Strange nonchaotic stars John F. Lindner,1,2 Vivek Kohar,1 Behnam Kia,1 Michael Hippke,3 John G. Learned,1 and William L. Ditto1 1Department of Physics and Astronomy, University of Hawai‘i at M¯anoa Honolulu, Hawai‘i 96822, USA 2Physics Department, The College of Wooster, Wooster, Ohio 44691, USA 3Institute for Data Analysis, Luiter Straße 21b, 47506 Neukirchen-Vluyn, Germany (Dated: February 5, 2015) The unprecedented light curves of the Kepler space telescope document how the brightness of somestarspulsatesatprimaryandsecondaryfrequencieswhoseratiosarenearthegoldenmean,the most irrational number. A nonlinear dynamical system driven by an irrational ratio of frequencies generically exhibits a strange but nonchaotic attractor. For Kepler’s “golden” stars, we present evidence of the first observation of strange nonchaotic dynamics in nature outside the laboratory. 5 This discovery could aid theclassification and detailed modeling of variable stars. 1 0 PACSnumbers: 05.45.Tp,05.45.Df,97.10.Sj,95.75.Wx 2 b e As a byproduct of its vastly successful search for ex- or nonchaotic motion on strange attractors. Although F oplanets [1–5], the Kepler spacecraft has revolutionized quasiperiodic forcing is not necessary for strange non- 4 stellar photometry by its high precision tracking of the chaotic dynamics, the first experiment to demonstrate brightness of 150 000 stars nearly continuously for four a strange nonchaotic attractor [12] involved the buck- ] years. Both the quantity and quality of the Kepler data D enable new stellar discoveries. The Kepler stars include C cosmic distance markers (or standard candles) such as a . n Cepheid and 41 RR Lyrae variable stars. While some of i these stars pulsate with a single frequency, Kepler ob- l n servations confirm that others pulsate with multiple fre- [ quencies. Severalofthesestars,includingtheRRcLyrae 2 starKIC5520878,pulsatewithtwoprincipalfrequencies, v which are nearly in the golden ratio [6]. As the most ir- 7 rational number, the golden ratio can have significant 4 dynamicalconsequences. Forexample,bythe KAMthe- 7 orem, dynamics with two frequencies in the golden ratio 1 0 maximally resistperturbations [7]. Furthermore, nonlin- . ear systems driven by two incommensurate frequencies, 1 forming an irrational ratio [8], and especially the golden 0 5 ratio, exhibit distinctive dynamics that exist between 1 order and chaos. Strange (because fractal) nonchaotic : attractors characterize this distinctive dynamics [9, 10]. v i Here we report evidence for strange nonchaotic behav- X ior in the pulsations of stars like KIC 5520878, making r this golden star the prototype of a new class. Strange a nonchaotic attractors have only been seen in laboratory experiments but never before in nature. The observa- tion of stellar strange nonchaotic dynamics provides a new window into variable stars that can improve their classification and refine the physical modeling of their interiors [11]. Periodically driven damped nonlinear systems can ex- hibit complexdynamicscharacterizedby strangechaotic attractors, where strange refers to the fractal geome- try of the attractor and chaotic refers to the exponen- FIG. 1. Spectral content. (a) Raster pattern of disks with tial sensitivity of orbits on the attractor. Quasiperiodi- brightness proportional to the KIC 5520878 stellar flux at cally driven systems forced by incommensurate frequen- intervals of δt≈ 1766 s≈0.5 h. (b) Fourier transform mag- cies are natural extensions of periodically driven ones nitudeofstellar fluxsampled at intervalsδthasprimaryand and are phenomenologically richer. In addition to pe- secondary frequencies (red arrows) at f1 ≈ 1/(0.266 d) and riodic or quasiperiodic motion, they can exhibit chaotic f2 ≈1/(0.169 d),wheref2/f1 ≈1.58≈ϕisthegolden ratio. 