We experimentally conduct an early detection of thermoacoustic instability in a staged single-sector combustor using a novel methodology that combines symbolic dynamics and machine learning. We propose two invariants in this study: the determinisms of the joint symbolic recurrence plots $ D J$ and the ordinal transition pattern-based recurrence plots $ D T$. These invariants enable us to capture the phase synchronization between acoustic pressure and heat release rate fluctuations associated with a precursor of thermoacoustic instability. The latent space consisting of $ D J$ and $ D T$, which is obtained by a support vector machine in combination with the $k$-means clustering method, can appropriately determine a transitional regime between stable combustion and thermoacoustic instability.

Thermoacoustic instability is an undesirable self-sustained nonlinear phenomenon that is encountered in the development of practical combustors. Studies to detect a precursor of thermoacoustic instability from the viewpoint of complex systems have recently attracted attention in not only combustion engineering^{1} but also nonlinear physics.^{2–4} We attempt to develop a novel detection methodology that combines symbolic dynamics and machine learning. We treat thermoacoustic instability in a staged single-sector combustor for aircraft engines. We simultaneously measure the acoustic pressure and $ OH \u2217$ chemiluminescence intensity fluctuations in steady-state and transient tests. The fuel flow rate of the main mixer in the combustor is increased to induce thermoacoustic instability. The latent space obtained from the steady-state test consists of two invariants we propose in this study: determinisms of joint symbolic recurrence plots and ordinal transition pattern-based recurrence plots. These recurrence plots are constructed from the acoustic pressure and $ OH \u2217$ chemiluminescence intensity fluctuations. The latent space plane is valid for determining a transitional regime from stable combustion to thermoacoustic instability. We clearly observe the critical slowing down phenomenon in the transitional regime with temporally increasing main fuel flow rate. The latent space successfully enables us to detect a precursor of thermoacoustic instability appearing near the critical slowing down during the transient test. Our study demonstrates the importance of a support vector machine as a supervised machine learning method for the determination of the boundary between the combustion states in a staged single-sector combustor.

## I. INTRODUCTION

The feedback coupling of acoustic pressure, flow velocity, and heat release rate fluctuations in a confined combustor results in the formation of self-excited thermoacoustic instability.^{5–7} The occurrence of thermoacoustic instability, which leads to the fatal damage of a combustor, is a serious problem encountered in the development of aircraft engines with reduced environmental impacts. Therefore, early detection of thermoacoustic instability is one of the long-standing important research topics in the fields of propulsion engineering and nonlinear physics. Many experimental studies^{8–12} have been carried out to examine the applicability of nonlinear invariants obtained by a complex-system approach as potential detectors of thermoacoustic instability in various turbulent combustors.

Since the seminal paper on permutation entropy^{13} was published by Bandt and Pompe, ordinal (permutation) patterns in a time series have attracted much attention in terms of their use in clarifying a rich variety of dynamic behaviors in nonlinear systems.^{14} Many experimental studies^{15–19} have clarified the dynamic behavior of acoustic pressure fluctuations during stable combustion and thermoacoustic instability using the permutation entropy and the permutation spectral test.^{20} Nowadays, symbolic recurrence plots (SRPs)^{21,22} are used to qualitatively evaluate the similarity between two time series and have been introduced to clarify the synchronized state of acoustic pressure and heat release rate fluctuations during a transitional state to thermoacoustic instability for a laboratory-scale swirl-stabilized turbulent combustor.^{23,24} The symbolic dynamics-based synchronization index,^{23,24} which is the product of a determinism^{21} of SRPs and the phase synchronization parameter,^{25} has been used to find the region driving thermoacoustic instability.

