Detection of Dynamical Regime Transitions with Lacunarity as a Multiscale Recurrence Quantification Measure
Tobias Braun, Vishnu R. Unni, R. I. Sujith, Juergen Kurths, Norbert Marwan
DD ETECTION OF D YNAMICAL R EGIME T RANSITIONS WITH L ACUNARITY AS A M ULTISCALE R ECURRENCE Q UANTIFICATION M EASURE P REPRINT
Tobias Braun
Complexity Science (RD4)Potsdam Institute for Climate Impact Research14473 Potsdam, GermanyTel.: +49-331-28820744 [email protected]
Vishnu R. Unni
Department of Mechanical and Aerospace EngineeringPrinceton UniversityNJ, USA
R. I. Sujith
Indian Institute of Technology MadrasChennai 600036India
Juergen Kurths
Complexity Science (RD4)Potsdam Institute for Climate Impact Research14473 Potsdam, Germany
Norbert Marwan
University of PotsdamInstitute of Geosciences14473 Potsdam, GermanyJanuary 26, 2021 A BSTRACT
We propose lacunarity as a novel recurrence quantification measure and illustrate its efficacy to detectdynamical regime transitions which are exhibited by many complex real-world systems. We carry outa recurrence plot based analysis for different paradigmatic systems and nonlinear empirical data inorder to demonstrate the ability of our method to detect dynamical transitions ranging across differenttemporal scales. It succeeds to distinguish states of varying dynamical complexity in the presenceof noise and non-stationarity, even when the time series is of short length. In contrast to traditionalrecurrence quantifiers, no specification of minimal line lengths is required and rather geometricfeatures beyond linear structures in the recurrence plot can be accounted for. This makes lacunaritymore broadly applicable as a recurrence quantification measure. Lacunarity is usually interpreted as ameasure of heterogeneity or translational invariance of an arbitrary spatial pattern. In application torecurrence plots, it quantifies the degree of heterogenity in the temporal recurrence patterns at allrelevant time scales. We demonstrate the potential of the proposed method when applied to empiricaldata, namely time series of acoustic pressure fluctuations from a turbulent combustor. Recurrencelacunarity captures both the rich variability in dynamical complexity of acoustic pressure fluctuationsand shifting time scales encoded in the recurrence plots. Furthermore, it contributes to a betterdistinction between stable operation and near blowout states of combustors.
Keywords
Recurrence Plots · Regime Shifts · Lacunarity · Nonlinear time series · Thermoacoustic Instability a r X i v : . [ phy s i c s . d a t a - a n ] J a n etection of Dynamical Regime Transitions with Lacunarity as a Multiscale Recurrence Quantification Measure Many efforts in nonlinear time series analysis have been dedicated to the challenge of detecting transitions betweendifferent dynamical states of a system [52, 7, 60]. A broad range of real–world systems undergo such shifts betweendistinct regimes and their identification provides a better understanding of the complex dynamics under study [49, 13,14, 41, 61, 41, 9, 67, 22]. The universality of transitions between different dynamical states for a broad spectrum ofdifferent systems elucidates why applications have been widely dispersed among many disciplines. For instance, timeseries in earth sciences usually require sophisticated approaches to determine abrupt changes in the complex dynamics[13, 46, 24]. Regime shift detection has also gained popularity in analysis of EEG data [51], neuroscientific time series[58, 26] and other medical research fields [69] where the identification of pathological regimes is crucial. Due to theircomplexity, financial and social time series offer interesting applications as well [43, 17, 40].Major challenges in detecing regime shifts in real-world data are often data related, e.g. by means of unevenly sampled[46], nonstationarity, noisy or short time series. In convenient cases, transitions are visible to the eye but usually, theexact localization of the occuring dynamic transition and, in particular, the identification of precursors poses a challenge.Segments of a time series may appear qualitatively similar at first glance but could turn out to show significantlydifferent dynamical features. With respect to climate systems, multiple spatial and temporal scales can also hamperclear distinctions between variations in complexity of a time series. Even though regime transitions occur in a broadclass of systems, data related peculiarities raise the need of a comprehensive box of tools rather than a single universalmethod.In contrast to linear methods such as autocorrelation or power spectrum analysis, nonlinear techniques are able touncover more subtle transitions in complex time series data. Multiple different approaches constitute the state-of-the-artstoolbox, ranging from complex networks [35], entropies [7], detrended fluctuation analysis [29] or symbolic dynamics[37]. Since many empirical time series are univariate and no prior knowledge is accessible about the true dimensionalityof the system, phase space reconstruction is a powerful approach to study the system’s dynamics [7]. Approaches basedon the phase space trajectory are closely related to the well-known Lyapunov exponents and have proven to be effectivein classifying different dynamical states [41, 40].Another technique with relatively low numerial effort is the analysis of nonlinear time series by recurrence plots (RPs)[16]. The basic idea behind this method relates back to the perspective that dynamical systems recur to states theyhave visited before [57]. As a representation of such recurrences, the binary recurrence matrix R ij of a phase spacetrajectory x i ∈ R d indicates times where the system recurs to formerly visited states by s and all other times by s. Itis widely used as a graphical tool but also allows for quantification of various dynamical aspects of the system understudy. Since its first conception the method was successfully extended and applied to various real-world systems [44].The detection of regime transitions has become a prototypical field of application for RPs since it enables us to analysecomplex temporal patterns of nonlinear time series in a simplified fashion [47]. The majority of measures in recurrencequantification analysis (RQA) are based on black or white line structures in an RP. For instance, diagonal lines resembleparallel segments of the phase space trajectory and thus entail a degree of predictability. Approaches to capture morecomplex features in RPs beyond the reductionist approach of measuring line lengths have been conceived [11, 9], e.g.by allowing for the entirety of possible permutations of recurrences in small submatrices. This leads to the observationthat the variety of distinct microstates in deterministic and stochastic systems occupies only a fraction of the possiblepermutations, yielding lower entropy values. Even though this technique generally shows good robustness and doesnot require specification of minimum line lengths, such an approach is limited to the information captured by smallsubmatrices on a restricted local level. We propose a method where recurrences are also evaluated regardless of theirexact orientation in the recurrence matrix, but with a surplus quantification of the scaling from the smallest to the largestpossible submatrices. This can be achieved by not analysing the permutations of recurrences but by the statistics oftheir locally determined count.To this extent, we make use of a measure called lacunarity [42, 56]. Traditionally, it has been applied to quantifycomplex spatial patterns. Perhaps, the clearest interpretation of lacunarity is that it quantifies the degree of heterogeneityof the studied pattern. Often, it is applied in the context of distinguishing fractal patterns [10] because objects of samefractal dimensionality can still exhibit different degrees of heterogeneity. As a rule of thumb, patterns with largergaps yield higher values of lacunarity. Beyond this straight-forward interpretation, it classifies patterns with respectto their deviation from translational invariance [10]. This highlights its applicability as a measure of heterogenity.Characterizing the heterogenity of RPs on different temporal scales using lacunarity yields meaningful information aboutthe complexity of the underlying time series. Besides, heterogenity has already been considered as a quantifier to analyserecurrence networks [27]. Lacunarity has succesfully been applied to various systems ranging from characterizations ofthe scaling properties of the Amazon rainforest [39] and urban areas [23] to heterogenous patterns in bone structures[48] and stellar mass distributions [19]. It is also a popular tool in Neuroscience [31]. To our best knowledge, it has notyet been applied to RPs. Combining both approaches as a powerful tool to detect dynamical regime shifts is the main2etection of Dynamical Regime Transitions with Lacunarity as a Multiscale Recurrence Quantification Measurecontribution of this work. In order to demonstrate the scope of the developed methodology, we showcase applicationsto both paradigmatic systems and nonlinear empirical time series.This work is organized as follows: in Sect. 2 we introduce our methodology by briefly summarizing the RP techniqueand describing the computation of lacunarity by a box-counting algorithm. In this context, we give a brief dynamicalinterpretation of our method. Subsequently, we study results for synthetic data from three paradigmatic systems in Sect.3, namely the Logistic Map, the Roessler system and a bistable noise-driven system. The robustness of our methodagainst noise and short time series length is examined. Finally, we provide first evidence that the proposed methodis capable of detecting regime transitions in complex empirical time series. In Sect. 4, we identify known dynamicalstates in time series of acoustic pressure in a turbulent combustor [65] and attempt to illustrate the distinction betweentwo dynamically similar but practically contrary regimes. We conclude our findings in Sect. 5. First, we introduce recurrence plots in Sect. 2.1 and briefly revise traditional recurrence quantification analysis.Afterwards, we define lacunarity and present detailed information for its application to RPs in Sect. 2.2. To gain a moreprofound understanding of the proposed method, we discuss its dynamical interpretation for the phase space trajectory.
