SSymbolic partition in chaotic maps
Misha Chai and Yueheng Lan
1, 2, a) School of Science, Beijing University of Posts and Telecommunications, Beijing,China 100876 State Key Lab of Information Photonics and Optical Communications,Beijing University of Posts and Telecommunications, Beijing, China 100876 (Dated: 26 January 2021)
In this work, we only use data on the unstable manifold to locate the partition boundaries bychecking folding points at different levels, which practically coincide with homoclinic tangencies(HTs). The method is then applied to the classic two-dimensional H´enon map and a well-knownthree-dimensional map. Comparison with previous results is made in the H´enon case and Lyapunovexponents are computed through the metric entropy based on the partition, to show the validity ofthe current scheme.PACS numbers: 05.45.-a, 05.45.AcKeywords: symbolic dynamics, topological structure, H´enon map, chaotic maps, folding points
Symbolic dynamics is a very effective descrip-tion of chaotic motion that captures robusttopological features but ignores coordinate-dependent metric properties of a system.However, it is difficult to produce a good sym-bolic partition, especially in high dimensionswhere the stable and unstable manifolds getentangled in a complex manner. In this pa-per, we propose a new scheme which onlyfocuses on the unstable manifold and thusavoid the computation of the possibly high-dimensional stable one. With this simplifi-cation, the scheme may be applied in moregeneral situations to efficiently carry out thesymbolic partitions.
I. INTRODUCTION
In the mid-1600s, Newton started the researchon differential equations and solved the two-bodyproblem. Later-generation mathematicians andphysicists tried to extend Newton’s method to thewell-known three-body problem but miserably faileduntil Poincar´e introduced a new point of view whichfocuses on qualitative rather than quantitativefeatures of the dynamics. When a motion turnschaotic, analytic approximation becomes less usefuland geometric description seems more natural. Avery useful tool to represent the topological feature a) Corresponding Author: [email protected] of chaos is symbolic dynamics, which translatespoints on the attractor into long sequences ofsymbols drawn from a given set labeling differentpatches of the attractor, and the dynamics intoa shift in the symbol sequence. The key to theconstruction of good symbolic dynamics is to find asimplest partition which is able to assign each pointon the attractor a unique symbol sequence .In flows, although it is possible to distinguishdifferent orbits with orbit topology , a commonpractice is converting it to a map on a well-chosenPoincar´e surface of section. If a 1D map is chaoticon an interval, it is possible to make partitionswith extremum points . In higher dimensions,things become much harder and we need to checkthe homoclinic tangencies (HTs) of the stable andunstable manifolds of particular invariant set. Forsome two-dimensional maps, many approacheswhich heavily rely on the geometry of phase space,have been successfully used to generate symbolicpartitions. H´enon map , for example, a classicaltwo-dimensional map, is used by many authorsfor symbolic partition , where the stable andunstable manifold of a fixed point is often built tosearch for the HTs, both of which are one dimen-sional and relatively easy to compute. Even so, atsome parameter values, the precise determinationof the primary homoclinic tangencies (PHTs) turnsillusive . For maps in three or more dimensions,the partition becomes even harder since the stableor the unstable manifold has a dimension higherthan one, which may be very difficult to describequantitatively. One interesting set of approaches a r X i v : . [ n li n . C D ] J a n IG. 1. The mapping of points in the process of stretching and folding: (a) point A will be mapped to point A after one iteration; (b) the points in area B and B will be mapped to area B after one iteration. rely on unstable periodic orbits densely embeddedin the chaotic attractor, to generate unique symbolsequence . However, in order to improve theaccuracy, those methods require a sufficient numberof unstable periodic orbits which is a challengeespecially with limited, noisy, time series data.Symbol states could also be constructed with anetwork to efficiently represent system dynamicsand sometimes approximation of the generatingpartition could be obtained by properly designedstochastic optimization techniques .Here we propose a new approach which focusesonly data on the unstable manifold. Since real-world maps including those on Poincar´e sectionsare quite dissipative, i.e. , the Lyapunov dimensionis low. In fact, the unstable manifolds of mostwell-known maps have a dimension less than threeand often are even one-dimensional in some pa-rameter range, which indicates a high dimension ofthe stable manifold and thus brings trouble to theconventional computation of HTs. In our method,however, the folding points as a subset of the HTscould be conveniently determined on the unstablemanifold constructed from the mapping data andthus, it would not cause a problem. For finiteresolution, the set of partition points can also beused to deduce the number of symbols that areneeded. The scheme is tested on the H´enon map atdifferent parameter