Ascertaining when a basin is Wada: the merging method
aa r X i v : . [ n li n . C D ] J un Ascertaining when a basin is Wada: the merging method
Alvar Daza, Alexandre Wagemakers, and Miguel A.F. Sanju´an
1, 2, 31
Nonlinear Dynamics, Chaos and Complex Systems Group,Departamento de F´ısica, Universidad Rey Juan Carlos,Tulip´an s/n, 28933 M´ostoles, Madrid, Spain Department of Applied Informatics, Kaunas University of Technology,Studentu 50-415, Kaunas LT-51368, Lithuania Institute for Physical Science and Technology,University of Maryland, College Park, Maryland 20742, USA (Dated: June 19, 2018)
Abstract
Trying to imagine three regions separated by a unique boundary seems a difficult task. However,this is exactly what happens in many dynamical systems showing Wada basins. Here, we present anew perspective on the Wada property:
A Wada boundary is the only one that remains unalteredunder the action of merging the basins . This observation allows to develop a new method to testthe Wada property, which is much faster than the previous ones. Furthermore, another majoradvantage of the merging method is that a detailed knowledge of the dynamical system is notrequired. . INTRODUCTION Wada basins are one of those unexpected encounters that often happen in science. Thestory begins when a topologist named Takeo Wada tried to answer the following question:Can three or more open regions be separated by a single boundary? Our daily experiencemakes us think that this is impossible. It suffices to look at a common political map torealize that the boundaries separate two different regions, except perhaps some isolatedpoints that separate three or more regions (think about the Four Corners in the USA, forexample). However, Takeo Wada devised an iterative process to make this counter-intuitivesituation possible, as reported by his student Kunizo Yoneyama [1, 2]. The Wada lakes wereconceived in a topological context as a way to separate three connected regions in a planeby means of a continuous boundary [1]. From a topological point of view, Wada boundarieshave intriguing properties. For example, the Polish topologist Kazimierz Kuratowski showedthat in the plane, continuous Wada boundaries must be indecomposable continua [3] (thoughthe situation in higher dimensions is quite different).This discovery remained as a mathematical curiosity until James Yorke and his collabo-rators found that the basins of attraction of some dynamical systems presented the Wadaproperty [4, 5]. From the dynamical point of view, the most interesting feature of Wadabasins is the fact that an arbitrarily small perturbation of a system with initial conditionslying in a Wada boundary can drive it to any of the possible attractors, which implies aspecial kind of unpredictability [6]. Therefore, in this context, Wada boundaries are usuallyreferred to as those that separate three or more basins at a time, but the basins need notto be connected. Since the earliest references to the Wada property in dynamical systems,many authors claim that the boundaries have the Wada property for disconnected basins[7–12]. In this work, we adopt this latter definition: Wada boundaries are those that sepa-rate three or more basins, no matter whether the basins are connected or not. Therefore,using this definition, we believe that the methodology and the results presented in this workare valid for any number of dimensions.Despite our primary intuition, Wada basins are a common feature appearing in manydynamical systems. Since its first report, Wada basins have been found in open Hamilto-nian systems [10], ecological models [11], delayed differential equations [12], hydrodynamicalsystems [13], and many engineering problems [14–16]. This is possible because Wada bound-2ries are related to iterative processes and fractal structures, which are a common featurein the basins of nonlinear dynamical systems [17].So far, two methods have been proposed to determine when the basins of a systempossess the Wada property. The first one was developed by Nusse and Yorke [18, 19], andinvolved the computation of the unstable manifold of a saddle point of the basin boundary.This method requires a detailed knowledge of the system and the computation of unstabletrajectories, which can be cumbersome in many cases. Indeed, many papers [10, 13, 14, 20]are devoted just to determine whether the Nusse-Yorke condition is fulfilled in a particulardynamical system and for a certain set of parameters. Years after the original works byYorke and collaborators, Daza et al. [21] developed a grid method based on the successiverefinement of the grid in order to determine whether all the boundary points were Wadapoints (points that separate three or more basins at a time). This latter method can beautomated and used for every dynamical system. As a drawback, it requires the precisecomputation of new trajectories at very high resolutions. Although it supposes a qualitativeand quantitative improvement with respect to the Nusse-Yorke method, the grid methodneeds several hours or even days of parallel computation to check the Wada property in agiven dynamical system. In this paper, we present a new method to determine when a basinis Wada, which is founded on the observation that:
A Wada boundary is the only one thatremains unaltered under the action of merging the basins . This new method, that we call the merging method , can test the Wada property in a few seconds, and furthermore it doesnot require a detailed knowledge of the system. Thus, the merging method supposes a newquantitative and qualitative leap with respect to the previous available methods to test theWada property.The paper is organized as follows. In Sec. II, we explain how Wada boundaries can bedefined as the only ones that remain unaltered after the action of merging the basins. InSec. III A, based on the previous definition, a new method to test the Wada property ispresented. Section III B is devoted to the detailed analysis of the merging method usingdifferent model examples. Finally, we discuss the results and present the main conclusionsof the paper in Sec. IV. 3
I. MERGING BASINS