Many real-world systems show a tendency to recur to states they have visited before. Such information can be capturedby a two-dimensional visualization that may yield striking patterns which resemble the recurrences at all time instancesof the time series. By studying regularities in such recurrence patterns, rich information can be obtained on thedynamics of the underlying system that go beyond the scope of linear statistical methods of time series analysis such asautocorrelation functions. In particular, different quantification measures of the visual representation prove powerful inclassifying differing systems, detecting non-linear correlations and identifying dynamical regime transitions. The basicconcept can be outlined by defining the recurrence matrix R ij = (cid:26) || x i − x j || ≤ (cid:15) || x i − x j || > (cid:15) (1)with a time series x at two arbitrary times i and j and a suitable norm (cid:107) · (cid:107) . The vicinity threshold (cid:15) needs to be fixedwith respect to the distances of time series values such that a meaningful expression of recurrences is obtained. Apopular approach to do so is, for example, to choose a value which entails a fixed recurrence rate for the RP [47]. Theresulting visual representation of a recurrence matrix is a binary image of black and white dots from which temporalpatterns can be inspected. For higher dimensional systems, recurrence analysis needs to be based on a phase spacerepresentation of the time series as a trajectory in d -dimensional space. Despite the fact that the dimension of theregarded system is often unknown, Takens’ theorem [63] ensures that a time–delay embedding can be found that givesan appropriate phase space reconstruction. To this extent, an embedding delay is often specified prior to fixing theoptimal embedding dimension for the system. We will apply the broadly used mutual information criterion and theFalse Nearest Neighbours (FNN) method [8, 55] to estimate both parameters.For many systems, RPs have been successfully applied to reveal a high degree of complexity both in terms of nonlineardynamics and stochastic fluctuations [47]. In the detection of regime transitions, a reliable quantifier that yieldsan unambiguous distinction of dynamical regimes is generally required. Traditional complexity indicators such asLyapunov Exponents [30] are not always robust against noise and require rather long time series. They are also oftennot able to capture the relevant time scales of the system’s shifting dynamics which is particularly important forsystems with multiple characteristic time scales. Yet, such information is contained in RPs and can be uncovered usingrecurrence quantification measures [71]. Most of them are based on the statistical distribution of line structures inRPs. For instance, diagonal lines of certain length indicate a similar evolution of different segments of a time series. Apopular quantification of an RP based on diagonal lines is defined as the fraction of lines that exceed some specifiedminimal line length. As it quantifies the degree of determinism in a time series, it is refered to as DET . Verticallines indicate that the system is trapped in a certain phase space region for subsequent times. White vertical gapsindicate transitions between different phase space regions while black square-like structures point at time intervalsin which the system remains confined in a small region of the phase space. If such patterns reoccur with statisticalsignificance, they uncover regularities of the time series and may yield characteristic recurrence time scales. Line–basedrecurrence measures have been applied to a diversity of complex real–world systems as complexity measures to uncovertransitions [47]. Yet, different embedding parameters can yield varying results and edge effects as well as high samplingrates might result in spurious quantifications [45], thus requiring corrections [34]. On top of that, the application ofline-based RQA measures is limited to systems that do not show more complex recurring patterns which are refered to3etection of Dynamical Regime Transitions with Lacunarity as a Multiscale Recurrence Quantification Measureas microstates of an RP [11]. Related complexity measures that go beyond this scope have shown that they can havesuperior performance [40, 9]. In this work, will put forward a novel RQA measure that characterizes the heterogenity ofan RP and is not based on certain microstructures.
Lacunarity is often illustrated as a measure of ‘gappiness’ or as a property that can characterize the heterogenity of aspatial pattern. It was introduced to distinguish between different fractal patterns of equivalent fractal dimensionality.However, its scope goes beyond the distinction of fractal structures. More formally, we can regard it as a measureof deviance of a pattern from translational homogenity [56, 59]. As lacunarity is usually derived for different spatialscales, it is possible to identify a certain scale above which a pattern is translational invariant and below which it is tooheterogenous to be regarded as such.Methods to calculate lacunarity for empirical patterns are often based on box–counting algorithms [32] similar to thoseused in estimating fractal dimensionality of a pattern. Essentially, a grid is applied to the studied pattern and pixels ineach box are counted. The specific algorithm to compute lacunarity for RPs is summarized in algorithmic form below.Even though box-counting algorithms can also be modified so that an analysis of grayscale- [12] or RGB–encoded [25]patterns is possible, in application to RPs a basic algorithm for binary patterns suffices. The quantity that enables us toanalyse scaling properties of a complex pattern is the size of boxes on the applied grid. Given an RP as a T × T –matrixand a fixed box size w , the mass M of each box is obtained by counting the black pixels inside the box. This results ina mass distribution P w ( M ) for all N boxes. From this distribution, we compute the moments Z ( q ) ( w ) = (cid:88) M M q P w ( M ) . (2)In the definition of lacunarity Λ , only the first and the second moments Z (1 , are considered: ˜Λ( w ) = Z (2) ( w ) (cid:2) Z (1) ( w ) (cid:3) = 1 + σ ( w ) µ ( w ) (3)with mean µ and standard deviation σ . To have a measure in [0 , , we normalize lacunarity by also computing thelacunarity ˜Λ † of the complement of the set ( s replaced by s and vice versa ). Consequently, we define it as Λ( w ) = 2 − (cid:18) w ) + 1˜Λ † ( w ) (cid:19) . (4)This refined definition of lacunarity also enhances the detection of significant gap sizes compared to eq. (3) and is thusprefered [48]. Various box–counting methods beyond the basic approach employed in this work are known (e.g. glidingbox-counting [3]) and generalized versions exist (e.g. for multifractal data [2]). The often used standard gliding boxapproach results in a higher number of boxes but is also known to cause biased values due to edge effects [18]. Notethat in any case, minimum and maximum box size have to be chosen based on the time series length.4etection of Dynamical Regime Transitions with Lacunarity as a Multiscale Recurrence Quantification Measure Algorithm 1
Recurrence Lacunarity if embedding required then Choose d, τ and embedd time series end if Compute RP R with specified vicinity threshold (cid:15) for box width w = 2 , . . . (cid:28) T do Split ( T × T ) -matrix R into N = (cid:98) T /w (cid:99) × (cid:98)