Machine learning, one of the artificial intelligence technologies, has been successfully applied in the fields of combustion engineering and nonlinear sciences^{24,26–31} and has the potential to make significant progress in the development of new combustion control technologies. Kobayashi *et al.*^{26} and Hachijo *et al.*^{27} have attempted an early detection and prevention of thermoacoustic instability in a swirl-stabilized turbulent combustor by combining a support vector machine (SVM) with a complexity–entropy causality plane^{32,33} and an ordinal partition transition network (OPTN).^{34,35} Waxenegger-Wilfing *et al.*^{28} have adopted an SVM^{36} for the early detection of thermoacoustic instability in a rocket thrust chamber by training the features obtained by recurrence quantification analysis.^{21} Methodologies using a SVM have a promising potential to predict the onset of thermoacoustic instability with high generalizability in various combustors.^{26–28}

The Japan Aerospace Exploration Agency (JAXA) has been developing a staged combustor^{37} using both diffusion combustion and lean premixed combustion to achieve low nitrogen oxide (NOx) emissions. Kobayashi *et al.*^{38} have attempted to detect a precursor of thermoacoustic instability using a modified version of the permutation entropy of acoustic pressure fluctuations in a staged combustor. The local Rayleigh index,^{5} which is estimated as a product of acoustic pressure and heat release rate fluctuations, is commonly used to identify the region driving thermoacoustic instability. On this basis, in addition to acoustic pressure fluctuations, heat release rate fluctuations should be taken into account to improve the performance of detecting thermoacoustic instability. However, no studies have been conducted on early detection considering both acoustic pressure and heat release rate fluctuations in a staged combustor under high-temperature and high-pressure conditions.

The main purpose of this study is to develop a novel methodology that combines symbolic dynamics-based recurrence quantification analysis and an SVM to detect a precursor of thermoacoustic instability in a staged single-sector combustor for aircraft engines,^{37} focusing on the mutual coupling between acoustic pressure and heat release rate fluctuations. Small perturbations affect the transition process from stable combustion to thermoacoustic instability. Around the tipping point of the combustion state, the critical slowing down phenomenon, in which the rate of recovery to stable combustion decreases in response to small perturbations, exists and has been observed in various combustors.^{39,40} Gopalakrishnan *et al.*^{39} have clarified that the critical slowing down can be observed before the onset of thermoacoustic instability in a Rijke tube. An *et al.*^{40} have also shown that the critical slowing down exists before the onset of well-developed thermoacoustic instability. In this study, we discuss the relevance of the critical slowing down to the transition process from stable combustion to thermoacoustic instability in a staged single-sector combustor.

## II. EXPERIMENTS

Figure 1 shows a single-sector staged combustor with a lean staged fuel nozzle (LSF). The LSF consists of the main mixer for low-NOx combustion by lean premixed combustion and the pilot mixer for flame stabilization by diffusion combustion. The main mixer is located around the pilot mixer. In the pilot mixer, there are two stages of fuel injection: (i) inward fuel injection and (ii) annular fuel injection from the edge of the pilot mixer. The fuel is atomized by airflows from the inner and outer swirlers of the pilot mixer to reduce NOx emissions. The main mixer uses triple swirlers located coaxially with only the center swirler having the opposite swirling direction. The direction is clockwise for the inner and outer swirlers viewed from downstream and counterclockwise for the center swirler. The two shear layers between the three flows promote the mixing of fuel and air.^{37} Tachibana *et al.*^{41} have numerically examined the flow structures during thermoacoustic instability in a staged single-sector combustor that is identical to our combustor. The swirlers produce four important recirculation flows consisting of two large inner and corner recirculation zones and two small pilot-cup tip and wall recirculation zones. The recirculation zones stabilize the main and pilot flames. The changes in flame shape and position induced by the flow velocity oscillations contribute to global heat release rate fluctuations during thermoacoustic instability. These flow dynamics are significantly associated with the sustainment of thermoacoustic instability in our combustor. Kerosene is used as fuel. We conduct two types of test: a steady-state test in which the fuel flow rate of both the pilot and main mixers is kept constant, and a transient test in which the fuel flow rate of the main mixer is gradually increased while that of the pilot mixer is kept constant. Acoustic pressure and $ OH \u2217$ chemiluminescence intensity fluctuations are measured simultaneously in these tests. Note that $ OH \u2217$ chemiluminescence intensity is an important physical quantity that represents the heat release rate. In the steady-state test, the inlet pressure and temperature in the combustor are set to 700 kPa and 760 K, respectively. The airflow rate $ W a$ and the fuel flow rate of the pilot mixer $ W f , p$ are kept at 364 and 1.2 g/s, respectively. The fuel flow rate of the main mixer $ W f , m$ is increased from 9.0 to 13.8 g/s at intervals of approximately 0.2 g/s. In the transient test, the inlet pressure and temperature are the same as those in the steady-state test. $ W a$ and $ W f , p$ are kept at 436 and 1.2 g/s, respectively. $ W f , m$ is linearly increased from 9.0 to 13.8 g/s over a period of 20 s. Note that the fuel flow rates in these tests are not applicable to cruise conditions and the landing and take-off cycle (LTO) or other flight operation conditions. In both tests, acoustic pressure fluctuations $ p \u2032$ are acquired using two pressure transducers: PT1 (Kulite, XTEH-10LAC-190-35BARA) and PT2 (MEGGITT, CP211). We analyze $ p \u2032$ at PT1 obtained by the semi-infinite tube method^{42} and use PT2 as the reference pressure signal to correct the time delay of $ p \u2032$ at PT1 due to the presence of a semi-infinite tube. $ OH \u2217$ chemiluminescence intensity fluctuations $ q \u2032$ are acquired using a water-cooled optical fiber at PT3. Here, PT3 is located at the plane normal to both PT1 and PT2. The sampling frequency of $ p \u2032$ and $ q \u2032$ measurements is 50 kHz. In this study, we adopt the analytical methods described in Sec. III for $ p \u2032$ at PT1 and $ q \u2032$ at PT3, considering the time delay ( $\u22484.0\xd7 10 \u2212 4$ s) due to the difference between measurement locations for $ p \u2032$ and $ q \u2032$. We determine the time delay by estimating the lag between the local maxima of $ p \u2032$ and $ q \u2032$ during thermoacoustic instability. All the combustion experiments are conducted utilizing the medium-pressure combustion test facility owned by JAXA.