The set of all initial conditions leading to a particular attractor is called the basin ofattraction of a dynamical system. We will focus on a very special set of initial conditionscalled the boundary. A point p is in the boundary of a basin B i if ∀ ε >
0, the open ballcentered in p of radius ε , b ( p, ε ), is such that b ( p, ε ) ∩ B i = ∅ and b ( p, ε ) ∩ B i ∁ = ∅ , where B i ∁ is the complement of B i . If the point satisfies the previous condition for all the basins B i with N a ≥ Wada point . If all the boundary points areWada points, then the basin of attraction has the
Wada property , and we call it a Wadabasin.However, we can formulate the Wada property in slightly different terms. Assume wehave N a ≥ ∂B i of eachbasin B i , but instead of using its complement B i ∁ to determine which points belong to theboundary, we will say that a point p is in the boundary if it is arbitrarily close to B i andalso arbitrarily close to at least one of the other basins B j . That is, p is in the boundary ∂B i if ∀ ε >
0, the open ball centered in p of radius ε , b ( p, ε ), is such that b ( p, ε ) ∩ B i = ∅ and b ( p, ε ) ∩ S j = i B j = ∅ . Then, we determine each boundary ∂B i as the boundary betweena basin B i and all the other merged basins S j = i B j , so that we end with as many differentboundaries as different possible attractors, i = 1 , . . . , N a . Thus, all the boundaries createdfollowing this previous procedure and the boundary of the original basins corresponding tothe N a attractors are exactly the same ∂B i = ∂B j for ∀ i = j, i = 1 , . . . , N a , if and only ifthe system is Wada.The two previous definitions of Wada basins are completely equivalent. However, thesecond definition emphasizes the striking idea that Wada basins can be merged and theboundary will still remain unaltered. To be more precise, given N a ≥ N a − N a basinsthen there would be only one basin and the boundary would be lost). This notable effect isbetter appreciated when Wada boundaries are compared to non-Wada boundaries. The time-2 π (Poincar´e) map of the forced damped pendulum defined by ¨ x + 0 . x + sin x = 1 .
66 cos t possesses three attractors, and consequently its phase space ( x, ˙ x ) contains three basins.This is a paradigmatic system showing Wada basins [4]. In the top-left panel of Fig. 1-(a),we display the original three-colored Wada basins of the forced damped pendulum described4 a) (b)(c) Figure 1.
Merging Wada basins.
The time-2 π (Poincar´e) map of the forced damped pendulumdefined by ¨ x + 0 . x + sin x = 1 .
66 cos t possesses three attractors, and consequently its phase space( x, ˙ x ) contains three basins. This system verifies the Wada property [4]. (a) The top-left panel (inred, green and blue) represents the Wada basins, where one color corresponds to one basin. Theother three panels are the result of the action of merging the basins: we merge two colors, and keepthe third unchanged. (b) In the top-left panel a disk is divided in three colors. The other threepanels show the action of merging in this non-Wada picture. (c) The color-code of the mergedbasins can be inferred from the bottom-right picture: yellow=red+green, magenta=blue+red,cyan=blue+green. in [4]. The other three plots show the result of merging the basins according to the colorcode sketched in Fig. 1-(c) (yellow=red+green, magenta=blue+red, cyan=blue+green). It isimportant to notice that each color represents a different basin, being impossible to establisha one-to-one correspondence between basins of different colors. However, even though thebasins are different, the boundaries are the same for the four panels of Fig. 1-(a).If we look at the colored disks in Fig. 1-(b), we can see how the action of merging affects5sual (non-Wada) basins. Here we can clearly notice that the boundaries change under theaction of merging the basins. In fact, the center of the disk is the only point that is in theboundary of the four panels, so that it is a Wada point.Despite the abundant research devoted to Wada basins, the effect that the Wada bound-aries remain unaltered after the action of merging the basins has been unnoticed. In the nextsections, we will use it to develop a new way of testing Wada basins in dynamical systems. III. MERGING BASINS TO TEST THE WADA PROPERTYA. Description of the merging method
The property that we have just described, that is, that Wada basins can be mergedwithout any change in their boundary, can be used to build a new method to test the Wadaproperty. From a purely mathematical point of view, given the basins of a system, it sufficesto check that the boundary remains unaltered under the merging of the basins. However, itwould require an arbitrarily high resolution of the basins to guarantee that the boundariesof the merged basins are exactly the same.Usually, the basins are computed by means of a regular grid of finite size. In this ap-proach, every pixel of the grid has a linear size ε and contains only one corresponding initialcondition, in such a way that the fate of this initial condition determines the color of thepixel. Therefore, the computation of the boundaries is limited by the size of the pixel ε .In Fig. 2-(a)-(b), we can see that the computed boundaries of the merged basins, whichwe call slim boundaries , are not exactly the same, even though they are Wada boundaries.It can be observed at naked eye that although the boundaries seem similar, they are notstrictly identical. It is noticeable that the boundary depicted in Fig. 2-(a) is thicker thanthe boundary depicted in Fig. 2-(b).To overcome this issue, we can try to fatten the boundaries for their subsequent com-parison (see the fattened boundary of Fig. 2-(c)). In the fattening procedure, we replaceeach pixel belonging to the boundary by a fat pixel defined by the fattening parameter r .This fattening parameter is the radius of the fat pixel according to the Chebyshev metricor maximum distance metric. This metric preserves the square shape of the pixels and it isdefined in the plane as r = max ( | x − x | , | y − y | ), where ( x, y ) are the usual Cartesian6 =1r=2r=3r=4r=5r=6r=7 (a) Slim boundary 1 (b) Slim boundary 2 (c) Fat boundary1 Pixel (d) Fattening procedure
FATTENING
Figure 2.