T /w (cid:99) disjunct boxes for each box b ( i,j ) w do Count recurrences: M = (cid:80) i,ji (cid:48) ,j (cid:48) δ ( R i (cid:48) j (cid:48) − end for Compute normalized lacunarity Λ for fixed w from eq. (2-4) Apply bootstrap by randomly drawing sufficient number of boxes b ( i,j ) w for significance testing if T % w (cid:54) = 0 then repeat iteration with varied grid position until robustness ensured end if end for In the application of lacunarity to RPs, we refer to it as recurrence lacunarity (RL). In view of RPs, black pixels areequivalent to recurrences of a trajectory in reconstructed phase space. Thus, we are effectively carrying out an analysisof local (in a temporal sense) recurrence statistics and quantification of local variations. This is essentially implementedvia the computation of variance of recurrence points contained in boxes which are located at different positions in theRP. An extension to higher statistical moments is also conceivable [68]. Our approach circumvents the necessity ofdefining any sort of microstate and is not restricted to the usual statistical analysis of line structures in the RP. Thescaling (sucessive increasing of time intervals) is expected to pinpoint relevant temporal scales related to averagerecurrence times and quasi–periodicities. This is confirmed by the displayed recurrence lacunarity curves in Fig. 2 fromwhich some will be studied in more detail in Sect. 3.Figure 1 shows examples of RPs of systems that show fundamentally different dynamics. While a white noise processentails an RP with randomly distributed black dots, a Logistic Map in the chaotic regime ( r = 3 . ) still results insome local structures such that RL is higher for almost all w , resembling higher complexity. A Roessler system in theperiodic regime generates well pronounced lines corresponding to deterministic periodic dynamics. As expected, thecharacteristic width between the lines is captured in the variation of RL as a local minimum (left dotted circle) since forboxes of the same size, homogenity is enhanced. The second visible minimum (right dotted circle) is located at twicethe period of the time series. Stochastic signals such as an AR(2)–process and Multifractal Gaussian Noise (MFGN) [6]yield a higher degree of complexity in terms of low translational invariance of the corresponding RPs. Both the MFGNand the bistable noise-driven system (see sec. 3.3) time series have a visible periodic modulation which is resembled bythe distinct gap sizes in the respective RPs. For the former time series, RL sharply drops when w reaches the gap size.For the latter, the stochastic component results in slight variations of the gap size but still, RL captures them as a localminimum at w ≈ . As standard RQA measures are based on line structures in the RP, they have clear interpretations in terms of the phasespace trajectory and their relationships to dynamical invariants (such as Lyapunov exponents and Renyi entropy) arewell established [47]. These questions also arise for RL. Picking a box within an RP and deriving some related statisticscan be interpreted as sampling two segments of a phase space trajectory. If we denote the starting point of the firsttrajectory by i and the second by j , the lower left corner of the box is located at ( i, j ) and its upper right corner at ( i + w, j + w ) . The number M i,j of recurrences contained in the box characterizes the similarity between the twotrajectory segments by means of their recurrences. For fixed (cid:15) , these are equivalent to the number of contained phasespace vectors that are nearest neighbours by means of a low distance in phase space. Figure 3 illustrates the relationbetween box-counting of recurrences and the phase space trajectory based on the Roessler attractor as a paradigmaticexample (see Sect. 3.2 ). RL quantifies the heterogeneity of recurrent temporal patterns representing different segmentsof the phase space trajectory. For increasing length of trajectory segments (from right to left in Fig. 3), their recurrenceas well as their divergence can be captured by means of recurrence patterns encoded in the RP.Apart from the dynamical interpretation, it is an open question how fractality of RPs is related to self-similarity in theunderlying time series or the attractor [5]. As a general conception, the very basic structures of RPs such as diagonal5etection of Dynamical Regime Transitions with Lacunarity as a Multiscale Recurrence Quantification Measure y t t (a) White noise . . . y t t (b) Logistic Map( r = 3 . ) − y t t (c) Roessler system (periodic, a =0 . , b = 0 . , c = 3 ) − y t t (d) AR(2)–process( µ = 1 , µ = − , θ = 0 . ) − . . . y t t (e) MFGN–process with sinusoidallytuned Hurst Exponent between [0 . , . − y t t (f) Bistable white noise-driven system( K = 100 , A = 300 , ω = 10 , D = 45 ) Figure 1: Recurrence plots for time series of different deterministic and stochastic systems.lines, vertical lines and blocks show some resemblance to typical 1- and 2-dimensional fractals like the Cantor set orthe Sierpinski carpet [38]. Self–similarity by definition arises from the spatial recurrence of patterns [70]. Even thoughideal monofractal patterns can not be expected to occur in a pure fashion in RPs, some degree of self–similarity appearsintuitive: small-scale recurring patterns often constitute the correlations on longer time scales which resembles whatis displayed in an RP. Self–similarity over some range of box sizes applied to an RP could thus be interpreted as asimilar tendency to recur to formerly visited states regardless of the respective time scale. On top of that, it is knownthat geometrical quantities computed from recurrence networks show relations to the phase space dimension and exhibitfractal scaling properties [15]. This ultimately raises the question whether there is a direct relation of RPs fractality tothe degree of fractality of the time series and the attractor. This should be explored in more detail in a future study andis beyond the scope of this work. 6etection of Dynamical Regime Transitions with Lacunarity as a Multiscale Recurrence Quantification Measure w (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) White NoiseLogistic MapRoesslerAR(2)Multifr. GNBistable
Figure 2: Recurrence lacunarity curves Λ( w ) in double-logarithmic plot, corresponding to displayed RPs in Fig. 1. empty half full Figure 3: Schematic illustration of the relation between box-counting on RPs and the underlying phase space trajectoryfor the Roessler attractor ( a = 0 . , b = 0 . , c = 14 ). The full RP is displayed on the left with two exemplary boxes(blue and red). Above, the box counts M w ( i, j ) of all boxes (although not indicated) located at grid positions ( i, j ) areshown with the average box count (vertical line) whereas their index is chosen as a y-coordinate for better visibility.From left to right, boxes of decreasing width w are zoomed in and additional boxes are indicated. The segments of thephase space trajectory that correspond to the boxes are color coded respectively. The scatter plots illustrate that fordecreasing box widths, heterogeneity by means of dispersion of the box counts increases.7etection of Dynamical Regime Transitions with Lacunarity as a Multiscale Recurrence Quantification Measure System Dynamical regimes Equations Param. Embedding
Logistic Map deterministic/chaotic x n +1 = rx n (1 − x n ) , r ∈ [3 . , . r / Roessler deterministic/chaotic ˙ x ( t ) = − ( y ( t ) + z ( t )) , a = 0 . y ( t ) = x ( t ) + ay ( t ) , b = 0 . z ( t ) = b + ( x ( t ) − c ) z ( t ) , c ∈ [2 , c d = 3 ,τ /∆ t = 18 Bistable noise-driven determ./stochastic ˙ x ( t ) = (cid:2) K (cid:0) x ( t ) − x ( t ) (cid:1) + A cos ωt (cid:3) + Dξ ( t )( K = 100 , A = 320) ω, D / Thermoacousticcombustor experimental / equiv.ratio φ d = 5 ,τ = 23 Table 1: Overview of studied systems for detecting dynamical regime shifts. The systems are distinguished based ontheir exhibited dynamical regimes, underlying equations, transition parameters and delay-embedding parameters.
