Recurrence Network Analysis of Exoplanetary Observables
RRNA in planetary dynamics
Recurrence Network Analysis of Exoplanetary Observables
Tamás Kovács a) Institute of Physics, Eötvös University, Pázmány P. s. 1A, Budapest 1117, Hungary (Dated: 7 August 2019)
Recent advancements of complex network representation among several disciplines motivated the investigation of exo-planetary dynamics by means of recurrence networks. We are able to recover different dynamical regimes by means ofvarious network measures obtained from synthetic time series of a model planetary system. The framework of complexnetworks is also applied to real astronomical observations acquired by recent state-of-the-art surveys. The outcome ofthe analysis is consistent with earlier studies opening new directions to investigate planetary dynamics.
The dynamical variety of known extrasolar planetary sys-tems stimulates their stability analysis as an importantcharacteristic of the system. The common procedure toperform such an investigation is to obtain the parametersof the best-fit planetary model and then integrate the equa-tions of motion numerically. In this work we propose amethod in order to describe system dynamics that is basedon the observed uni-variate time series and the topologyof complex network reconstructed from the signal itself.We show that the procedure is capable to distinguish theordered and chaotic motion in a synthetic 2-planet systemwith a given level of significance. We also find that differ-ent kind of data sets, produced by the currently used de-tection methods, works well within the framework of themethod. The analysis of real-world observations also pro-vides results consistent with former studies. The techniqueis computationally very efficient since it does not requirethe phase space trajectories and, therefore, costly n-bodysimulation can be avoided.
Thus this new strategy can beused as a complementary tool to extract the dynamical be-havior of extrasolar planetary systems.
I. INTRODUCTION
Dynamical stability is one of the most relevant physicalcharacteristics (beside the bulk density, atmospheric composi-tion, interiors of a planet, etc.) to describe extrasolar planetarysystems. In order to study the dynamical evolution of a systemthe initial conditions of the numerical integration are essential.In planetary science this input includes the orbital elementsfor a given epoch. Nowadays the ground and space-based ob-servations provide a large amount of extremely precise radialvelocity (RV) and transit timing data. These information canbe transformed by using comprehensive statistical methods into the desired initial conditions.The majority of known exoplanetary systems harbourmore than one planet resulting in complex non-Kepleriandynamics . The possible instability of the system is an in-ternal attribute of the deterministic dynamics when three ormore celestial objects participate in the motion. To explore a) Electronic mail: [email protected] the chaotic nature one has to have the integrated trajecto-ries and/or the structure of the relevant part of the phasespace assisted by a chaos detection method (Lyapunov expo-nent, MEGNO ( M ) – Mean Exponential Growth of NearbyOrbits , etc.). These requirements can be fulfilled by havingthe best-fit planetary model incorporating the orbital elementsand the masses as it has already been shown . The resultsof these studies indicate the heterogeneity of exoplanetary dy-namics. Interestingly, there are also candidates with chaoticproperties beside the larger number of stable resonant con-figurations.We propose an alternative method to perform stability anal-ysis of exoplanetary systems that requires only a scalar timeseries of the measurements, e.g. RV, transit timing variation(TTV) , or astrometric positions. The fundamental concept ofPoincaré recurrences in closed Hamiltonian systems and thepowerful techniques of nonlinear time series analysis com-bined with complex network representation allow us to in-vestigate the underlying dynamics without having the equa-tions of motion. That is we can ignore the orbital elementsas initial conditions and other system parameters from the en-tire analysis. Moreover, the procedure completely disregardsthe use of numerical n-body integration which is an essen-tial and sometimes fairly time consuming part of the stabilityanalysis. We present that this new scheme, first time appliedin exoplanetary research, works well with signals acquired bythe currently available observational techniques (see above). II. METHOD