2 FIG.2. Attractorreconstruction. Two viewsofathree-dimensionalplot ofnormalized fluxFN att=nδtsuccessively delayed byτ =2δt. Thisdelaycoordinateembedding“unfolds”thetimeseriesintoawarpedtorus,suggestingtwo-frequencynonlinear dynamics. Equal-sized spheres locate data. Rainbow colors code time, from red to violet. Flux triplets straddling data gaps appear far from torus. ling of a magnetoelastic ribbon driven quasiperiodically addition, events such as cosmic ray strikes, monthly sci- by two incommensurate frequencies in the golden ratio ence data downlinks, and safe modes caused both small f /f = ϕ = (1 + √5)/2 1.62, which is the irra- andlargegapsinthe data. The bestdetrending practice 1 2 ≈ tional number least well approximated by rational num- for planet searches [15] cannot be employed for variable bers, as its continued fraction expansion ϕ= 1+1/ϕ= stars,suchasKIC5520878,becauseeachpulsationisin- 1+1/(1+1/(1+ ))isall1s. Hereweapplythetimese- trinsically different. Instead, we manually processed the ··· riesanalysistechniquesofthe ribbonexperimentto ana- raw time series F by detrending and rescaling the data lyzethedynamicsofseveralKepler multi-frequencyvari- segments to zero mean and unit variance to obtain a re- able stars,including the RRc Lyrae star KIC 5520878,a fined or normalized time series FN. blue-white star16 000lightyearsfromEarthinthe con- A natural strategy to test the flux time series for stellation Lyra whose brightness varies as in Fig. 1(a). strange nonchaotic behavior is to estimate both the di- MostclassicalCepheidsandRRLyraearesingle-mode mensionofthe underlying attractorandthe largestLya- periodic and radial pulsators. For many years, the only punov exponent of the orbits. A fractional dimension knownexceptionwaslong-termBlazhkolightcurvemod- would imply fractal geometric scaling and strangeness, ulationfoundamongsomeRRLyrae[13]. Recently,bet- while a positive Lyapunov exponent would imply the terqualityphotometryhasuncoveredvariousmulti-mode divergence of nearby orbits and chaos [7]. Although pulsationswithinthesevariablestars,possiblyfromnon- these metrics are difficult to estimate [16], we investi- radial modes. A few dozen such stars have been docu- gated both. mented [14] for mysterious behavior involving frequency For the fractal geometry [17], we calculated the infor- ratios near 0.62, apparently without connection to the mation dimension, which typically delivers the most ac- goldenratioϕ 1/0.62orthepossibilityofstrangenon- curatedimensionestimatesforshortnoisytimeseries[18] ≈ chaotic behavior. We refer to the former stars as golden by quantifying how the information needed to specify a stars and to the latter stars as strange nonchaotic stars, setof points scales with the area thatcontains them. To and we recognize that these classes may overlap. estimate it, wepartitionedasectionofthe attractorinto We downloaded the Kepler Input Catalogue 5520878 manytinyboxes,computedthefractionofpointsineach data from the Mikulski Archive for Space Telescopes. box,andcalculatedthecorrespondinginformationoren- The light curve, or flux F (in detector electrons per sec- tropy. The logarithmic scaling of information with slope ond)versustimet(indays),wassampledeveryδt=1766 d 1.8 in the range 1 < d < 2 is consistent with some ∼ seconds or about once each half hour, for most of four fractalstructure. (Incontrast,forpurequasiperiodicmo- years, including over 5000 primary periods. To keep its tion,thesectionisasinusoidalcurveofdimensiond=1, solar cells in sunlight, the Kepler spacecraft rotated 90◦ and for completely noisy data, the section is a uniform every quarter solar orbit causing the star to illuminate area of dimension d=2.) different pixels of its charge-coupled device. Different For the largest Lyapunov exponent [17], we imple- pixels have different responses and appear to have con- mentedboththeWolf[19]andRosenstein[20]algorithms tributed to shifts and skews in the data segments. In and found the latter more accurate and stable when 3 (a) (b) r m e b u m r t u c e n p s frequency threshold FIG.3. Spectralscaling. (a)FouriertransformmagnitudeofstellarfluxstrobedattheprimaryperiodT1hasawidedistribution ofpeaks. Peaksabovethresholdσ (horizontalline)arehighlighted(gold). (b)Log-logplotofnumberofsuper-thresholdpeaks N versusthethresholdσ hasalinearregime(inlegbelowkneeofcurve)thatindicatespowerlawscalingandfractalstructure. Spectral exponent slope −2 < s < −1 indicates strange nonchaotic dynamics. Deviation for low thresholds arises from light curve’s finitetemporal resolution. tested on known time series. Beginning with the Fig. 2 strange