We demonstrate the ability of RL to capture different kinds of regime shifts by its application to model data fromdifferent dynamical systems (Table 1). These systems cover some of the important aspects that need to be accounted forif transitions in real data should be identified. In Sect. 3.1, we study transitions between chaotic and periodic dynamicsfor the Logistic Map. In Sect. 3.2, transitions in the Roessler System as a three-dimensional continuous system areanalysed. A bistable noise-driven stochastic process is examined in Sect. 3.3 in order to demonstrate the ability of ourapproach to uncover rather subtle transitions. Finally, we examine an experimental dynamical systems in Sect. 4.RPs reveal various structures [11] on a broad range of time scales. RL captures these by means of variations in thecharacteristics of the power law scaling of RL with w . We will visualize this by showing the variation of the powerlaw for each of the systems and refer to it as the RL curve . Moreover, we characterize the scaling of RL curves bycomputing the slope α of logΛ against log w by linear regression. Note that if multiple scaling regions coexist for oneRP, a single slope will yield ambiguous results and other indicators should be chosen (e.g. the regression error).Complexity measures indicate transitions by changing to significantly high or low values. In order to assess significance,we apply the following bootstrapping method that entails confidence intervals: we draw a random sample (withrepetitions) from all boxes the RP was divided into and derive RL only from this sample (see Algorithm 1). Thisresampling procedure is repeated N times to obtain a distribution of RL values that jitter around the true value. The / – quantiles characterize the width of this distribution and are used as confidence bounds. Similar approches areusually employed with other complexity measures [64, 36, 50]. In most applications, confidence bounds should reflectsignificance equally sufficient for all values obtained for the respective complexity measure (i.e. for each configurationof the parameters that control regime transitions). In our case, this consequently raises the question whether to samplefrom the global or local distribution of box counts. In the local case, box counts for a single window are used to computeconfidence bounds whereas in the global case, box counts from all windows are joined. In both cases, the calculation ofquantiles is performed such that only a single parameter-independent value is obtained, yielding horizontal lines thatindicate the separation between ‘regular’ dynamics and dynamical transitions. Extensive comparative analysis of theresulting confidence bounds indicates that the latter approach yields more convincing results for the studied systems.8etection of Dynamical Regime Transitions with Lacunarity as a Multiscale Recurrence Quantification MeasureBootstrapping from the global box-count distribution generally yields more narrow bounds that seem to overestimatethe true number of regime shifts. Consequently, for an underlying series of length T , we obtain T locally bootstrappedRL distributions of size N from which we compute – confidence bounds. As a first paradigmatic example, we employ the Logistic Map [54] as a system that is well-known to producepronounced transitions between periodic, chaotic and laminar behaviour. The variation of the parameter r introducesbifurcations between regimes of different complexity as displayed in Fig. 3. While increasing r entails a tendencyof less predictability, windows of periodic dynamics arise in between. For each parameter value r ∈ [3 . , . with δr = 0 . we generate a time series of suitable length such that after discarding transients, we have T = 1000 points in time. For each such time series x r ( t ) , we calculate an RP. We choose the threshold (cid:15) = 0 . σ ts with σ ts beingthe standard deviation of x r ( t ) . From each RP, we compute RL for a number of k = 70 different box sizes w ∈ [2 , T / .This results in a single RL curve like in Fig. 2. A standard measure to detect regime transitions for such paradigmaticsystems is the largest Lyapunov exponent λ [7]. We use it as a reference for our results to evaluate the detection oftransitions.Figure 4: Bifurcation diagram of Logistic Map and RL curves for varying r ∈ [3 . , . with n = 2000 . Dashedvertical lines indicate chaos-chaos transitions. Each time series has T = 1000 values after discarding transients. For theRL curves, w –axis and color coding are scaled logarithmically. The black curve illustrates the variation of λ .Below the bifurcation diagram in Fig. 4, RL is displayed using color coding. Each RL curve is plotted in double-logarithmic coordinates. Periodic windows are clearly detected as the corresponding RPs are homogenuous at all timescales. Furthermore, it appears that e.g. around r ≈ . (red vertical line) it identifies a transition to a regime not wellcaptured by λ which is known to arise from the intersection of the supertrack functions [51]. At these, an unstablesingularity results in laminar behaviour i.e. the time series becoming ‘trapped’ in a certain range of values for sometime intervals. RL is able to detect such chaos–chaos transitions.In real–world data sets, several effects can impede the detection of regime transitions. We test the robustness of RLcompared to DET as a standard RQA measure for two such effects, i.e. contamination with white noise and short timeseries lengths. Figure 5a shows bifurcations for varying noise strength. We plot both DET and the slope α of the RLcurve for r ∈ [3 . , . . RPs are generated as described above but for time series that are contaminated by uncorrelatedwhite noise of different strength. The largest Lyapunov exponent λ shows that for σ n = 0 . , only few of thechaos-periodic transitions are detected. For σ n = 0 . , some transitions are still well depicted by both measureswhile others become less prominent already. We evaluate the performance of detecting transitions for σ n = 0 . bycalculating confidence intervals for the noise-free case as a rather strict bound. Both measures perform with similarsuccess. Yet we point out that RL seems to resolve transitions close to strongly periodic dynamics more clearly inpresence of noise, e.g. at around r ≈ . and r ≈ . . Analogously to this analysis, we plot results for different timeseries lengths in Fig. 5a. As expected, it appears that the number of false detections increases for both measures forshorter time series. However, both still succeed to pinpoint chaos–order transitions even for very short time series with T = 100 . Beyond that, the laminar parameter range is also still identified. Anyway, α indicates more false shifts than DET which may be due to the rather basic box-counting approach that limits the computation of RL to a few boxes incase of very short time series. 9etection of Dynamical Regime Transitions with Lacunarity as a Multiscale Recurrence Quantification Measure -1.0-0.50.00.5 λ σ = 0 σ = 0 . σ = 0 . D E T α (a) λ , DET and the slope α of the RL curve for for varying r ∈ [3 . , . with n = 1000 . Different curves correspond to differentnoise intensities. Each time series has T = 1000 values afterdiscarding transients. – confidence bounds (gray dashed lines)for both RQA measures are obtained via bootstrapping and referto the noise-free case. Significant transitions outside of these areindicated by blue/red rectangles for noise with σ = 0 . . -1.0-0.50.00.5 λ T = 500 T = 100 D E T -3-2-1 α D E T α (b) λ , DET and the slope α of the RL curve α for varying r ∈ [3 . , . with n = 1000 . Different curves correspondto different time series lengths. – confidence bounds (graydashed lines) for both RQA measures are obtained via bootstrap-ping. Significant transitions outside of these are indicated byblue/red rectangles. Figure 5: Robustness of RL to varying noise intensity and time series length for the logistic map.
We further explore the ability of RL to detect regime transitions for continuous dynamical systems with the Roesslersystem as a standard example (see Tab. 1). We vary c ∈ [2 , with n = 2000 different values. Increasingly dominantchaotic behaviour is expected whereas the average distance between unstable periodic orbits decreases for increasing c .The nonlinearly coupled x –component is embedded with parameters given in Tab. 1. The vicinity threshold is fixed as (cid:15) = 0 . σ ts .Instead of α , we analyse single scale-specific values of Λ( w ) since in general, multiple scaling regions can be identifiedthat are not well represented by a single scaling exponent (see Fig. 2). In the lower panel of Fig. 6 λ (black) and thesecond Lyapunov exponent λ (white) serve as a reference and enable us to localize periodic windows in the regardedparameter range. The color coded RL curves display rich information on various characteristic scales underlying thechaotic and periodic dynamics. Note that RL detects bifurcations only captured by both of the displayed Lyapunovexponents. For instance, this can be seen in the range c ∈ [2 , where RL corresponds well to λ for larger box sizeswhereas pronounced variations are neither captured by λ nor by determinism. Above c = 5 , the transitions captured byRL match those indicated by DET . Particularly for the smaller box sizes, a trend in overall complexity for increasing c is present. We subtract this quadratic trend from three lacunarities for fixed box sizes w in the upper panel of of Fig. 6and compute confidence bounds via the introduced bootstrapping procedure. Almost all of the occuring transitionsare detected as indicated by the colored vertical lines. Interestingly, certain periodic windows are most prominentlycaptured by distinct box sizes such as at c ≈ . or c ≈ . . This is due to the fact that as soon as a box size thatcorresonds to a characteristic scale is reached, the RP becomes more homogenuous on this scale and entails low RL.From this perspective, RL may additionally provide similar spectral information on the time series as usually obtainedfrom Wavelet analysis [1] in some cases. 