A scalar time series carries relevant dynamical informationif the measured physical quantity is coupled to other statespace variables of the system. Based on this fact Takens’ em-bedding theorem ensures that the phase space trajectory canbe reconstructed from the measured data. The procedure iscalled time delay embedding and requires two parameters, thetime lag ( τ ) and the embedding dimension ( d ). There are stan-dard techniques to find the appropriate values of these param-eters, e.g. false nearest neighbours help to estimate the em-bedding dimension while the first minimum of the mutualinformation function yields the time delay .Once the reconstructed trajectory is ready to use, one cansearch for Poincaré recurrences, the events when the trajec-tory x comes back at time t j into an ε -ball around its earlierposition at time t i ( t i < t j ). The idea of recurrence plot (RP) a r X i v : . [ a s t r o - ph . E P ] A ug NA in planetary dynamics 2
FIG. 1. Stability and RN measures for the SJS system. (a) Stability map ( a Sat , e Sat ) of two-planet system according to the indicator M . Semimajor axis is measured in astronomical units (au). The green area denotes the stable realm while for larger eccentricities the dynamicsis chaotic (red). The dominant low-order mean motion resonances (MMR) are indicated at the top of the panel. Blue triangles are referencetrajectories from different dynamical regimes for further analysis. The blue dot represents the Saturn’s current location in the parameter plane.(b) and (c) Two RN measures L , T are pictured as the results of RNA on the same grid as in (a) taking into account two observables TTV ofJupiter and RV of the Sun, respectively. The color bars illuminate the heat map values in each case. is a 2D binary matrix representation of these events. Moreprecisely, the matrix element R i , j ( ε ) = || x j − x i || < ε , and 0 otherwise. The || . || can denote various norms suchas Euclidean, Maximum, Manhattan. In this work the Max-imum norm is used through the whole analysis. The R ma-trix is symmetric by its nature. First Refs. proposedthe method of recurrence quantification analysis (RQA) whichidentifies various measures based on the diagonal and hori-zontal/vertical texture of an RP. The advantage of RQA in dy-namical analysis has been demonstrated in many applicationsfrom climatology through neuroscience to astrophysics. Theidea of recurrence networks (RN) revolutionized the RPrepresentation of dynamical systems using the fact that an RPcan be thought of as the adjacency matrix of a complex net-work embedded in phase space. Quantitatively A i , j = A i , j ( ε ) = R i , j ( ε ) − δ i j , (1)where A i , j is the adjacency matrix, R i , j the recurrence plot,and δ i j the Kronecker delta. The role of δ i j is to exclude theloops. The nodes of these graphs are the state space vectorsthat are connected by edges if their proximity is smaller thana threshold ε . The main advantage of the complex networkframework is that the temporal correlations can be avoidedsince the dynamical feature of the underlying system is pre-served via its topology, thus, the explicit time ordering is notrelevant. The standard quantities known in graph theory (suchas degree centrality, average path length, transitivity, etc.) arealso applicable to RNs and they are capable to describe differ-ent dynamical regimes and dynamical transitions . This factwill be utilized in the following. III. RESULTSA. The model system
First a well-defined model is investigated in order to presentthe reliability of the method. Consider the Sun-Jupiter-Saturn(SJS) two-planet system in our own Solar System with their
FIG. 2. CDF differences between the chaos indicator M and RNmeasure T RV in the parameter plane ( a Sat , e Sat ). actual orbital elements. One can explore, for example withthe REBOUND package , the stability of the system by vary-ing, say, Saturn’s initial semimajor axis ( a Sat ) and eccentricity( e Sat ) in a grid and quantify the dynamics. Figure 1(a) showsthe result of dynamical stability of the system in the ( a Sat , e Sat )initial condition plane for 100x100 different trajectories char-acterized by the chaos indicator MEGNO ( M ). The valuesof M ≈ a Sat ≈ . , Kepler-412 , make thousands of revolutions on humantime scales producing time series with desired length to beanalyzed: ∼ TABLE I. The embedding parameters of various time series in SJS system used to construct RNs. Initial conditions in Fig 1(a) are consideredwith two different length, the original 950 data point signals and longer one containing 3500 measurements. The time delay parameters d , τ , and ε are the embedding dimension, time lag, and threshold, respectively. The threshold in each case is obtained from recurrence matrix withfixed RR = . . Regular Chaotic ResonantData 950 3500 950 3500 950 3500RV-Syn a d τ ε . e − . e − . e − . e − . e − . e − b d
10 11 14 13 11 11 τ ε . e − . e − . e − . e − . e − . e − d τ ε . . .
05 6814 .
92 340 .
35 1019 . d τ ε .
05 53 .
51 634 .
87 987 .
25 47 .
38 374 . a Synthetic data are obtained directly from numerical integration. b Spline interpolation has been made on synthetic (Syn) time series after adding noise and removing some data points as described in the text.