## III. ANALYTICAL METHODS

### A. Symbolic recurrence plots and SVM

Recurrence plots considering ordinal patterns in a time series, which is based on Bandt and Pompe’s idea,^{13} the so-called SRPs,^{21,22} are helpful for examining the similarity of two dynamic behaviors related to synchronization phenomena.^{43} We first embed $ p \u2032$ and $ q \u2032$ into the $d$-dimensional phase space: $ {p} \u2032( t i)={ p \u2032( t i), p \u2032( t i+\tau ),\u2026, p \u2032( t i+\tau (d\u22121))}$ and $ {q} \u2032( t j)={ q \u2032( t j), q \u2032( t j+\tau ),\u2026, q \u2032( t j+\tau (d\u22121))}$. Here, $i,j=1,2,\u2026, N p$, $ N p[=N\u2212d+1]$ is the number of data points in the phase space and $N$ is the number of data points of $ p \u2032$ and $ q \u2032$, $d$ is the embedding dimension, and $\tau $ is the embedding delay time. The elements of $ {p} \u2032$ and $ {q} \u2032$ are transformed into $d!$ rank-order patterns. A plot is displayed on a two-dimensional plane when the rank-order pattern $ \pi d , p \u2032( t i)$ at time $ t i$ equals $ \pi d , q \u2032( t j)$ at time $ t j$. In our preliminary test, we have adopted the permutation spectral test^{20} to determine a suitable $d$. The missing patterns^{44} do not appear when $d$ exceeds 5. On this basis, we set $d$ to 5 in this study. $\tau $ is set to $2.0\xd7 10 \u2212 5$ s, which corresponds to the sampling period. In this study, we construct SRPs at every time window length of 0.05 s. This time length corresponds to $ N p=2496$.