10etection of Dynamical Regime Transitions with Lacunarity as a Multiscale Recurrence Quantification MeasureFigure 6: DET , single scale-specific lacunarities Λ ( w ) and RL curves for varying c ∈ [2 , with n = 2000 forthe Roessler system. – confidence bounds are obtained via boostrapping and excursions outside the indicatedhorizontals are indicated. Each time series has T = 1000 values after discarding transients. In the lower panel, w –axis and color coding are scaled logarithmically. Black (cyan) curves display the first (second) Lyapunov exponentnormalized to their maximum absolute values for better visibility. Finally, we demonstrate that RL can also uncover subtle transitions in the time evolution of a nonstationary, stochasticsystem. To this extent, we study a bistable system which is driven by uncorrelated noise ξ ( t ) and a periodic component.Such a system is often illustrated as a Brownian particle trapped in a double-well potential with a periodic driving force[28]. It can be described by the Langevin equation given in Tab. 1 where K controls the shape of the deterministicpotential function, A yields the strength of the periodic component with its frequency ω and D controls the noisestrength. ξ ( t ) is chosen as Gaussian white noise. Equation 2.2 is solved using the Euler–Maruyama method with asampling interval of ∆ t = 0 . and subsequent downsampling to ∆ t = 0 . . The interplay of the noise strength andthe periodic force determines the transition dynamics between the two stable fixed points as described by the generalphenomenon of stochastic resonance [20]. For fixed A , an optimal D exists such that the signal-to-noise ratio (SNR) ismaximized.There are several notions of how different regime shifts can be classified [4]. A basic differentiation may be madebetween transitions which are induced by a stochastic or a deterministic component of the respective system. We varyboth the frequency ω of the driving force and noise strength D to test whether RL is able to detect these two differenttypes of transitions. Both increasing ω and decreasing D result in a lower SNR since it triggers more high-frequencyvariability and impedes regular switching. We generate time series of total length T = 1 . · whereas the frequency ω is first abruptly decreased from ω = 20 to ω = 8 at t = T / while noise strength remains constant at D = 36 . At t = 2 T / , noise strength σ is decreased from D = 36 to D = 28 while retaining ω = 8 . realizations x i ( t ) withrandom initial conditions are generated. The height of the potential barrier is h = K / (cid:29) A , therefore jumpsbetween the potential minima can not be solely caused by the periodic forcing. However, both noise strengths can resultin purely noise-driven jumps. RL is computed for RPs with a fixed recurrence rate of on sliding windows of width with a overlap for each sample. In total, the following two shifts are studied: X → X → X | K =100 ,A =320 :( ω = 20 , D = 36) t −→ ( ω = 8 , D = 36) t −→ ( ω = 8 , D = 28) between states X , X and X . The upper panel of Fig. 7 shows an example of a sampled time series. We can observethe subtlety of the two parameter shifts which can not be solely localized through visual inspection. In the lower panel,the variation of color-coded RL curves is displayed whereas the average over all generated samples is shown. Asexpected, the transitions are most striking for the largest considered box sizes w since these have the same order ofmagnitude as the switching time scales between the two fixed points. Such switches are encoded as gaps in the RPsand can thus be well identified by RL. The decrease of ω after X → X results in more homogenuous RPs on these11etection of Dynamical Regime Transitions with Lacunarity as a Multiscale Recurrence Quantification Measuretime scales since the system on average resides for shorter time periods within one potential minimum. Within the X regime, SNR is enhanced yielding a more regular switching behaviour. This regularity further stands out as an almoststable switching period as indicated by RL at w ≈ . Recurrences within this regime can consequently be regardedas less complex for time periods exceeding this switching cycle. A similar increase in large-scale RL can be observedfor the decrease in noise intensity X → X : as noise-driven transitions are less likely, the RPs are more ‘gappy’ onlonger time scales.Figure 7: Single time series realization and averaged RL curves on sliding windows for bistable noise-driven system.Sample averaged SNR (gray) is indicated in the upper panel, sample averaged DET (purple) and α (cyan) are comparedin the center panel. Vertical dashed lines mark the two transitions X → X and X → X . For the RL curves, w –axisand color coding are scaled logarithmically.Both states X and X thus share the feature that lower frequency variability is more complex in terms of more diverserecurrent patterns. Anyway, they differ with respect to their high–frequency variability. For small box sizes (mostlyreflecting intra-well dynamics), RL is higher in the X regime as indicated by the gray contours.When both the driving frequency and noise intensity are low, recurrences are less eratically clustered, yielding morecoherent, variable recurrent periods for short durations. The semi-stable period is preserved in X but less pronouncedthan in X since lower noise strength results in a weakened SNR through the mechanism of stochastic resonance.Finally, we compute three regime-specific RL curves for each of the 100 samples and calculate the sample averagedslopes α for the different regimes. With α X = − . ± . , α X = − . ± . and α X = − . ± . , thetwo regimes X and X can be well distinguished from X but are similar to each other in the range of their respectivestandard errors. Anyway, the middle panel in Fig. 7 shows that α still captures both transitions more clearly comparedto DET . While
DET performed superior for short time series (see 5b), RL gives more convincing results for theserather subtle shifts.
In order to evaluate the performance of RL as a complexity indicator for empirical data, we apply it to acousticpressure time series from a laboratory combustor with turbulent flow, operating at atmospheric pressure. The univariatetime series was measured with a resolution and is known to undergo a rich variety of transitions betweenchaotic, intermittent and periodic dynamics [22]. Practical relevance arises e.g. from the use of gas turbine engines forpropulsion and power generation. The two main acting subsystems are the unsteady heat release and the acoustic field.Positive feedbacks between both may lead into a state called thermoacoustic instability, enhancing heat transfer to thewalls of the combustion chamber and resulting in increased mechanical stress. This can cause severe damages to anaircraft’s engine or shutdown in the case of a power plant [33]. The parameter which can drive the turbulent system intoan unstable state is the fuel/air ratio. If it falls below a critical threshold, flame blowout can be the result which leads toan abrupt drop of thrust for an aircraft. This raises the need of effective monitoring to detect impending instabilities andthermoacoustic transitions such including thermoacoustic instability and blowout. In the following, we will apply RLanalysis to RPs of the system in order to idenfity the different regimes and localize the shifts between them. Severalstudies have been carried out in this context, classifying dynamical states with fractal measures [67], employing complex12etection of Dynamical Regime Transitions with Lacunarity as a Multiscale Recurrence Quantification Measurenetworks [66, 35] and RPs [53, 21, 22, 62]. Special emphasis will be given to the distinction of normal operatingconditions (combustion noise) and an impending blowout situation. We first introduce the experimental setup in Sect.4.1. In Sect. 4.2, we investigate regime shifts in the system in terms of varying RL of acoustic pressure time series.
The acoustic pressure ( p (cid:48) ) data used for the study was obtained from a turbulent combustor with a bluff body stabilizedflame. The combustor consists of a rectangular chamber (length = 140 cm, cross section= 9 cm × × samples per second. The fuel used is LPG ( Butane,
Propane). The flow rate of partially premixedair-fuel mixture is varied in a quasi steady manner by increasing the air flow rate for a fixed fuel flow rate (1.04 g/s).This also reduces the equivalence ratio, φ , of the fuel-air mixture defined as the ratio of actual fuel-air mass flow rateratio to the stoichiometric fuel-air mass flow rate ratio. As φ reduces by increasing the airflow rate, initially the pressurefluctuations change from aperiodic oscillations to thermoacoustic instability via intermittency. On further reduction of φ ,the pressure oscillations exhibits intermittency post thermoacoustic instability. When φ is reduced further, we approachflame blowout. Flame blowout is a phenomena where the flame loses its ability to stabilize inside a combustion chamberand hence undergoes extinction. Prior to flame blowout, the pressure oscillations have low amplitude and are aperiodic.Further details of the experimental setup (Fig. 8) and different dynamic regimes are detailed in [67]. Pressure sensor
Bluff body
Combustion chamberAir-fuel mixture
Figure 8: Schematic illustration of the combustion chamber employed in the experimental setup.