In order to apply recurrence network analysis (RNA) to SJSsystem, synthetic scalar time series were acquired from thefull phase space trajectories in the following way. Radial ve-locity data is the x-component of the Sun’s velocity vectorwhile transits are taken when the planet crosses the positivex-axis imitating the position of a real occultation. In caseof the RV signals the trajectories were integrated until 1050Jupiter orbits and 950 data points were sampled in order toset the sampling frequency larger than the mean motion of theJupiter. For transits exactly 950 events have been taken. Thisalso implies that the trajectories examined via transits havesomewhat different length, namely, 950 Jupiter transits mighttake shorter or longer time in different dynamical regimes.Figure 1(b) and (c) depict the network measures averagepath length ( L ) for TTV signals and transitivity ( T ) for RVdata, respectively, in the ( a Sat , e Sat ) parameter plane. Thephase space reconstruction and the network analysis has beendone by the
TISEAN and PYUNICORN packages, respec-tively. The similarity is evident between the panels (a) and(b), (c). In panel (b) for TTV signals the larger the L , themore chaotic the dynamics. This result is in good agreementwith general findings in discrete Hamiltonian systems like thestandard map . The RNA of RV time series produces similarresults for the measure T , panel (c). In this case transitiv-ity closer to 1 yields weaker instability according to the gen-eral picture that regular trajectories show smaller divergencein phase space. This fact results in a more coherent topologyof the network that produces higher transitivity. One can alsonotice that the stability map of TTVs is more sensitive to theedge of chaos inducing more details between the green andred domains. This is also important because weak chaos, alsoknown as stickiness in Hamiltonian dynamics, emerges justat the edge of stability islands. The stickiness effect was alsoidentified by using complex network measures in low dimen-sional conservative systems . The method of RNA appliedto single variable time series is, thus, able to reconstruct the stability map what is obtained from numerical integration andphase space trajectories. Moreover, it is also remarkable thatachieving this outcome the length of the measured data mustnot be longer than that of the phase space trajectory.Table I contains the embedding parameters for the referencetrajectories introduced in Figure 1. In addition, time delay em-bedding has been done for longer time series containing 3500data points in all three cases. This analysis verifies that 950RV and TTV measurements provide a reliable RNA. In orderto demonstrate that RN measures ( L , T ) are suitable to dis-tinguish regular and chaotic orbits, further statistical analysesare performed.First, a point-wise difference based on the cumulative dis-tribution functions (CDF) of M and ( L , T ) in the ( a Sat , e Sat )parameter plane is designed ∆ P ( M , x ) = P ( M ) − P ( x ) , (2)where P ( x ) , x ∈ { L RV , T RV , L TTV , T TTV } , is the correspond-ing value of CPD at each combinations of 10000 ( a Sat , e Sat )pairs. An example of CDF differences, ∆ P ( M , T ) , for RVdata is depicted in Figure 2. The difference is close to zeroalmost in the entire parameter plane. Remarkable deviationfrom zero can be observed in chaotic region.Another more sophisticated quantitative comparison of RNmeasures can be obtained as follows. Let us define two dis-joint subsets of MEGNO distribution defined by a criticalvalue of M ∗ S ( M ∗ ) : = { ( a , e ) | M ( a , e ) ≤ M ∗ } , S ( M ∗ ) : = { ( a , e ) | M ( a , e ) > M ∗ } , (3)with group size n and n = n − n ( n = α -quantile Q α ( M ) with α = n / n of the distribution M for a given M ∗ and constructthe same division for the corresponding ( a Sat , e Sat ) parametersNA in planetary dynamics 4
FIG. 3. (a) Distribution of chaos indicator MEGNO using the 100x100 grid of ( a , e ) parameter plane. (b) and (c) Two particular subsets of S and S (cid:48) obtained from the same quantile defined by M ∗ = . FIG. 4. The relative frequency of false detection of various charac-teristics vs. M ∗ based on the same quantile. based on various RN measures S (cid:48) ( Q α ( x )) : = { ( a , e ) | x ( a , e ) ≤ Q α ( x ) } , S (cid:48) ( Q α ( x )) : = { ( a , e ) | x ( a , e ) > Q α ( x ) } , (4)where x again incorporates certain network characteristics.This analysis allows one to compare the difference of two dis-tributions based on distinct measures. Figure 3(b) and (c) de-pict one particular example of two subsets of ( a Sat , e Sat ) pairsassociated to S ( M ∗ = ) and S (cid:48) ( Q α ( T RV )) , α ≈ . . The relative frequency p of those parameter pairs that donot belong to the same group based on the two different mea-sures indicate the false detection of dynamical nature. Slightlyvarying the chaos indicator close to the border of regular andchaotic feature, 2 ≤ M ∗ ≤ , the relative frequency of ”group-ing errors” can be quantified. Figure 4 collects the frequenciesof the average path length and transitivity obtained from RVand TTV signals representing that the classification error re-mains under 10% in all cases.Although, the RN measures give qualitatively reasonablesuccess for either observables, it is clear that their values de-pend on the choice of the