values and successfully appliedto a well-known three-dimensional map.In the following, we explain the basic idea of thenew approach based on an observation in the 1D map in Section 2, emphasizing the importance ofthe stretching and folding mechanism of chaos gen-eration. In Section 3, H´enon map is used as an ex-ample to show how partition points are detected onthe unstable manifold of a fixed point, when fold-ings are strong or not so strong. To further checkthe validity of the new method, we will extend theapplication to a three-dimensional map in Section 4,which has a very complex attractor and is proposedby A. S. Gonchenko and S. V. Gonchenko , andalso mentioned in ref . Compared with 2D maps,new troubles emerge concerning the 3D structure,but our method still works very well. In the end, wecompute the metric entropy to justify our partition,like what other authors did . The results aresummarized in Section 5. II. FOLDING POINTS AND SYMBOLICPARTITION
First, we have a close look at the folding pointsthat determine the symbolic partition in the 1Dmap. As we know, the action of a map leading tochaos consists of two steps : stretching and folding.For example, in the logistic map x n +1 = rx n (1 − x n )with r = 4, the points in the interval [0, 1] canbe viewed as being stretched twice to its originallength and then folded back as shown in Fig. 1. Asa result, the interval [0, 1] is decomposed into twosubintervals on either of which the map is monotoneand extends the subset to the full interval. Asthe iteration goes on, each subinterval is dividedinto smaller and smaller intervals to ensure the2 IG. 2. H´enon map Eq. (1) with a=1.4, b=0.3: (a) part of the unstable manifold of the fixed point (black star)with six maximum curvature points (red star). (b) the folding and the critical points: by one iteration of the baselineS1, a new segment S2 emerges with the folding point P3 and the critical point A on S1 is the preimage of P3; oneiteration of S2 results in two new segments S3 and S4 along with the folding point P7. The critical point B on S2is the preimage of P7. monotonicity. In general, our method starts witha “baseline” which plays a similar role to theinterval [0, 1] but needs to be determined (seeFig. 1 and Observation III below), which may notbe so obvious in high-dimensional maps, especiallywhen the folding is not strong enough. Iterationsof the baseline leads to layers of segments, eachof which looks more or less similar to the baselineand is sequentially connected to each other with”folding points” to be characterized in detail below.Therefore, each segment is bounded by two foldingpoints at the two ends and can be viewed as somekind of ”maximally stretched” piece of the manifold.Hence, precisely determining the location offoldings is essential to the symbolic partition sincein the process of stretching and folding, two areas B and B will be mapped to the same area B after one iteration as shown in Fig. 1, so thatthe symbolic sequences after one shift would bethe same for the corresponding points in B and B . Therefore, these pairs of points have to liein different symbol regions before the mapping.Thus the folding point should play the role ofthe partition point since any neighborhood of itcontains points approaching each other after one orseveral iterations, which belongs to the set of HTsmentioned before in two or higher dimensions. Asdisplayed in Fig. 1, A is the “folding point” andthe partition should be made at its preimage — the“critical point” A , which is also called “primaryturning point” in the literature. Thus, we have Observation I: folding points emerge frommultiple iterations of the baseline and a segmentis part of the manifold between two consecutivefolding points which is maximally stretched locally.Images of folding points are still folding points andfor each genealogy group there is a starting onewhose preimage is called a critical point where theradius of curvature is about the size of the attractor.To locate a folding point precisely, therefore, wemay do a few more iterations in the relevant smallneighborhood to get a fully folded structure and un-ambiguously pick up the unique point with the max-imum curvature and then make a few inverse itera-tions of this point to get it. The key of our method isto define a proper baseline and to locate proper fold-ing points to separate these segments, which is simi-lar to the search for PHTs in other algorithms . Inthe current scheme, only a well-selected segment onthe unstable manifold of a chosen fixed point is em-ployed as the baseline, which will be iterated a fewtimes to get layers of unstable manifold segmentsthat are separated by the folding points. Each fold-ing point defines a family of points and we need topick up one as the first folding point and the preim-ages of these folding points can be chosen as thecritical points for partition. In the following, if notstated otherwise, the term “folding point” is usuallyreferring to the first folding point in its family.In addition, we also want to know how manysymbols is required for a chaotic map by simplychecking the number of critical points on each3
IG. 3. The symbolic partition for H´enon map at different parameters and the partition line (black dot-dashed line)connects the critical points (red star): (a) a = 1 . , b = 0 .