The different operating conditions of a thermoacoustic combustor have been extensively studied in the literature both inlaboratory conditions and model system data [33]. Measures based both on RPs and recurrence networks have beensuccessfully applied to detect dynamical transitions between different regimes [21, 22]. Prior to our analysis, we canalready give a brief classification of the different dynamical states based on these insights. Figure 9 shows the full timeseries in the bottom panel and enlarged segments with normalized amplitude of it in the top panel. The first segmentshows combustion noise ( X ) which is the general term for stable operating conditions. It is comprised of low amplitudeaperiodic pressure fluctuations which can be classified as chaotic dynamics. Every dotted gray line in the lower panelmarks a decrease in φ . In this sense, each three second sub–time series should be regarded as a separate experimentwith constant parameters and will be evaluated as such in the following. As φ is discontinuously increased along thetime axis, we observe a dynamical state characterized by aperiodic oscillations interrupted by large amplitude harmonicoscillations. This state is generally refered to as intermittency ( X ) and is observed as a transition state betweeenaperiodic oscillations and thermoacoustic instability. The third zoomed segment shows thermoacoustic instability ( X ) which is constituted by periodic large amplitude pressure fluctuations. As φ is increased further, the periodic oscillationssubside and a different state of intermittency ( X ) is observed which we will refer to as intermittency after instability(in contrast to intermittency prior to instability). The last segment again shows aperiodic oscillations of low amplitudewhich are a precursor of an impending blowout situation where the flame can not longer be sustained ( X ) .In order to carry out our analysis, an adequate phase space embedding is required for the measured time series as theunderlying system should be regarded as high-dimensional. We apply the mentioned standard methods to fix a suitable13etection of Dynamical Regime Transitions with Lacunarity as a Multiscale Recurrence Quantification Measureembedding delay and dimension. Since we aim at analysing the dynamics of the combustor for a range of parameters,we estimate common embedding parameters appropiate for all different dynamical states. To also evaluate significanceof our results for RL, we apply a sliding window analysis to the time series. We choose a window size of
900 ms with overlap between consecutive windows while no overlap is allowed between the different sub-time series withfixed φ . We ensured that all of the following results are robust in a reasonable range of window and overlap widths.A sufficient tradeoff for all time series segments is obtained by analysing how strongly both embedding parametersfluctuate for the different regimes in time. Embedding delay is maximum for combustion noise and it decreasestowards enhanced periodic oscillations. Highest average embedding dimensions are estimated for intermittency prior toinstability. We conclude that our global parameter choice of d = 5 , τ = 23 is suitable for further analysis.Based on this choice, we compute RPs on sliding windows based on the delay-embedded phase space trajectory ofthe system. We choose a threshold (cid:15) such that it yields constant recurrence rate of for all RPs. All results arequalitatively sustained for reasonable variations of (cid:15) . For a visual impression of RPs of a similar system, the reader ispointed to [53, 21]. The procedure is now carried out as follows: we first calculate a RL curve for each obtained RPwhich refers to a certain time instance for fixed φ . We concatenate the entire set of RL curves to illustrate them in thesame fashion as for the synthetic data examples to display variations of complexity on all time scales. Additionally,we classify the different dynamical regimes by scale-averages of the RL curves. In order to estimate scale averages ofRL, we first average all RL curves for fixed φ obtained from the sliding window analysis. Next, we average RL valuesfor time scales w ≤ , < w <
100 ms and w ≥
100 ms separately. Note that these groups cover differentnumbers of RL values. The results are shown in Fig. 9.In the lower panel, we observe a gradual descent of RL at all time scales w from combustion noise into the impendinginstability, interrupted by spikes of varying amplitude. The overarching trend until about shows a reduction indynamic complexity from the aperiodic, chaotic oscillations during stable operation into periodic oscillations duringthermoacoustic instability. By means of RL, we can interpret this decrease as weakened heterogenity of recurrences,entailing that the system shows temporal patterns with less strong variability as it approaches the instability regime.The intermittency route into instability becomes well visible by the shape of the RL curves: the cut-off value of thetruncated power law decay is continuously reduced, displaying a significant shift of characteristic time scales in thepressure fluctuations of the combustor. Cut-off values can be infered approximately from the graph by tracking suddencolor switches. In the range between − , intermittency manifests itself in the episodic reduction of large scaleheterogenity in the RPs. The system jumps between harmonic and aperiodic oscillations and thus shifts its characteristictime scale discontinuously until multiple coexisting periods (horizontal yellow lines) are reduced to a single dominantperiod. A sharp rise of RL at all scales marks the slow intermittency regime prior to the near blowout situation. Theaverage level of complexity during this lean blowout state appears close to that observed for combustion noise at thebeginning of the measurement series. In the upper panel, the scale-averages yet uncover a difference in sub-ms RLbetween combustion noise and lean blowout state: the aperiodic oscillations in the former regime seemingly occur in amore heterogenous fashion than for the latter. The two other scales-averages do not indicate a remarkable differencebetween these two regimes. Note that the scale averages are rescaled to [0 , for better comparison. Consequently,RL enables us to detect five distinct regimes and to track respective variations in the characteristic time scale of thesystem. These findings generally corroborate those from earlier studies. Furthermore, it detects a subtle difference inthe complexity between the dynamically similar combustion noise state and the near blow out situation. We have put forward a novel recurrence quantification measure to quantify the degree of complexity of nonlineartime series, namely recurrence lacunarity. The identification of different dynamical regimes in multiple real–worldapplications has attracted a lot of interest in the literature. The method we propose contributes to this toolbox ofcomplexity indicators by representing a time series as a recurrence plot and characterizing its heterogenity on all timescales relevant to the system. Even though both recurrence plots and lacunarity are broadly acknowledged as powerfulstand-alone tools, we have demonstrated that their combination can yield valuable insights in the context of regime shiftdetection. The method’s main advantage is that it is able to capture multiscale features of the recurrence plot whiletraditional measures are based on line structures which can always only encode a certain aspect of the dynamics. Ourapproach does not require specification of minimum line lengths.14etection of Dynamical Regime Transitions with Lacunarity as a Multiscale Recurrence Quantification Measure -1-0.500.51 p / p m a x X X X X X t [ s ] -4-2024 p [ k P a ] (a) Enlarged, normalized time series segments of different dynamical regimes and the entiremeasurement time series of acoustic pressure in kPa . Each fixed parameter window covers aduration of three seconds. The displayed dynamical states are refered to as combustion noise ( X ) , intermittency prior to instability ( X ) , thermoacoustic instability ( X ) , intermittencyafter instability ( X ) and near-blowout oscillations ( X ) respectively. . . . . . . Λ w w < ms w ∼ ms w > ms1.1 1.03 0.96 0.9 0.83 0.76 0.69 0.63 0.56 0.49 0.43 0.36 φ t [s]110100 w [ m s ] (b) Scale-averaged lacunarities Λ w and RL curves for full time series computed on slidingwindows of
900 ms width. In the upper panel, green and red shading indicate combustionnoise ( X ) and near blowout oscillations ( X ) respectively. Air flow rate is increased in aquasi-static manner after maintaining it steady for a duration of 3 seconds. In the lower panel, w –axis and color coding are scaled logarithmically. Figure 9: Application of RL to acoustic pressure fluctuation time series from a thermoacoustic combustor.15etection of Dynamical Regime Transitions with Lacunarity as a Multiscale Recurrence Quantification MeasureIt naturally yields information both on the heterogenity of recurrence plots and their scaling properties, opening up anew perspective of analysing point statistics in partitioned RPs rather than constricted structures. We have shown thatthe applicability of lacunarity to general nonlinear dynamical systems by means of the well established recurrence plotapproach can be fruitful.As this work’s main focus was on the identification of dynamical regime shifts, we have studied three differentparadigmatic systems in detail to showcase the potentials of the method. We have found that the proposed method isable to uncover transitions of different origin even in presence of noise and for short time series which makes it broadlyapplicable to many real–world phenomena. We have further demonstrated that recurrence lacunarity may also be usefulin characterizing subtle transitions of different nature. In comparison to DET as a traditional recurrence quantifier, itshowed comparable well performance for noisy time series. Even though the results were less convincing for a shorttime series, RL performed superior for subtle transitions in the bistable system.Finally, we have employed a system exhibiting thermoacoustic instability to study whether our approach enables usto detect regime shifts in an (experimental) real world system. Our method has enables us to identify the differentdynamical transitions that the system undergoes. We found that the intermittency route from stable operation intothermoacoustic instability manifests itself as a continuous transition from aperiodic to harmonic oscillations. We haveultimately addressed the challenge of differentiating between the dynamically similar but practically different statesof stable operation and a near blowout situation. It appeared that short-term acoustic pressure fluctuations show lessvariable temporal recurrent patterns during a near-blowout situation than during regular operating mode. How this canbe interpreted and whether it can also be captured in terms of (nonlinear) serial dependence should be addressed infuture work.Furthermore, it appears as a promising direction for future work to investigate in more detail the relations between thefractal scaling of RPs and fractality of the underlying time series and the attractor dimension. Including RL in featureselection approaches may also improve the performance of machine learning techniques that classify nonlinear data[72]. Another line of research should be concerned with the robustness of the proposed method against spurious effectsintroduced by erroneous embedding and RP related pitfalls when compared to traditional RQA measures [34, 45].