threshold ε . To avoid this weak-ness two possibilities are known. First, one can derive dy-namical invariants (e.g. Rényi entropy, correlation dimension,maximal Lyapunov exponent) from recurrence plots. How- ever, to get reliable feedback about the dynamics by these in-variants one has to have much longer time series. And thisis against our will, to make stability analysis based on realobservables that contain only several hundreds/thousands ofdata points. The second option is to perform hypothesis testswith surrogate time series . Since in planetary dynamicsthe signals show predominantly quasi-periodic variations, thenull hypothesis in surrogation method we want to test shouldbe that the original data comes from quasi-periodic process.Pseudo-periodic twin surrogates (PPTS) combine the powerof Pseudo-periodic and Twin surrogate methods and fitperfectly to test quasi-periodicity in planetary dynamics.More concretely, time series generated by the PPTS methodprovide different RP matrices for periodic/quasi-periodic or-bits and the chaotic ones preserving the phase space structure.This ensures that these surrogates are relevant to test the nullhypothesis that the original signal comes from quasi-periodicprocess. The question, however, still remains open: What arethe dynamical properties of the underlying system in case ofrejecting the null hypothesis. In view of the fact that plan-etary dynamics is deterministic, high dimensional and defi-nitely nonlinear and by declining the null hypothesis of PPTS,meaning that the motion is not quasi-periodicity, the dynamicsshould be chaotic according to certain confidence level.The hypothesis tests in Figure 5 comprise also the robust-ness of the RNA against measurement noise and missing datapoints. In order to reproduce a realistic astronomical observa-tion, first Gaussian white noise with zero mean and unit vari-ance is superimposed onto the synthetic time series in the SJSsystem with Signal-to-Noise ratio ≈ ,spline interpolation is performed on the unevenly spaced databefore the time delay embedding and RNA is applied.To obtain a 1% level of significance for a one-sided test 99surrogate time series must be generated and then the RNmeasures calculated from the original time series has to becompared to those acquired from the surrogates. Usually wedo not know whether the distribution of RN measures is Gaus-sian, therefore, a rank-based statistics is preferred instead ofNA in planetary dynamics 5 FIG. 5. Outcome of hypothesis tests in three different dynamical regimes. Panels (a)-(c) show the results for L based on TTV signals. Panels(d)-(f) present the measures T in case of RV data sets. The rank-based statistics involves noisy and irregularly sampled reference trajectories(red solid line) and 100 PPT surrogates (blue triangles). The green dashed lines mark the ± a normal. A rank-based statistics of the RNA specifies eitherto keep or reject the null hypothesis. Figure 5 illustrates thehypothesis tests for three different kind of dynamics markedby blue triangles in Fig. 1(a), and two observables (RV, TTV)in the SJS system. Let us concentrate on the first column,panels (a) and (d). The analysis here is devoted to show thatthe leftmost blue triangle ( a Sat , e Sat )=(7.2,0.02) corresponds toregular quasi-periodic motion. The red solid lines representthe RN measures L and T gained from the synthetic timeseries while the blue triangles symbolize the same RN mea-sures based on the generated 100 PPTSs. Both plots indicatethat L and T associated to the original time series fall intothe ensemble of blue triangles. Thus, one can keep the nullhypothesis, i.e. the original signal comes from quasi-periodicdynamics.Due to strong gravitational perturbation taking place pri-marily at large eccentricities the system is destroyed, i.e. one of the planets escapes. Therefore, an initial condition closeto the border of the regular part has been considered wherethe motion is chaotic while bounded for the integration time,( a Sat , e Sat )=(8.0,0.2). Executing the RNA one finds the resultsportrayed in panels (b) and (e). The scheme is the same asbefore, in turn, it can be clearly seen that the red lines arelocated outside the blue triangle zoo suggesting that the nullhypothesis can be rejected. That is, the original time serieswas generated by chaotic dynamics. For the RNA we pick upa third, resonant, initial condition in the stability map at po-sition ( a Sat , e Sat )=(9.6,0.4). The reassuring results of the hy-pothesis tests are summarized in panels (c) and (f).In order to verify that the spline interpolation does not causeany artificial effect during time delay embedding, the sameanalysis has been made on synthetic, i.e. noiseless uniformlysampled, signals as well. Figure 6 presents the outcome ofhypothesis tests that perfectly match those based on noisy andNA in planetary dynamics 6
FIG. 7. Transit timing variation of Kepler-36b and c. The two sig-nals are in anti-phase as is expected from the dynamical considera-tions. Upper panel: Kepler-36b the inner planet with smaller mass(0 . M Jupiter ) and consequently larger TTV amplitude, 72 mea-surements during 103 epochs yield 30% missing data points. Lowerpanel: Kepler-36c the more massive outer planet (0 . M Jupiter )with better coverage, 77 observations out of 89 epochs, 15% missingdata. scanty analysis in Figure 5.