3; (b) a = 1 . , b = 0 . segment in different layers. For example, in thelogistic map with r=4, there is one folding pointafter one iteration of the baseline, so the baselinecan be decomposed to two intervals represented bytwo symbols, 0 and 1. As later iterations exactlyoverlap the same interval, there would appearno new critical point and hence two symbols areenough for a good partition. Thus we have ObservationII: the number of symbols for amap is determined through the following two steps: • If the number of critical points is N on a seg-ment, the number of symbols is N + 1 locally. • The number of symbols for a map may be cho-sen as the maximum number of symbols amongall segments.
III. GENERATING SYMBOLIC PARTITION IN 2DMAPS
In this section, we will apply the new scheme tothe classical H´enon map for different parameter val-ues. The equation of the H´enon map is x n +1 = − ax n + y n + 1 , y n +1 = bx n , (1)where a is a parameter that controls the folding and b for the dissipation. With the conventional value a = 1 . , b = 0 .
3, the folding is strong enough forus to crisply locate the critical points, while with a = 1 . , b = 0 .
54 where the folding is insufficient,troubles emerge which we will show how to deal within our scheme. Just like what we did in 1D maps, we first need to find a baseline on the unstablemanifold. Hobson proposed a numerical schemewhich computes stable or unstable manifolds quiteaccurately and is hence utilized in the following. Itstarts from a short line near the fixed point alongthe unstable eigenvector, and an approximationof the unstable manifold results from multipleiterations of this short line. A. Parameter: a=1.4 b=0.3
In this case, following Hobson’s procedure, wechoose a short line | x − x ∗ | < .
001 near the fixedpoint along the unstable direction, which producesthe unstable manifolds in Fig. 2(a). The structureof the H´enon attractor suggests that the attractoris Cantor-like in the transverse direction and thesaddle point sits on the edge of the attractor.Benedicks and Caeleson and Sim´o proved thatthe attractor is the closure of the unstable manifoldof the saddle point. From the attractor of theH´enon map, we have the following observation. Observation III:
The baseline satisfies the fol-lowing conditions: • The chosen fixed point lies on the baseline. • The baseline is part of the unstable manifoldof the saddle point. • It stretches continuously in both directions un-til touching the folding points.To determine the baseline, we calculate curva-tures along the unstable manifold in Fig. 2(a). The4
IG. 4. H´enon map Eq. (1) with a = 1 . , b = 0 .