Acknowledgements
This research was supported by the Deutsche Forschungsgemeinschaft in the context of the DFG project MA4759/11-1‘Nonlinear empirical mode analysis of complex systems: Development of general approach and application in climate’,the DFG project MA4759/9-1 ‘Recurrence plot analysis of regime changes in dynamical systems’ and it has receivedfunding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No820970 as a TiPES contribution. RIS acknowledges the Science and Engineering Research Board (SERB) of theDepartment of Science and Technology, Government of India for the funding under the grant Nos.: DST/SF/1(EC)/2006(Swarnajayanti Fellowship) and JCB/2018/000034/SSC (JC Bose Fellowship). VRU thanks University of CaliforniaSan Diego for the postdoctoral fellowship.
Conflict of interest
The authors declare that they have no conflict of interest.
References [1] A. Aballe, M. Bethencourt, F. Botana, and M. Marcos. Using wavelets transform in the analysis of electrochemicalnoise data.
Electrochimica Acta , 44(26):4805–4816, 1999.[2] M. Alber and J. Peinke. Improved multifractal box-counting algorithm, virtual phase transitions, and negativedimensions.
Physical Review E , 57(5):5489, 1998.[3] C. Allain and M. Cloitre. Characterizing the lacunarity of random and deterministic fractal sets.
Physical reviewA , 44(6):3552, 1991.[4] P. Ashwin, S. Wieczorek, R. Vitolo, and P. Cox. Tipping points in open systems: bifurcation, noise-induced andrate-dependent examples in the climate system.
Philosophical Transactions of the Royal Society A: Mathematical,Physical and Engineering Sciences , 370(1962):1166–1184, 2012.[5] P. Babinec, M. Kuˇcera, and M. Babincová. Global characterization of time series using fractal dimension ofcorresponding recurrence plots: from dynamical systems to heart physiology.
Harmon Fractal Image Anal ,1:87–93, 2005. 16etection of Dynamical Regime Transitions with Lacunarity as a Multiscale Recurrence Quantification Measure[6] E. Bacry, J. Delour, and J.-F. Muzy. Multifractal random walk.
Physical Review E , 64(2):026103, 2001.[7] E. Bradley and H. Kantz. Nonlinear time-series analysis revisited.
Chaos: An Interdisciplinary Journal ofNonlinear Science , 25(9):097610, 2015.[8] L. Cao. Practical method for determining the minimum embedding dimension of a scalar time series.
Physica D:Nonlinear Phenomena , 110(1-2):43–50, 1997.[9] Y. Chen and H. Yang. Heterogeneous recurrence representation and quantification of dynamic transitions incontinuous nonlinear processes.
The European Physical Journal B , 89(6):155, 2016.[10] Q. Cheng. Multifractal modeling and lacunarity analysis.
Mathematical Geology , 29(7):919–932, 1997.[11] G. Corso, T. d. L. Prado, G. Z. d. S. Lima, J. Kurths, and S. R. Lopes. Quantifying entropy using recurrencematrix microstates.
Chaos: An Interdisciplinary Journal of Nonlinear Science , 28(8):083108, 2018.[12] P. Dong. Test of a new lacunarity estimation method for image texture analysis.
International Journal of RemoteSensing , 21(17):3369–3373, 2000.[13] J. F. Donges, R. Donner, N. Marwan, S. F. Breitenbach, K. Rehfeld, and J. Kurths. Non-linear regime shifts inholocene asian monsoon variability: potential impacts on cultural change and migratory patterns.
Climate of thePast , 11(5):709–741, 2015.[14] J. F. Donges, R. V. Donner, K. Rehfeld, N. Marwan, M. H. Trauth, and J. Kurths. Identification of dynamicaltransitions in marine palaeoclimate records by recurrence network analysis.
Nonlinear Processes in Geophysics ,18(5):545–562, 2011.[15] R. V. Donner, J. Heitzig, J. F. Donges, Y. Zou, N. Marwan, and J. Kurths. The geometry of chaotic dynamics—acomplex network perspective.
The European Physical Journal B , 84(4):653–672, 2011.[16] J. Eckmann, S. O. Kamphorst, D. Ruelle, et al. Recurrence plots of dynamical systems.
World Scientific Series onNonlinear Science Series A , 16:441–446, 1995.[17] A. Fabretti and M. Ausloos. Recurrence plot and recurrence quantification analysis techniques for detecting acritical regime. examples from financial market inidices.
International Journal of Modern Physics C , 16(05):671–706, 2005.[18] R. Feagin, X. Wu, and T. Feagin. Edge effects in lacunarity analysis.
Ecological modelling , 201(3-4):262–268,2007.[19] J. Gaite. Fractal analysis of the large-scale stellar mass distribution in the sloan digital sky survey.
Journal ofCosmology and Astroparticle Physics , 2018(07):010, 2018.[20] L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni. Stochastic resonance.
Reviews of Modern Physics ,70(1):223, 1998.[21] V. Godavarthi, S. A. Pawar, V. R. Unni, R. I. Sujith, N. Marwan, and J. Kurths. Coupled interaction betweenunsteady flame dynamics and acoustic field in a turbulent combustor.
Chaos: An Interdisciplinary Journal ofNonlinear Science , 28(11):113111, 2018.[22] V. Godavarthi, V. R. Unni, E. Gopalakrishnan, and R. I. Sujith. Recurrence networks to study dynamical transitionsin a turbulent combustor.
Chaos: An Interdisciplinary Journal of Nonlinear Science , 27(6):063113, 2017.[23] A. V. M. Gomides, L. J. de Paula Gonçalves, L. R. Silva, and A. R. Backes. Lacunarity as a tool for analyzingsatellite images of urban areas. In , pages 307–311.IEEE, 2018.[24] B. Goswami, N. Boers, A. Rheinwalt, N. Marwan, J. Heitzig, S. F. Breitenbach, and J. Kurths. Abrupt transitionsin time series with uncertainties.
Nature communications , 9(1):48, 2018.[25] M. Ivanovici, N. Richard, and H. Decean. Fractal dimension and lacunarity of psoriatic lesions-a colour approach. medicine , 6(4):7, 2009.[26] E. M. Izhikevich.
Dynamical systems in neuroscience . MIT press, 2007.[27] R. Jacob, K. Harikrishnan, R. Misra, and G. Ambika. Measure for degree heterogeneity in complex networks andits application to recurrence network analysis.
Royal Society open science , 4(1):160757, 2017.[28] Y. P. Kalmykov, W. Coffey, and S. Titov. On the brownian motion in a double-well potential in the overdampedlimit.
Physica A: Statistical Mechanics and its Applications , 377(2):412–420, 2007.[29] J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde, and H. E. Stanley. Multifractaldetrended fluctuation analysis of nonstationary time series.
Physica A: Statistical Mechanics and its Applications ,316(1-4):87–114, 2002. 17etection of Dynamical Regime Transitions with Lacunarity as a Multiscale Recurrence Quantification Measure[30] H. Kantz. A robust method to estimate the maximal lyapunov exponent of a time series.
Physics letters A ,185(1):77–87, 1994.[31] A. Karperien, H. Jelinek, N. Milosevic, and P. Cracow. Reviewing lacunarity analysis and classification ofmicroglia in neuroscience. In , 2011.[32] A. L. Karperien and H. F. Jelinek. Box-counting fractal analysis: a primer for the clinician. In
The Fractalgeometry of the brain , pages 13–43. Springer, 2016.[33] P. Kasthuri, I. Pavithran, S. A. Pawar, R. I. Sujith, R. Gejji, and W. Anderson. Dynamical systems approach tostudy thermoacoustic transitions in a liquid rocket combustor.