B. Real-world measurements
Recently the number of known extrasolar planetary sys-tems outstandingly increased due to the cutting-edge technol-ogy. Extremely precise light curves with transit timing mea-surements might shed light on the dynamical diversity of theobserved systems as shown above. The catalog of the full-cadence data set of the Kepler mission contains a large num-ber of dynamically interesting systems especially from theTTV point of view. Some of them have been extensively stud-ied by means of stability . One of these systems is Kepler-36 wherein two planets (Kepler-36b and c) orbiting the cen-tral Sun-like star nearly in 7:6 MMR. The orbital separationof the two ”Super-Earths” is fairly small, 0.013 au. Due to thetightly packed configuration, the mutual gravitational interac-tion causes large TTV, see Figure 7. The irregular dynamicsof the system was proposed first by Deck et al. , further anal-ysis revealed that the system might be close to the border ofthe 7:6 MMR and the dynamics is governed by the stickinessof the resonance.Recurrence network analysis has been carried out for theTTV data of Kepler36b and c. It should also be noted that thedata points in Kepler-36 time series are less with a factor of8-10 than those in the SJS system. That is, the stability analy-sis is based on ∼
100 orbital periods of the planets. Moreover,the number of missing transit points is somewhat larger thanin the synthetic SJS, roughly 13% for inner planet and 25%for the outer one. The 99% significance level hypothesis testinvolving RN measures T and L is presented in Figure 8 and FIG. 8. Hypothesis tests including RN measures T for both planetsin Kepler-36 systems. The embedding parameters are d = , τ = , and ε = . The red solid lines represent the transitivity valuesobtained from the original signals, T b = . , T c = . . The 100surrogates, blue points, permit one-sided 99% level of significance.The null hypothesis can be rejected in case of Kepler-36b as the rankbased statistics suggests.
9. Results depicted in Fig. 8 lower panel show that in the caseof the more massive outer planet the dynamics is regular. Incontrast, for the inner planet, upper panel, the null hypothesiscan be rejected in accordance with the rank-based statisticsdescribed above. The outcome of the RNA implies that thesystem’s behaviour is irregular. However, average path lengthfor the same system in Fig. 9 stipulates regular dynamics forboth planets. Comparing the results with Figure 1, the statisti-cal description of Kepler-36 fits to the picture of stable chaosappearing at the border of the resonances. Moreover, it alsosupports the view published in based on different methodsof dynamical analysis.
IV. CONCLUSION
We conclude that the RNA method is suitable to reproducethe dynamical behaviour of a planetary system with reliablesignificance. This result is based on the fact that the topologyof recurrence networks preserves the underlying dynamics ofthe system producing the time series under study. Thus, theRN measures, L and T , are adequate to distinguish regularand chaotic nature of the motion. However, one should keep inmind that these measures are geometric characteristics ratherthan dynamical invariants of the dynamics. Therefore, theabove mentioned contrast between quasi-periodic and chaoticbehavior can only be made by appropriate hypothesis tests.Further advantage of the present idea is that it uses directlythe measured data set and requires neither Monte-Carlo simu-lation to achieve the best-fitting planetary model nor costly n-body integration. Consequently, the operation needs substan-tially shorter time to achieve the result. For example, the anal-ysis of a time series with 950 data points in addition with theNA in planetary dynamics 7 FIG. 9. The same as in Figure 8 The embedding parameters are d = , τ = , and ε = . . The average path length values obtainedfrom the original signals, L b = . , L c = . . According tonull hypothesis the dynamics is quasi-periodic.
100 surrogates requires no more than 10 minutes on a mediumdesktop machine.We emphasize that 950 measurements for both signals (RV,TTV) seem to be a sufficient amount of data to restore thedynamical operation given by the same length of numericalintegration. Extending the data set with the length of the inte-gration time makes change to the embedding parameters, nev-ertheless, the recurrence network analysis provides the sameresults for longer signals. It has to be noted that this methoddepends on the measured time series which is the past of theplanetary dynamics and has a limited length evidently shorterthan those obtained from numerical integration. Therefore,the conclusions drawn from the analysis should be treated inplace and interpreted correctly, especially taking into accountthe time scales.Moreover, the future surveys will serve further measure-ments to the existing ones generating longer time series thatare going to be the basis of more trusty dynamical analysis.The proposed scheme can be generalized to more than twoplanets in a system , thus, it can be used as a supportivemethod to the current efficient stability investigations. ACKNOWLEDGMENTS
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