^{21}The other is SRPs incorporating ordinal transition patterns of $ p \u2032$ and $ q \u2032$, namely, ordinal transition pattern-based recurrence plots (TRPs). JSRPs consisting of $ S J(i,j)$ are obtained by taking the product of the elements $ S p \u2032(i,j)$ and $ S q \u2032(i,j)$,

^{34,35}based on the Markov chain, we define the ordinal transition patterns to the next combination of rank-order patterns $ \Pi l$ as $ \Pi k l$, where $k,l=1,2,\u2026,d ! 2$. When the ordinal transition patterns $ \Pi k l( t i)$ at time $ t i$ and $ \Pi k l( t j)$ at time $ t j$ correspond to each other, a plot is displayed on a two-dimensional plane in accordance with Eq. (6),

^{21}we estimate them as the ratio of the diagonal line structures to the total recurrence points as follows:

SVM^{36,45} is one of the supervised machine learning methods for class classification. We apply the SVM to a two-dimensional plane consisting of $ D J$ and $ D T$. We first prepare an $ N s$ data set ${ x i}(i=1,2,\u2026, N s)$ and labels. Here, $ x i=( D J, D T)$ and $ N s=15574$. The $k$-means method^{45} is adopted in this study to cluster ${ x i}$ into three combustion states: stable combustion, transitional state, and thermoacoustic instability. We consider an evaluation function defined as the sum of the squared distances between noncentral data points and the centroid of each cluster. The cluster of each $ x i$ is determined by minimizing the sum of the squared distances. On the basis of the concept of margin maximization, a latent space is obtained by solving an optimization problem for the decision function to determine the classification boundary along the clustered $ D J\u2212 D T$ plane. The Lagrangian function is introduced to transform the problem into a dual problem when solving the optimization problem. Details of the SVM are given in Ref. 27. The obtained classification results are input to the SVM, and a Gaussian kernel is used to determine the nonlinear classification boundaries in the latent space.

### B. Symbolic dynamics-based synchronization index

*et al.*

^{23}and Mori

*et al.*

^{24}have reported the availability of the symbolic dynamics-based synchronization index as one of the measures used to find the region driving thermoacoustic instability in a laboratory-scale swirl-stabilized turbulent combustor. In this study, we adopt the symbolic dynamics-based synchronization index $ S S I$ for the combustion state from stable combustion to well-developed thermoacoustic instability in a staged single-sector combustor. We first estimate the determinism $ D S$ on an arbitrary diagonal line in SRPs obtained using Eq. (1),

*et al.*

^{25}proposed a phase synchronization parameter based on the phase equation known as the Kuramoto model.

^{46}We here estimate the phase synchronization parameter $ r p \u2032 q \u2032$, which is used to evaluate the degree of phase synchronization between $ p \u2032$ and $ q \u2032$,

### C. Critical slowing down

^{47}Nonlinear dynamical systems exhibit a reduced recovery from small perturbations near the tipping point, a phenomenon known as critical slowing down, which is a type of critical transition.

^{48,49}The presence of critical slowing down has also been reported in combustion phenomena.

^{40}An

*et al.*

^{40}have shown the presence of the critical slowing down before the onset of well-developed thermoacoustic instability by estimating the rolling window variance, the autoregressive coefficient, and Kendall’s rank correlation coefficients for acoustic pressure fluctuations. The existence of critical slowing down can be captured from the increases in the variance $\sigma $ and autoregressive coefficient $\alpha $. We estimate Kendall’s rank correlation coefficient $ K u$ to identify increasing trends as follows:

## IV. RESULTS AND DISCUSSION

Figure 2(A) shows the time variations in acoustic pressure fluctuations $ p \u2032$ at various main fuel flow rates $ W f , m$ for the steady-state test. $ p \u2032$ at $ W f , m=9.0$ g/s corresponding to stable combustion exhibits low-amplitude and highly aperiodic fluctuations. The low-amplitude and aperiodic fluctuations of $ p \u2032$ alternate with high-amplitude and periodic oscillations at $ W f , m=11.8$ g/s, indicating the formation of intermittent oscillations. The appearance of the intermittent oscillations is a precursor of thermoacoustic instability,^{9,50,51} in which burst and limit cycle oscillations appear alternately. Strong high-amplitude periodic oscillations are observed at $ W f , m=13.8$ g/s. Time variations in $ OH \u2217$ chemiluminescence intensity fluctuations $ q \u2032$ at different $ W f , m$ are shown in Fig. 2(B). $ q \u2032$ at $ W f , m=9.0$ g/s exhibits low-amplitude and aperiodic oscillations similarly to $ p \u2032$. High-amplitude and noisy-periodic oscillations emerge at $ W f , m=13.8$ g/s. Time variations in $ p \u2032$ and $ q \u2032$ with temporally increasing $ W f , m$ are shown in Fig. 3, together with the power spectral density (PSD) distribution during the transient test. $ p \u2032$ and $ q \u2032$ are gradually amplified when $ W f , m$ starts to increase at $t\u226515.0$ s. They transition into well-developed thermoacoustic instability at $t\u226535.9$ s. We clearly observe two dominant frequencies in PSDs of $ p \u2032$ and $ q \u2032$: $f\u2248750$ Hz corresponding to the longitudinal first-order acoustic resonance mode of the combustor and $f\u22481500$ Hz corresponding to the secondary acoustic resonance mode in thermoacoustic instability.