nonchaotic experiment [12], we strobed the data attractor reconstruction,we found the nearest neighbors attheprimaryperiodT andwrappeditbythesecondary 1 of all points not too close together. We then computed period T to produce the section. To qualitatively check 2 theaverageseparationofallneighboringpointsasafunc- thesectiondynamics[17],wetrackedthenormalizedflux tionofiteration. Alinearincreaseinthelogarithmofthe differences∆FN =FN[(n1+n)T1] FN[(n2+n)T1]oftwo − average separation versus iteration number would have points n and n as a function of iteration n. For many 1 2 indicated a positive largest Lyapunov exponent. How- steps, this difference is small, briefly increases to large, ever,we didnotobservesucha linearity,whichis consis- and rapidly returns to small, a characteristic nonchaotic tent with a zero largest Lyapunov exponent [20]. behavior. (Incontrast,forstrangechaotic attractors,the Since these two obvious metrics are difficult to esti- differencewouldstartsmall,increasetolarge,andthere- mate, we focused instead on the spectral scaling of the after remain uncorrelated, without the repeated rapid dynamics,whichis a morereliablemeasurethat hastra- close returns.) ditionally been used to identify strange nonchaotic mo- To quantify the section geometry, we examined the tion[12,21,22]. We analyzedthe spectralscalingintwo magnitude of the discrete Fourier transform of the sec- independentdatapipelineswithcomparableresults. One tion (or strobed) data, as in Fig. 3(a), and recorded the pipeline used Mathematica 10 software while the other number of peaks N above a threshold σ. The power law s used MATLAB, Period04,and custom C++ software. scaling of this spectral distribution N = N σ , over 2 0 FromaFouriertransformFˆN[f]ofthenormalizedtime decades of thresholds and 3 decades of number, with an series FN[t], we identified the two most significant fre- exponent s 1.5 in the range 2 < s < 1, as in ≈ − − − quencies in the light curve, as shown in Fig. 1(b), where theFig.3(b)log-logplot,isaclassicsignatureofstrange red arrows indicate primary and secondary frequencies nonchaotic dynamics [9, 10]. The power law means that at f 1/(0.266 d) and f 1/(0.169 d), correspond- the distribution of peaks is scale free, with a range of 1 2 ≈ ≈ ing to primary and secondary periods of T 6.41 h large and small peaks, presumably reflecting the fractal 1 ≈ and T 4.05 h, where f /f = T /T 1.58 ϕ is nature of the underlying strange attractor [24, 25]. The 2 1 2 2 1 ≈ ≈ ≈ nearly the incommensurate golden ratio, in agreement deviationat low thresholds results fromthe lightcurve’s withpreviouswork[14,23]. (Otherspectralpeakscorre- finite frequency resolution. spond to harmonics of f and linear combinations of f The cumulative case for strange nonchaotic dynamics 1 1 and f .) We next plotted three-dimensional delay coor- includes the strobed or section spectrum, the spectral 2 dinates FN[t],FN[t τ],FN[t 2τ] attimest=nδtand exponent, the distinctive orbit separation behavior, and { − − } delay τ = 2δt to reconstruct the time series’ underlying to a lesser extent the section dimension and Lyapunov attractor [24]. The resulting Fig. 2 torus is consistent exponent. The identification is challenging because of with quasiperiodic forcing of the dynamics. Points far inevitable noise, limited and imperfect (despite unprece- fromthetorusstraddlegapsinthedata,andthewarpof dented) data, and the smallness of the secondary fre- the torus reflects the nonlinearity of the light curve [17]. quency amplitude compared to the primary frequency OnewaytotakeaPoincar´esectionofastatespaceflow amplitude (as suggested by the Fig. 1(b) spectral peak istointerceptitwithaplane,butasecondwayistosam- heights). Quasi-periodic dynamics naturally flows on a ple or strobe it at a fixed frequency. As in the original two-dimensional torus, with the two frequencies driving 4 curve having either the KIC 5520878 near-golden fre- quency ratio or the ideal golden frequency ratio, with or without noise added to the fluxes or the times. The spectraldistributionsoftheseartificialdatasetswerenot strangenonchaotic,demonstratingthattheKIC5520878 pulsations are not simply noisy quasiperiodic. To check