54: (a) part of the unstable manifold of the fixed point (blackstar) with three maximum curvature points (red star); (b) the folding and the critical points: by one iteration of thebaseline S1, a new segment S2 emerges with the folding point P1 and the critical point A on S1 is the preimageof P1; one iteration of S2 leads to two new segments S3 and S4 along with the folding point P4. The critical point B on S2 is the preimage of P4; one iteration of S3 leads to a new segment S5 along with three maximum curvaturepoints; one iteration of S4 leads to a new segment S6 along with one maximum curvature point; (c) the points P4,P5, P6 will be mapped to points P4 (cid:48) , P5 (cid:48) , P6 (cid:48) after one iteration. local maximum curvature at P1 is 3 . × ;P2: 30 . . × ; P4: 31 . . × ; P6: 32 . . × is reached byiterating nine more steps and P3 is obtained by nine inverse iterations of this point. The preimage of P3is the critical point A where the radius of curvatureis of the same order of the whole attractor. For S2,a similar procedure is followed to locate the foldingpoint P7 and the critical point B . In this process,two new segments S3 and S4 are emerging, thecritical points of which could be detected similarly.Putting all this information together, we get thepartition in Fig. 3(a) at this level. Here, we wouldlike to mention that the proper folding points areeasy to select in the current example since thedissipation of the map is strong enough. In thegeneral case where the dissipation is not enough,ambiguity could arise as in the following example. B. Parameter: a=1.0 b=0.54
Compared to the previous case, here we deal witha situation in which the folding is not strong, leadingto uncertainty that entails different partitions .5 IG. 5. The entropy h N (red circles) Eq. (3) for the Henon map Eq. (1) with a=1.0 b=0.54 and the 3D map Eq. (4)with a=-1.86 b=0.72 c=0.03. N denotes the length of the considered symbol sequences. The horizontal blue lineindicates the benchmark value of the Lyapunov exponent: (a) for the partition in Fig. 3(b); (b) for the partition inFig. 7. With the current parameter values, Grassbergerand Kantz produced a partition by searching forPHTs, but there is no precise definition on what is“primary”. Hansen arrive at a different partitionby employing the same method but also investigat-ing changes of critical points with the parameters.Moreover, he explained why his partition is better.Later various definitions of PHTs were proposed,but may only be used on specific occasions .Giovannini and Politi used a new method which isalso focused on the changes of critical points withthe parameters to explore what’s characteristic ofPHTs and found interesting bifurcations in thegeneration of symbolic partitions. But for PHTs,they finally gave a conclusion “... it is not possibleto give a priori a nonambiguous definition of theprimary homoclinic tangency” and “We think thatthe only meaningful way to determine a PHT is viaa trial-and-error procedure”. Biham and Wenzel also obtained a different partition through a set ofunstable periodic orbits. Grassberger compared hisresult in ref with that in ref , and delivered anexplanation in ref .As we did before, a few iteration of the short linein the unstable direction of the fixed point resultsin the structure in Fig. 4(a), on the initial part ofwhich we find three points with local maximumcurvature. The curvature at P1 is 8 . × ; P2:7 . . × . According to ObservationIII, the points between P1 and P3 can be definedas the baseline. In Fig. 4(b), by one iteration of thebaseline S1, a new segment S2 emerges on which weget the folding point P1 according to Observation I and then the critical point A . Then one iteration ofS2 results in two new segments S3 and S4 emerges.However, the folding is not so obvious and it’s onlymildly folded after one iteration. The curvaturesat P4 is 8.2807; P5: 7.3609; P6: 1.3997. Here it’sa little bit hard for us to figure out which partwill be folded. To determine the exact foldingposition, we do one more iteration of S2 and see inFig. 4(c) that the curve is folded at P (cid:48) which thusdistills P4 as the folding point. Then the criticalpoint B on S2 is the preimage of P4. By oneiteration of S3 and S4, the critical points could bedetected similarly. However, the difference is, withone iteration of S3, a new segment S5 is producedwhich is still stretching and no folding point tobreak it. As a result, no critical point exists onS3. With a similar argument, we conclude thatno folding point could be defined on S6 and hencethere is no critical point on S4, either. This processcould be carried on for more iterations. Finally,after dealing with 180 segments, we get the parti-tion in Fig. 3(b) and the accuracy reaches 1 . × − .In order to justify our partition, we compute themetric entropy h and compare it with the Lya-punov exponent which is supposed to be equal toeach other if the partition is correct, as suggestedby Grassberger and Politi . Also L. Jaegerand H. Kantz had shown that wrong partitionswould not have a trend of convergence. More ex-plicitly, we assign a weight p ( s , . . . , s N ) of occurringprobability (evaluated by counting) to each sequence6 = ( s , . . . , s N ), and H N = − (cid:88) S p ( S ) ln p ( S ) (2)defines an average amount of information needed tospecify sequences S. The difference with an increaseof N h N = H N +1 − H N (3)should converge to the Lyapunov exponent if thepartition is a true symbolic one. We choose atrajectory with length 2 for the calculation.In this example, with the current setting, the h N converges well to the Lyapunov exponent λ = 0 . ± . . IV. SYMBOLIC PARTITION IN A 3D MAP
In two-dimensional maps, we got nice partitionsusing our scheme. In this section, we will extendits application to three-dimensional maps. The mapthat we deal with is proposed by A.S. Gonchenkoand S. V. Gonchenkon , which could be writtenas x n +1 = y n y n +1 = z n z n +1 = bx n + az n + cy n − . z + 0 . yz − y (4)where a = − . , b = 0 . , c = 0 .