Figure 4 shows the two-dimensional plane consisting of the determinism of joint symbolic recurrence plots $ D J$ and the determinism of ordinal transition pattern-based recurrence plots $ D T$ for the steady-state test. Here, $ D J$ and $ D T$ are estimated at intervals of 0.05 s from $ p \u2032$ and $ q \u2032$. $( D J, D T)$ changes nonlinearly from the lower left to the upper right region on the plane with increasing $ W f , m$. This indicates that $ D J$ and $ D T$ capture a transition from stable combustion to thermoacoustic instability with increasing $ W f , m$. The latent space obtained by adopting the $k$-means method and SVM for Fig. 4 is shown in Fig. 5(a), where the black data points are obtained from transient test data. Combustion states are classified into (i) stable combustion (blue), (ii) transitional state (yellow) from stable combustion to thermoacoustic instability, and (iii) thermoacoustic instability (red). Kobayashi *et al.*^{26} have proposed the detection index $ R t(= L t/ L a)$ for a swirl-stabilized turbulent combustor, which focuses on the ratio of yellow to blue labels in the transitional state. Here, $ L t$ denotes the number of label data points that are determined to be in the transitional state, and $ L a$ denotes the total number of label data points. In this study, we quantify a detection time of a precursor of thermoacoustic instability using $ R t$. Time variations in $ p \u2032$, $ q \u2032$, and $ R t$ are shown in Fig. 5(b) as a function of $ W f , m$. Stable combustion is formed at $t<24.0$ s. At 24.0 s $\u2264t\u226429.0$ s, it changes to the transitional state prior to thermoacoustic instability. The formation of thermoacoustic instability is observed at $t>29.0$ s. These results demonstrate that the methodology proposed in this study can clearly determine the formation regimes of the three combustion states. $ R t$ begins to rapidly increase when $t$ exceeds approximately 26.80 s. This time is determined as the detection time of a precursor of thermoacoustic instability and nearly corresponds to $ R t=25%$.

We here examine the instantaneous phase differences between $ p \u2032$ and $ q \u2032$, the geometrical structure of SRPs, and the symbolic dynamics-based synchronization index to deepen our understanding of the detection time. The instantaneous phase differences between $ p \u2032$ and $ q \u2032$ are shown in Fig. 6 for stable combustion, transitional state, and thermoacoustic instability. Note that the instantaneous phase differences are displayed as red round plots on the circle, and 2500 points corresponding to 0.05 s are plotted at intervals of 30 points. The blue rectangular plots represent the barycenter of all instantaneous phase differences within 0.05 s, and the magnitude of the vector from the origin to the barycenter corresponds to the phase synchronization parameter $ r p \u2032 q \u2032$. The red round plots for stable combustion are scattered in the range of $0\u2264 \varphi p \u2032 q \u2032\u22642\pi $, and $ r p \u2032 q \u2032$ is approximately 0.1, where $ \varphi p \u2032 q \u2032$ is the phase difference. The permutation entropy values^{13} for $ p \u2032$ and $ q \u2032$ are approximately 0.95 and 0.62, respectively. This indicates the formation of a nearly non-synchronized state during stable combustion. $ \varphi p \u2032 q \u2032$ remains almost unchanged in the transitional state, but we observe a significant increase in $ r p \u2032 q \u2032$. This implies a signature of the intermittent in-phase synchronization state between $ p \u2032$ and $ q \u2032$. $ \varphi p \u2032 q \u2032$ for thermoacoustic instability takes a value from $\u2212\pi /2$ to $\pi /2$ and $ r p \u2032 q \u2032$ increases approximately to 0.7, showing the formation of the in-phase synchronization state. Variations in SRPs for stable combustion, transitional state, and thermoacoustic instability are shown in Fig. 7, together with the expanded regions of SRPs for the same times as those in Fig. 6. The recurrence points corresponding to various ordinal patterns are distributed during stable combustion. The number of points in SRPs during stable combustion is less than those during the transitional state and thermoacoustic instability. This indicates that $ p \u2032$ and $ q \u2032$ during stable combustion do not exhibit the same dynamic behavior. The structures corresponding to ordinal patterns representing monotonically increasing and decreasing trends start to appear during the transition, indicating an intermittent in-phase synchronization state between $ p \u2032$ and $ q \u2032$. Many rectangular structures indicating the formation of limit cycle oscillations are formed at 44.95 s $\u2264t\u2264$ 45.00 s owing to the emergence of the in-phase synchronization state. Marwan *et al.*^{21} have proposed various recurrence plots, such as the cross recurrence plots and joint recurrence plots. These recurrence plots, including the original version of the recurrence plots,^{52} rely on the appropriate radius of the hypersphere in the phase space. The geometric structure of these plots strongly depends on the radius threshold. In contrast, SRPs can reduce the parameter dependence for the construction of recurrence plots and appropriately capture the changes in the combustion state.

