb {X}}$ interactions are continuously phase locked for only C1, and are intermittently phase locked otherwise (figure 3(e1)). This behaviour is similar to that observed in network I (figure 2(e1)), demonstrating that continuous phase locking of $p^{\prime }_{\mathbb {X}}$–$p^{\prime }_{\mathbb {Y}}$ across the entire network does not necessarily imply continuous phase locking of $p^{\prime }_{\mathbb {X}}$–$q^{\prime }_{\mathbb {X}}$ in each oscillator. Nevertheless, an examination of $\zeta _{p^{\prime }_{\mathbb {X}}q^{\prime }_{\mathbb {X}}}$ shows that in all four oscillators, $p^{\prime }_{\mathbb {X}}$ and $q^{\prime }_{\mathbb {X}}$ remain in phase on a time-averaged basis, with C1 showing the tallest $\zeta _{p^{\prime }_{\mathbb {X}}q^{\prime }_{\mathbb {X}}}$ peak (figure 3(e2)), in line with its $\Delta \psi _{p^{\prime }_{\mathbb {X}}q^{\prime }_{\mathbb {X}}}$ being continuously phase locked (figure 3(e1)).

Figure 3( f1) shows that $R_K$ remains close to zero for the entire sampling interval, which would typically suggest a disordered state of randomly aligned phasors. Here, however, the low values of $R_K$ are caused not by randomly aligned phasors, but by a systematic cancellation of phasors between the two halves of the network: the phasors of cluster {C1,C3} cancel those of cluster {C2,C4}. A similar cancellation of phasors, arising also from anti-phase synchronized clustering, has been reported previously in a large population of strongly coupled relaxation oscillators (Călugăru et al. Reference Călugăru, Totz, Martens and Engel2020). The spectrum of $R_K$ is weak and flat, with no noticeable peaks (figure 3(f2)). Recent analysis of a network of coupled Stuart–Landau oscillators by Joseph & Pakrashi (Reference Joseph and Pakrashi2020) has shown that the conditions favourable to anti-phase synchronization include a small number of oscillators (typically less than 20), low connectivity, weak coupling over distance, and strong symmetry in the network topology. Our thermoacoustic system satisfies all of these conditions: it has only four oscillators (four combustor nodes), each oscillator is directly coupled to only its two adjacent neighbours (i.e. nearest neighbour coupling), the coupling is achieved with a single annular cross-talk section, and the network topology is a symmetric ring. The discovery of anti-phase synchronization in such a network of four ring-coupled thermoacoustic oscillators is thus consistent with the low-order network analysis of Joseph & Pakrashi (Reference Joseph and Pakrashi2020).

4.1.3. Network III: a weak anti-phase chimera

Figure 4 shows the collective dynamics of network III, whose oscillators are identical in cross-talk position ($\xi _{1,2,3,4} = 1000$ mm) but not in equivalence ratio ($\phi _{1,3} = 0.61$, $\phi _{2,4} = 0.57$). On examining the time traces of $p^{\prime }$ (figure 4(a1)), we find that the network is stationary but divided into two halves: each of the two pairs of indirectly coupled identical oscillators (C1–C3, C2–C4) is in anti-phase synchronization, producing push–pull modes (Moon et al. Reference Moon, Yoon and Kim2021), but the two pairs are not synchronized with each other. In other words, half of the network (C1–C3) evolves at one frequency ($f_1 = 262$ Hz), while the other half (C2–C4) evolves at a different frequency ($f_2 = 77$ Hz), as can be seen in the PSDs of both $p^{\prime }$ (figures 4(b1–4)) and $q^{\prime }$ (figures 4(c1–4)). In recent experiments, Moon et al. (Reference Moon, Jegal, Yoon and Kim2020b) observed a similar frequency-based partitioning of the spatial domain and attributed it to mode localization induced by a loss of rotational symmetry in the network.

Figure 4. The same as in figure 2 but for network III, which exhibits a weak anti-phase chimera.