theeffectsofthemanysmallandthefewlargedatagaps, we linearly or sinusoidally interpolated to close them. This produced some artifacts in the section plot, but still produced a spectral exponent in the strange non- chaotic regime. Finally, we introduced the exact experi- mental gaps into the ideal quasiperiodic curve. For lim- itedrangesofadditivenoiseandthresholds,thisdidpro- duce spectral power law scaling, but the strobed spectra remained qualitatively distinct, and strange nonchaotic dynamics remained the best explanation. We expanded this analysis to five other multi- frequency variable stars in Kepler’s field of view. The three additional golden RRc Lyrae stars KIC 4064484, KIC 8832417 and KIC 9453114, whose frequency ratios wellapproximatethe goldenmean, allexhibit signatures of strange nonchaotic dynamics, and their spectral scal- ings nearly collapse onto each other when the axes are proportionally scaled, as in Fig. 4(a). In contrast, two non-golden RRab Lyrae stars KIC 4484128 and KIC 7505345 [27], whose frequency ratios well approximate thesimplefraction3/2,exhibitqualitativelydifferentdy- namics, as in Fig. 4(b). The possibility of other golden starswithstrangenonchaoticbehavior,discoveredbyon- FIG. 4. Other stars. (a) Spectral scalings of the four Ke- going large sky surveys like OGLE [28] or ASAS [29], pler goldenstarsallexhibitpowerlawscaling,over2decades promises to further classify the RR Lyrae and Cepheid of thresholds, and nearly collapse onto each other when the variable stars. axes are proportionally scaled (as denoted by primes). (b) By contrast, spectralscalings of twoKepler non-goldenstars withprimaryandsecondaryfrequencyratiosnear3/2donot Our model-independent nonlinear analysis of the light exhibit power law scaling but are in fact concave or multi- scaled. curves of such variable stars is complementary to de- tailednonlinearhydrodynamicmodelsofthestarsthem- selves[30]. Thestrangenonchaoticsignaturesofvariable the toroidal and poloidal motions; quasi-periodic forcing stars like KIC 5520878 may elucidate phenomena like ofanonlinearsystemnaturallyinhabitsawrinkledtorus the Eddington valve mechanism [31] thought to under- with fractal cross section, but in this case the wrinkles lie their pulsations: variations in the opacity of the star may be small (or the strangeness higher dimensional), mightquasiperiodicallymodulatethenormalhydrostatic and standard analysis might miss them. balance between pressure outward and gravity inward, We tested our time series analysis by analyzing a va- therebygeneratinglightcurvessomewherebetweenorder riety of null hypotheses that generated surrogate data and chaos that the best models will need to reproduce. setsofartificiallightcurvesoffluxversustime,including both nonparametric and parametric models. Nonpara- metric models included a classic phase randomization of Strange nonchaotic attractors have been observed in the original time series, which was the inverse Fourier laboratory experiments involving magnetoelastic rib- transform of the phase randomized Fourier transform of bons [12], electrochemical cells [32], electronic cir- the light curve [26]. The strobed section and spectral cuits [33], and a neon glow discharge [34], but never be- exponenteasilydiscriminatedbetweenthesurrogateand fore in non-experiments in nature. The pulsating star original data. KIC 5520878 may be the first strange nonchaotic dy- Parametric models included an ideal quasiperiodic namical system observed in the wild. 5 We gratefully acknowledge the entire Kepler team, [17] See Supplemental Material at [URL will be inserted by whose outstanding work has made possible our results, publisher] for rotating 3D animations of the attractor as well as support from the Office of Naval Research and details of our fractal and Lyapunovexponent calcu- lations. under Grant No. N00014-12-1-0026 and STTR grant [18] J. Doyne Farmer, “Chaotic Attractors of an Infinite- No. N00014-14-C-0033. J.F.L. thanks The College of dimensional Dynamical System”, Physica D 4, 366 Wooster for making possible his sabbatical at the Uni- (1982). versity of Hawai’i at Ma¯noa. [19] A. Wolf, J. Swift, H. Swinney, J.Vastano, “Determining Lyapunov exponents from a time