Chaos: An Interdisciplinary Journal of NonlinearScience , 29(10):103115, 2019.[34] K. H. Kraemer and N. Marwan. Border effect corrections for diagonal line based recurrence quantification analysismeasures.
Physics Letters A , 383(34):125977, 2019.[35] A. Krishnan, R. I. Sujith, N. Marwan, and J. Kurths. On the emergence of large clusters of acoustic power sourcesat the onset of thermoacoustic instability in a turbulent combustor.
Journal of Fluid Mechanics , 874:455–482,2019.[36] G. Lancaster, D. Iatsenko, A. Pidde, V. Ticcinelli, and A. Stefanovska. Surrogate data for hypothesis testing ofphysical systems.
Physics Reports , 748:1–60, 2018.[37] C. Letellier. Estimating the shannon entropy: recurrence plots versus symbolic dynamics.
Physical review letters ,96(25):254102, 2006.[38] B. Lin and Z. Yang. A suggested lacunarity expression for sierpinski carpets.
Journal of Physics A: Mathematicaland General , 19(2):L49, 1986.[39] Y. Malhi and R. M. Román-Cuesta. Analysis of lacunarity and scales of spatial homogeneity in ikonos images ofamazonian tropical forest canopies.
Remote Sensing of Environment , 112(5):2074–2087, 2008.[40] N. Malik, N. Marwan, Y. Zou, P. J. Mucha, and J. Kurths. Fluctuation of similarity to detect transitions betweendistinct dynamical regimes in short time series.
Physical Review E , 89(6):062908, 2014.[41] N. Malik, Y. Zou, N. Marwan, and J. Kurths. Dynamical regimes and transitions in plio-pleistocene asian monsoon.
EPL (Europhysics Letters) , 97(4):40009, 2012.[42] B. B. Mandelbrot.
The fractal geometry of nature , volume 173. WH freeman New York, 1983.[43] R. N. Mantegna and H. E. Stanley.
Introduction to econophysics: correlations and complexity in finance .Cambridge university press, 1999.[44] N. Marwan. A historical review of recurrence plots.
The European Physical Journal Special Topics , 164(1):3–12,2008.[45] N. Marwan. How to avoid potential pitfalls in recurrence plot based data analysis.
International Journal ofBifurcation and Chaos , 21(04):1003–1017, 2011.[46] N. Marwan, D. Eroglu, I. Ozken, T. Stemler, K.-H. Wyrwoll, and J. Kurths. Regime change detection in irregularlysampled time series. In
Advances in Nonlinear Geosciences , pages 357–368. Springer, 2018.[47] N. Marwan, M. C. Romano, M. Thiel, and J. Kurths. Recurrence plots for the analysis of complex systems.
Physics reports , 438(5-6):237–329, 2007.[48] N. Marwan, P. Saparin, and J. Kurths. Measures of complexity for 3d image analysis of trabecular bone.
TheEuropean Physical Journal Special Topics , 143(1):109–116, 2007.[49] N. Marwan, S. Schinkel, and J. Kurths. Significance for a recurrence based transition analysis. In
Proceedingsof the 2008 International Symposium on Nonlinear Theory and its Applications NOLTA08, Budapest, Hungary ,pages 412–415, 2008.[50] N. Marwan, S. Schinkel, and J. Kurths. Recurrence plots 25 years later—gaining confidence in dynamicaltransitions.
EPL (Europhysics Letters) , 101(2):20007, 2013.[51] N. Marwan, N. Wessel, U. Meyerfeldt, A. Schirdewan, and J. Kurths. Recurrence-plot-based measures ofcomplexity and their application to heart-rate-variability data.
Physical review E , 66(2):026702, 2002.[52] S. Michael.
Applied nonlinear time series analysis: applications in physics, physiology and finance , volume 52.World Scientific, 2005.[53] V. Nair, G. Thampi, and R. I. Sujith. Intermittency route to thermoacoustic instability in turbulent combustors.
Journal of Fluid Mechanics , 756:470–487, 2014.[54] E. Ott.
Chaos in dynamical systems . Cambridge university press, 2002.18etection of Dynamical Regime Transitions with Lacunarity as a Multiscale Recurrence Quantification Measure[55] L. M. Pecora, L. Moniz, J. Nichols, and T. L. Carroll. A unified approach to attractor reconstruction.
Chaos: AnInterdisciplinary Journal of Nonlinear Science , 17(1):013110, 2007.[56] R. E. Plotnick, R. H. Gardner, W. W. Hargrove, K. Prestegaard, and M. Perlmutter. Lacunarity analysis: a generaltechnique for the analysis of spatial patterns.
Physical review E , 53(5):5461, 1996.[57] H. Poincaré. Sur le problème des trois corps et les équations de la dynamique.
Acta mathematica , 13(1):A3–A270,1890.[58] T. d. L. Prado, S. Lopes, C. Batista, J. Kurths, and R. Viana. Synchronization of bursting hodgkin-huxley-typeneurons in clustered networks.
Physical Review E , 90(3):032818, 2014.[59] Y. Quan, Y. Xu, Y. Sun, and Y. Luo. Lacunarity analysis on image patterns for texture classification. In
Proceedingsof the IEEE conference on computer vision and pattern recognition , pages 160–167, 2014.[60] M. Scheffer, J. Bascompte, W. A. Brock, V. Brovkin, S. R. Carpenter, V. Dakos, H. Held, E. H. Van Nes,M. Rietkerk, and G. Sugihara. Early-warning signals for critical transitions.
Nature , 461(7260):53, 2009.[61] D. Smirnov, S. Breitenbach, G. Feulner, F. Lechleitner, K. Prufer, J. Baldini, N. Marwan, and J. Kurths. A regimeshift in the sun-climate connection with the end of the medieval climate anomaly.
Scientific reports , 7(1):11131,2017.[62] R. I. Sujith and V. R. Unni. Complex system approach to investigate and mitigate thermoacoustic instability inturbulent combustors.
Physics of Fluids , 32(6):061401, 2020.[63] F. Takens. Detecting strange attractors in turbulence. In
Dynamical systems and turbulence, Warwick 1980 , pages366–381. Springer, 1981.[64] R. J. Tibshirani and B. Efron. An introduction to the bootstrap.
Monographs on statistics and applied probability ,57:1–436, 1993.[65] J. Tony, E. Gopalakrishnan, E. Sreelekha, and R. I. Sujith. Detecting deterministic nature of pressure measurementsfrom a turbulent combustor.
Physical Review E , 92(6):062902, 2015.[66] V. R. Unni, A. Krishnan, R. Manikandan, N. B. George, R. I. Sujith, N. Marwan, and J. Kurths. On the emergenceof critical regions at the onset of thermoacoustic instability in a turbulent combustor.
Chaos: An InterdisciplinaryJournal of Nonlinear Science , 28(6):063125, 2018.[67] V. R. Unni and R. I. Sujith. Multifractal characteristics of combustor dynamics close to lean blowout.
Journal ofFluid Mechanics , 784:30–50, 2015.[68] N. Valous, W. Xiong, N. Halama, I. Zörnig, D. Cantre, Z. Wang, B. Nicolai, P. Verboven, and R. Rojas Moraleda.Multilacunarity as a spatial multiscale multi-mass morphometric of change in the meso-architecture of plantparenchyma tissue.
Chaos: An Interdisciplinary Journal of Nonlinear Science , 28(9):093110, 2018.[69] J. G. Venegas, T. Winkler, G. Musch, M. F. V. Melo, D. Layfield, N. Tgavalekos, A. J. Fischman, R. J. Callahan,G. Bellani, and R. S. Harris. Self-organized patchiness in asthma as a prelude to catastrophic shifts.
Nature ,434(7034):777, 2005.[70] C. Webber. Recurrence quantification of fractal structures.
Frontiers in physiology , 3:382, 2012.[71] C. L. Webber Jr and N. Marwan. Recurrence quantification analysis.
Theory and Best Practices , 2015.[72] Q. Ye, Y. Xia, and Z. Yao. Classification of gait patterns in patients with neurodegenerative disease using adaptiveneuro-fuzzy inference system.