03. The non-linear term only exists in the equation for the z − component. But the dynamics is chaotic at thecurrent parameter values and the strange attractorlooks much more complicated than that of the H´enonmap.As what we did in 2D maps, we first need to finda proper baseline. However, for 3D maps, it’s notas easy as in 2D maps since the unstable manifoldappears entangled in a complex way in three dimen-sions. But Observation III still works. Five pointswith locally maximum curvatures are identified onthe initial part of the manifold as shown in Fig. 6(a).The curvature at P1 is 178 . . . . . A is the critical point shownin Fig. 6(b). In all the subfigures of Fig. 6(c)-(f),black segments are mapped to red segments afterone iteration. In (c), there is a maximum curvatureof 6.0620 at P6. But because the red segment isstill stretching locally and P6 does not break it, soaccording to Observation I, it can’t be defined as afolding point. Therefore, there is no critical pointon this black segment. In (d), the red segment willbe folded at P7 with a curvature of 25.0883 andthe new segments which are separated by P7 havereached the maximum stretching locally until P7breaks it, so it can be regarded as a folding pointand thus locates the critical point B as a preimage.In (e), the maximum curvature at P8 is 8.9726 andthe red segment is still stretching locally and P8does not break it, and thus there is no critical pointon the black segment for the same reason as in (c).In (f), there are four maximum curvatures at P9:2.9522, P10: 161.9620, P11: 2.9731, P12: 1.0919.P10 can be defined as a folding point because ofthe segments which are separated by P10 bothreach maximum stretching locally until P10 breaksit. Hence its preimage C is the critical point.After dealing with 250 segments, we get the sym-bolic partition of the attractor as displayed in Fig. 7.Like what we did in 2D maps, in order to justifyour partition, we compute h N and compare it withthe Lyapunov exponent if the partition is correct.Here the calculation was performed with a series oflength 2 and we find that h N converge well to theLyapunov exponent λ = 0 .
157 with the increase ofN, as shown in Fig. 5(b).
V. CONCLUSION
Symbolic partition is essential for a topologicaldescription of orbits in nonlinear systems butremains a challenge for long. In this paper, we focuson the unstable manifold of certain invariant setand carry out the partition based on the stretchingand folding mechanism of chaos generation. Threeobservations are listed as our guidelines, whichstarts with the determination of folding points,since folding points not only define the baseline ofthe manifold but also gives critical points as theirpreimages. Critical points serve as boundary pointsfor a symbolic partition. Our scheme is successfully7
IG. 6. 3D map Eq. (4) with a = − . , b = 0 . , c = 0 .
03: (a) part of the unstable manifold of the fixed point(black star) with five maximum curvature points (red star). The projection (in red line) on the ( x, y )-plane is alsodisplayed; (b) the folding and the critical points: by one iteration of the baseline S1, a new segment S2 emerges withthe folding point P4 and the critical point A on S1 is the preimage of P4; (c) a new red segment emerges with P6by one iteration of the black segment; (d) two new red segments emerges with folding point P7 by one iteration ofthe black segment and the critical point B is the preimage of P7; (e) a new red segment emerges with P8 by oneiteration of the black segment; (f) two new segments emerges with folding point P10 by one iteration of the blacksegment and the critical point C is the preimage of P10. demonstrated on the H´enon map with different setsof parameters and on a well-known 3D map.The focus on the unstable manifold in our ap-proach avoids the study of possibly high-dimensionalstable manifold, which may accelerate computationin an essential way. As a result, we do not haveto search HTs in the full phase space but insteadpin down the critical points by iterations on theunstable manifold. After the iteration genealogy offolding points is sorted out, the partition seems easy to do. However, the determination of the precisestarting point in the genealogy could be a problemif the folding process is slow, just as defining theHTs in the literature which could be a source ofconfusion . Nevertheless, the organization ofthe layered structure in the current approach mayhelp alleviating difficulties as shown in the examples.In the current computation, the determination ofthe baseline and individual segments is essential tothe success of the application. In all the examples,8 IG. 7. The unstable manifold and the symbolic partition of the 3D map of Eq. (4): (a) the red curves are givensymbol 0; the black curves are given symbol 1; (b) the projection to the ( x, y )-plane. we utilized the unstable manifold of a well-chosenfixed point. Whether this is generally applicable isa question that needs further exploration. Also, weonly applied the scheme to maps with just one unsta-ble direction. How to extend it to high-dimensionalmaps with multiple unstable directions is key for itsapplication in real-world problems. For flows in thephase space, a common practice is to choose a properPoincar´e section and construct the return map sothat the current technique may still apply. How-ever, in general, it is near impossible to select a goodsection that works for all orbits and thus a globalmap is hard to obtain. It appears very rewarding toinvestigate the possibility of carrying out symbolicpartition directly in the full phase space of a flowwith current scheme.