Examining the inter-combustor $p^{\prime }_{\mathbb {X}}$–$p^{\prime }_{\mathbb {Y}}$ interactions, we find that $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ for both pairs of indirectly coupled identical oscillators (C1–C3, C2–C4) is approximately constant in time, with values hovering near odd integer multiples of ${\rm \pi}$, interrupted by occasional phase slips (figure 4(d1)). In the $\zeta _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ curve (figure 4(d2)), this gives rise to peaks at $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}} = \pm {\rm \pi}$, indicating that both pairs of indirectly coupled identical oscillators (C1–C3, C2–C4) are undergoing intermittent phase locking in an anti-phase manner (Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2003). By contrast, in all four pairs of directly coupled non-identical oscillators (C1–C2, C2–C3, C3–C4, C4–C1), $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ drifts unboundedly in time (figure 4(d1)), indicating desynchronization associated with phase drifting. This is caused by the oscillator frequency alternating between $f_1$ and $f_2$ around the network. Thus $\zeta _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ is uniformly distributed across all possible values of $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ (figure 4(d2)). Collectively, these observations indicate that network III is in a stationary state of weak anti-phase chimera. As alluded to in § 1.1, a weak anti-phase chimera is a special type of chimera in which two or more oscillators in a network evolve in anti-phase synchronization, while at least one other oscillator evolves at a frequency different from that of the synchronized ensemble (Ashwin & Burylko Reference Ashwin and Burylko2015). Weak anti-phase chimeras have been predicted theoretically by Maistrenko et al. (Reference Maistrenko, Brezetsky, Jaros, Levchenko and Kapitaniak2017) in a minimal network of three identical pendulum-like nodes. Here, we present experimental evidence showing that a weak anti-phase chimera can emerge in a thermoacoustic system, namely a small network of four non-identical thermoacoustic oscillators coupled in a ring configuration.

Turning now to the intra-combustor $p^{\prime }_{\mathbb {X}}$–$q^{\prime }_{\mathbb {X}}$ interactions, we find that intermittent phase locking occurs in all four oscillators, but to different degrees (figure 4(e1)): the two oscillators at $\phi _{1,3} = 0.61$ (C1 and C3) exhibit only occasional phase slips, whereas the two oscillators at $\phi _{2,4} = 0.57$ (C2 and C4) exhibit frequent phase slips. In the former two oscillators (C1 and C3), the peak in $\zeta _{p^{\prime }_{\mathbb {X}}q^{\prime }_{\mathbb {X}}}$ sits near the boundary between in-phase and anti-phase dynamics (figure 4(e2)), indicating that the Rayleigh criterion is barely satisfied in C1 and C3. By contrast, in the latter two oscillators (C2 and C4), the peak in $\zeta _{p^{\prime }_{\mathbb {X}}q^{\prime }_{\mathbb {X}}}$ sits well within the in-phase limits (figure 4(e2)), indicating that the Rayleigh criterion is satisfied in C2 and C4, with appreciable energy being transferred from the flames to the acoustic field. This increase in energy transfer is believed to be the physical cause of the higher pressure amplitudes observed in C2 and C4 relative to C1 and C3 (figure 4(a1)).

Figure 4( f1) shows that $R_K$ for network III is higher than that for network II (figure 3(f1)), indicating greater phase coherence. Although phasor cancellation occurs in both pairs of anti-phase synchronized oscillators (C1–C3, C2–C4), the process is not perfect owing to phase slips and minor fluctuations in $\Delta \psi _{p^{\prime }_{\mathbb {X}}p^{\prime }_{\mathbb {Y}}}$ (figure 4(d1)). Given that the phasors in C1–C3 rotate at $f_1$ while those in C2–C4 rotate at $f_2$, a new spectral component emerges at their linear combination ($f_1 - f_2$), as can be seen in the spectrogram and PSD of $R_K$ (figure 4(f2)).