Variation in the symbolic recurrence rate $ S R$ on an arbitrary diagonal line with delay time $ \tau a$ ( $= \tau l\Delta t$) is shown in Fig. 8, where 20 s $\u2264t\u226440$ s. $ S R$ is nearly zero during stable combustion. It gradually increases at intervals of $ \tau a\u22481.3$ ms as the combustion state undergoes the transition, showing the onset of the intermittent in-phase synchronization state. These results support the finding that a precursor of thermoacoustic instability captured by $ D J\u2212 D T$ is strongly associated with the emergence of the intermittent phase synchronization state. Time variation in the symbolic dynamics-based synchronization index $ S S I$ is shown in Fig. 9. $ S S I$ is approximately 0.05 on average at $t<$ 26.00 s, indicating a weak interaction between $ p \u2032$ and $ q \u2032$. In contrast, $ S S I$ begins to increase as $t$ exceeds approximately 26.80 s, which reasonably corresponds to the emergence time of an intermittent synchronized state. $ S S I$ takes approximately 0.8 on average at $t>35.95$ s, showing the formation of thermoacoustic instability owing to strong phase synchronization between $ p \u2032$ and $ q \u2032$. As mentioned in Sec. I, the symbolic dynamics-based synchronization index has been proposed to identify the region driving thermoacoustic instability in a swirl-stabilized turbulent combustor.^{23,24} Similarly to this study, recent experimental studies^{23,24} have shown that the acoustic pressure and heat release rate fluctuations are desynchronized (strongly synchronized) during stable combustion (thermoacoustic instability) in a swirl-stabilized turbulent combustor, and the symbolic dynamics-based synchronization index sensitively responds to the changes in combustion state. These results show the importance of symbolic dynamics-based synchronization index as a potential measure to find a precursor and the subsequent onset of thermoacoustic instability in a staged combustor. Note that we should carefully examine in our future work if the synchronization index is a useful detector of a precursor by comparing it with other detectors.^{4}