ACKNOWLEDGEMENT
This work was supported by the National Natu-ral Science Foundation of China under Grant No.11775035, and also by the Fundamental ResearchFunds for the Central Universities with Contract No.2019XD-A10.
VI. DATA AVAILABILITY STATEMENT
The data used to support the findings of this studyare available from the corresponding author uponrequest. P. Collet and J.-P. Eckmann,
Iterated maps on the intervalas dynamical systems (Springer Science & Business Media,2009). C. Dong and Y. Lan, “A variational approach to connectingorbits in nonlinear dynamical systems,” Phys. Lett. A ,705–712 (2014). P. Cvitanovic, R. Artuso, R. Mainieri, G. Tanner, G. Vat-tay, N. Whelan, and A. Wirzba, “Chaos: classical andquantum,” ChaosBook. org (Niels Bohr Institute, Copen-hagen 2005) (2005). M. H´enon, “A two-dimensional mapping with a strange at-tractor,” Commun. Math. Phys. , 69–77 (1976). F. Giovannini and A. Politi, “Generating partitions inh´enon-type maps,” Phys. Lett. A , 332–336 (1992). P. Grassberger and H. Kantz, “Generating partitions forthe dissipative h´enon map,” Phys. Lett. A , 235–238(1985). K. T. Hansen,
Symbolic dynamics in chaotic systems , Ph.D.thesis, University of Oslo (1993). P. Grassberger, H. Kantz, and U. Moenig, “On the sym-bolic dynamics of the henon map,” J. Phys. A: Gen. Phys. , 5217 (1989). L. Jaeger and H. Kantz, “Structure of generating partitionsfor two-dimensional maps,” J. Phys. A-math. Gen. , L567(1997). P. Cvitanovi´c, G. H. Gunaratne, and I. Procaccia, “Topo-logical and metric properties of h´enon-type strange attrac-tors,” Phys. Rev. A , 1503 (1988). H. Kantz and L. Jaeger, “Improved cost functions for mod-elling of noisy chaotic time series,” Physica D , 59–69(1997). A. Politi, “Symbolic encoding in dynamical systems,” in
From Statistical Physics to Statistical Inference and Back (Springer, 1994) pp. 293–309. O. Biham and W. Wenzel, “Characterization of unstableperiodic orbits in chaotic attractors and repellers,” Phys.Rev. Lett. , 819 (1989). R. Badii, E. Brun, M. Finardi, L. Flepp, R. Holzner,J. Parisi, C. Reyl, and J. Simonet, “Progress in the analysisof experimental chaos through periodic orbits,” Rev. Mod.Phys. , 1389 (1994). O. Biham and W. Wenzel, “Unstable periodic orbits andthe symbolic dynamics of the complex h´enon map,” Phys.Rev. A , 4639 (1990). P. Cvitanovi´c, “Periodic orbits as the skeleton of classicaland quantum chaos,” Physica D , 138–151 (1991). R. L. Davidchack, Y.-C. Lai, E. M. Bollt, and M. Dhamala,“Estimating generating partitions of chaotic systems by un-stable periodic orbits,” Phys. Rev. E , 1353 (2000). J. Plumecoq and M. Lefranc, “From template analysis togenerating partitions: I: periodic orbits, knots and symbolicencodings,” Physica D , 231–258 (2000). J. Plumecoq and M. Lefranc, “From template analysis togenerating partitions: Ii: Characterization of the symbolicencodings,” Physica D , 259–278 (2000). M. B. Kennel and M. Buhl, “Estimating good discrete par-titions from observed data: Symbolic false nearest neigh-bors,” Phys. Rev. Lett. , 084102 (2003). M. Buhl and M. B. Kennel, “Statistically relaxing to gener-ating partitions for observed time-series