series” Physica D 16, 285 (1985). [20] M. T. Rosenstein, J. J. Collins, C. J. DeLuca. “A prac- tical method for calculating largest Lyapunovexponents [1] W. J. Borucki, et al., “Kepler planet-detection mission: from small data sets”, Physica D 65, 117 (1993). introduction and first results” Science 327, 977 (2010). [21] M. Ding, C. Grebogi, E. Ott, “Dimensions of strange [2] W. J. Chaplin et al., “Ensemble asteroseismology of nonchaotic attractors”, Phys. Lett. A137, 167 (1989). solar-typestarswiththeNASAKepler mission”,Science [22] F.J.Romeiras,E.Ott,“Strangenonchaoticattractorsof 332.6026 213 (2011). thedampedpendulumwithquasiperiodicforcing”,Phys. [3] A.Finkbeiner,“Planets in chaos”, Nature511(7507), 22 Rev. A35, 4404 (1987). (2014). [23] M.Hippke,etal.,“PulsationperiodvariationsintheRRc [4] D. G. Koch et al., “Kepler mission design, realized pho- Lyrae star KIC 5520878” Astrophys.J. 798, 42 (2015). tometric performance, and early science”, Astrophys. J. [24] F. Takens, “Detectingstrange attractors in turbulence”, 713(2), L79 (2010). InD.A.RandandL.S.Young,Dynamical Systems and [5] J.M.Jenkinsetal.,“OverviewoftheKepler sciencepro- Turbulence,LectureNotesinMathematics898.Springer- cessingpipeline”,Astrophys.J.Lett.713(2),L87(2010). Verlag. (1981). [6] M.Livio,The golden ratio: The story of phi, the world’s [25] B.R.Hunt,E.Ott,“Fractalpropertiesofrobuststrange most astonishing number, Random House LLC, New nonchaotic attractors”, Phys. Rev. Lett. 87, 254101 York(2008). (2001). [7] R.C.Hilborn,Chaosandnonlineardynamics: Anintro- [26] J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, J. duction for scientists and engineers, Oxford University Doyne Farmer. “Testing for nonlinearity in time series: Press (1994). themethodofsurrogate data”, PhysicaD58, 77(1992). [8] D.Cubero,J.Casado-Pascual,F.Renzoni,“Irrationality [27] J.M.Benko˝,E.Plachy,R.Szab´o,L.Molna´r,Z.Koll´ath. andquasiperiodicityin drivennonlinearsystems”,Phys. “Long-timescale Behavior of the Blazhko Effect from Rev.Lett. 112, 174102 (2014). RectifiedKepler Data”, TheAstrophysicalJournal Sup- [9] C.Grebogi,E.Ott,S.Pelikan,J.A.Yorke,“StrangeAt- plement Series 213(2), 31 (2014). tractorsthatarenotchaotic”,PhysicaD13,261(1984). [28] I. Soszyn´ski, et al., “A. The Optical Gravitational Lens- [10] U. Feudel, S. Kuznetsov, A. Pikovsky, “Strange Non- ing Experiment. The OGLE-III Catalog of Variable chaoticAttractors: Dynamicsbetween Orderand Chaos Stars. III. RR Lyrae Stars in the Large Magellanic in Quasiperiodically Forced Systems”, World Scientific Cloud”, Acta Astronomica, 59, 1 (2009). Series on Nonlinear Science, Series A, Vol. 56 (World [29] G. Pojmanski et al., “The All Sky Automated Survey”, ScientificSingapore, 2006). Acta Astronomica 55(3), 275 (2005). [11] O. Regev, Chaos and Complexity in Astrophysics, Cam- [30] E. Plachy, Z. Koll´ath, L. Molna´r, “Low-dimensional bridge UniversityPress, New York (2006). chaos in RR Lyrae models”, Mon. Not. R. Astron. Soc. [12] W.L.Dittoetal.,“Experimentalobservationofastrange 433(4), 3590-3596 (2013). nonchaotic attractor”, Phys. Rev.Lett. 65, 533 (1990). [31] A. S. Eddington, “On the cause of Cepheid pulsation”, [13] R. Blazhko, “Mitteilungenbervernderliche Sterne”, As- Mon. Not. R.Astron. Soc. 101, 182 (1941). tronomische Nachrichten 175, 325 (1907). [32] G. Ruiz, P. Parmananda, ”Experimental observation of [14] P. Moskalik, “Multi-mode oscillations in classical strange nonchaotic attractors in a driven excitable sys- Cepheids and RR Lyrae-type stars”, Proc. Int. Astron. tem”, Phys.Lett. A367, 478 (2007). Union 9(S301), 249 (2013). [33] T.Zhou,F.Moss, A.Bulsara, “Observationofastrange [15] D.M.Kippingetal.,“TheHuntforExomoonswithKe- nonchaotic attractor in a multistable potential”, Phys. pler (HEK).II.AnalysisofSevenViableSatellite-hosting Rev. A45, 5394 (1992). Planet Candidates”, Astrophys.J. 770(2), 101 (2013). [34] W.X.Ding,H.Deutsch,A.Dinklage,C.Wilke,“Obser- [16] E.Plachy,J.M.Benko˝,Z.Koll´ath,L.Molna´r,R.Szab´o, vation of a strange nonchaotic attractor in a neon glow “NonlineardynamicalanalysisoftheBlazhkoeffectwith discharge”, Phys.Rev.E55, 3769 (1997). theKepler spacetelescope: thecaseofV783Cyg”,Mon. Not.R. Astron. Soc. 445(3), 2810 (2014).