We finally examine the relevance of the critical slowing down to an early detection of thermoacoustic instability. Figure 10 shows the time variations in the variance $\sigma $, autoregressive coefficient $\alpha $, and Kendall’s rank correlation coefficients $ K \sigma $ and $ K \alpha $. $\sigma $ and $\alpha $ of $ p \u2032$ are estimated in the rolling window, where the rolling window size $ t w=200000[=4 s]$. $ K u$ is estimated by keeping $n$ at 400 ( $=4$ s/0.01 s) and using the number of data points in the previous 4 s. $\sigma $ and $\alpha $ remain constant, whereas $ K \sigma $ and $ K \alpha $ increase and decrease alternately around zero at $t<27$ s. This clearly shows the persistence of stable combustion and the decrease in recovery rate. $\sigma $ and $\alpha $ increase at $t\u226527$ s, and $ K \sigma $ and $ K \alpha $ are approximately unity, indicating an increasing trend. $\sigma $ and $\alpha $ increase rapidly as the recovery rate decreases approaching the tipping point. This suggests that the critical slowing down exists in the transitional regime from stable combustion to thermoacoustic instability. On the basis of the study by An *et al.*,^{40} we conclude that the critical slowing down appears before the onset of thermoacoustic instability in a staged single-sector combustor. In the transient test, the existence of critical slowing down is captured at $t\u226527$ s. Our results clearly show that the critical slowing down starts to appear in proximity to the emergence of the intermittent phase synchronized state. A versatile approach to detecting a precursor of thermoacoustic instability within the framework of complex systems still remains a challenge, although many sophisticated methodologies for various turbulent combustors have been comprehensively summarized in a book^{3} and review articles.^{2,4} An important point to emphasize here is that the appropriate capture of the intermittent phase synchronized state between acoustic pressure and noisy heat release rate fluctuations is significant in an early detection of thermoacoustic instability appearing near the critical slowing down. In this sense, we should examine if our ordinal pattern-based methodologies are applicable to the capture of a possible intermittent phase synchronized state in a multisector combustor.^{53} Shinchi *et al*.^{53} have analyzed only the acoustic pressure fluctuations. Our methodology has a physical significance, in the sense that a precursor of thermoacoustic instability can be detected by phase synchronization between the pressure and heat release rate fluctuations, which is considered to be important for driving thermoacoustic instability.

## V. SUMMARY

We have experimentally conducted an early detection of thermoacoustic instability in a staged single-sector combustor for aero engines using a novel methodology with symbolic dynamics and machine learning. We conduct two types of test: a steady-state test and a transient test. Acoustic pressure and $ OH \u2217$ chemiluminescence intensity fluctuations are measured simultaneously in these tests. The fuel flow rate of the main mixer in the combustor is increased to induce thermoacoustic instability, keeping the airflow rate and the fuel flow rate of the pilot mixer constant. A two-dimensional plane constructed from the joint symbolic recurrence plots $ D J$ and the ordinal transition pattern-based recurrence plots $ D T$, called the $ D J\u2212 D T$ plane during a steady-state test, can capture a transition from stable combustion to thermoacoustic instability. The latent space obtained by an SVM for the two-dimensional plane, after clustering into three combustion states by the $k$-means method, enables early detection of thermoacoustic instability during a transient test in a single-sector combustor. The variance, the autoregressive coefficient, and Kendall’s rank correlation coefficients of acoustic pressure fluctuations in the rolling window clarify the existence of the critical slowing down in the transitional regime. The onset time of the critical slowing down reasonably corresponds to the detection time of thermoacoustic instability extracted from the latent space. Our methodology has potential use for finding a precursor of thermoacoustic instability appearing near the critical slowing down.

## ACKNOWLEDGMENTS

H.G. was partially supported by a Grant-in-Aid for Scientific Research (B) 22H01420.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Kento Baba:** Conceptualization (equal); Data curation (equal); Formal analysis (lead); Investigation (equal); Methodology (lead); Validation (lead); Writing – original draft (lead). **Sena Kishiya:** Data curation (equal); Validation (equal); Writing – original draft (equal). **Hiroshi Gotoda:** Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Supervision (lead); Writing – original draft (equal); Writing – review & editing (equal). **Takeshi Shoji:** Data curation (lead); Investigation (equal); Methodology (equal); Writing – review & editing (equal). **Seiji Yoshida:** Data curation (lead); Investigation (equal); Methodology (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

Acoustic pressure and $ OH \u2217$ chemiluminescence intensity data that support the findings of this study are not available. The other data obtained by analytical methods are available from the corresponding author upon reasonable request.

## REFERENCES

*Thermoacoustic Instability*

*Combustion Instabilities in Gas Turbine Engines: Operational Experience, Fundamental Mechanisms, and Modeling*

*Unsteady Combustor Physics*

*Permutation Complexity in Dynamical Systems*

*The Nature of Statistical Learning Theory*

*Transient Chaos*