data,” Phys. Rev.E , 046213 (2005). Y. Hirata and K. Aihara, “Estimating optimal partitionsfor stochastic complex systems,” Eur Phys J Spec Top ,303–315 (2013). Y. Hirata, K. Judd, and D. Kilminster, “Estimating a gen-erating partition from observed time series: Symbolic shad-owing,” Phys. Rev. E , 016215 (2004). N. S. Patil and J. P. Cusumano, “Empirical generating par-titions of driven oscillators using optimized symbolic shad-owing,” Phys. Rev. E , 032211 (2018). Z. Gao and N. Jin, “Complex network from time seriesbased on phase space reconstruction,” Chaos , 033137(2009). X. Sun, M. Small, Y. Zhao, and X. Xue, “Characterizingsystem dynamics with a weighted and directed network con-structed from time series data,” Chaos , 024402 (2014). L. Hou, M. Small, and S. Lao, “Dynamical systems in-duced on networks constructed from time series,” Entropy , 6433–6446 (2015). R. V. Donner, Y. Zou, J. F. Donges, N. Marwan, andJ. Kurths, “Recurrence networks—a novel paradigm fornonlinear time series analysis,” New J. Phys. , 033025(2010). T. Nakamura, T. Tanizawa, and M. Small, “Construct-ing networks from a dynamical system perspective for mul-tivariate nonlinear time series,” Phys. Rev. E , 032323(2016). M. McCullough, K. Sakellariou, T. Stemler, and M. Small,“Regenerating time series from ordinal networks,” Chaos , 035814 (2017). T. Tanizawa, T. Nakamura, F. Taya, and M. Small, “Con-structing directed networks from multivariate time seriesusing linear modelling technique,” Physica A , 437–455(2018). K. Sakellariou, T. Stemler, and M. Small, “Markov mod-eling via ordinal partitions: An alternative paradigm fornetwork-based time-series analysis,” Phys. Rev. E ,062307 (2019). A.S.Gonchenko and S.V.Gonchenko, “Variety of strangepseudohyperbolic attractors in three-dimensional general-ized h´enon maps,” Physica D (2016). A. Gonchenko, S. Gonchenko, A. Kazakov, and D. Turaev,“Simple scenarios of onset of chaos in three-dimensionalmaps,” Int. J. Bifurcat. Chaos , 1440005 (2014). S. V. Gonchenko, I. Ovsyannikov, C. Sim´o, and D. Turaev,“Three-dimensional h´enon-like maps and wild lorenz-likeattractors,” Int. J. Bifurcat. Chaos , 3493–3508 (2005). G. Fahner and P. Grassberger, “Entropy estimates for dy-namical systems,” Complex. Syst. (1987). R. Badii and A. Politi, “Hausdorff dimension and unifor-mity factor of strange attractors,” Phys. Rev. Lett. ,1661–1664 (1984). E. Ott,
Chaos in dynamical systems (Cambridge UniversityPress, 2002). D. Hobson, “An efficient method for computing invariantmanifolds of planar maps,” J. Comput. Phys. , 14–22(1993). M. Benedicks and L. Carleson, “The dynamics of the h´enonmap,” Ann. Math. , 73–169 (1991). C. Sim´o, “On the h´enon-pomeau attractor,” J. Stat. Phys. , 465–494 (1979). G. d’Alessandro, P. Grassberger, S. Isola, and A. Politi,“On the topology of the h´enon map,” J. Phys. A-math.Gen. , 5285 (1990). J.-P. Eckmann and D. Ruelle, “Ergodic theory of chaosand strange attractors,” in
The theory of chaotic attrac-tors (Springer, 1985) pp. 273–312. F. Giovannini and A. Politi, “Homoclinic tangencies, gen-erating partitions and curvature of invariant manifolds,” J.Phys. A: Gen. Phys. , 1837 (1999)., 1837 (1999).