The fattening procedure. (a)-(b) Even for Wada basins, the boundaries are notexactly the same for all the merged basins because of the finite resolution. (c) To avoid this effect,the boundaries must be fattened. (d) In the fattening procedure, each pixel in the boundary issubstituted by a fat pixel of radius r . In the plot, each color corresponds to a different radius inthe Chebyshev or chessboard distance. We call r the fattening parameter . r . Now, let us move forward to thelast part of the procedure.For the moment, we have the original boundaries of the merged basins, the slim boundaries ∂B i , and their fattened versions, the fat boundaries ∂B i . The final step of the procedureis to compare all the slim boundaries with all the fat boundaries. If all the slim boundariesfit in all the fat boundaries ∂B i ⊂ ∂B j ∀ i, j = 1 , . . . , N a ; then we will say that the basin isWada. Otherwise, we will say that the system is not Wada, and the method will determinewhich points are Wada and which ones are not. This last step verifies if each pixel of the slimboundaries ∂B i lies in the set ∂B j . To connect with our previous definition of a basin withthe Wada property, the algorithm checks if the points p i in the boundaries B i are within aball b ( p j , r ) of radius r ( r is the fattening parameter) around the points p j of the boundary B j .In the case of partially Wada basins [16], where Wada and non-Wada boundaries coexist,we can characterize them by the Wada parameter W N a defined in the grid method of Daza et al. [21]. This parameter W N a provides the ratio of Wada points to boundary points (Wadaand non-Wada), in such a way that W N a = 1 means that the system has the full Wadaproperty, whereas W N a < n b , and wecan also register the number of boundary points which are not Wada n NW . Then, the Wadaparameter for a fixed resolution can be calculated simply as W N a = 1 − n NW /n b .Again, for a better understanding of the comparison between slim and fat boundaries, itis convenient to observe an example of non-Wada basins, such as the disks of Fig. 1-(b). Wewould have to fatten the boundaries by a very large amount (comparable to the size of thedisks) in order to make the fat boundaries able to contain the slim ones. We can conclude,as mentioned before, that the only Wada point is the center of the disk.The whole procedure described before can be fully automated and the only input neededis a finite resolution basin. For basins with a resolution of 1000 × a) Original basin(b) Merged basins(c) Boundaries fromthe merged basins:"Slim boundaries"(d) Fatten the boundaries replacingeach pixel by a fatpixel of fattening parameter r: "Fat boundaries" Do all the slim boundaries fit into all the fat boundaries? YES NO (e) If all the slimboundaries fit into all the fatboundaries, thenthe system is Wada. The basin is Wada The basin is not Wada
Figure 3.
Flowchart of the merging method to test Wada basins. (a) Originally we havethe picture of a basin at a given resolution. (b) We merge the basins, so that we have as manymerged basins as different colors in the original basin. (c) We find the boundaries of the mergedbasins, which we can see they are similar but not exactly the same. (d) We fatten the boundariesusing fat pixels of fattening parameter r . (e) We check if all the slim basins are contained in thefat basins. If this is the case, then the basin is Wada, otherwise, it is not. ε . As we will discuss later,the higher the resolution the more reliable the determination of the Wada propertywill be.(b) For each basin B i , we merge the other basins obtaining two-color basins of attractionmade of the original basin B i and the merged basin S j = i B j . By this process, we get acollection of N a pictures with only two colors.(c) We compute the slim boundaries of the merged basins ∂B i . In order to do this, wecan simply see if a pixel has pixels of different colors around itself. Given the finiteresolution of the basins ε , these boundaries may appear slightly different even forWada basins.(d) The slim boundaries ∂B i obtained in the previous step are fattened by fat pixels offattening parameter r , becoming the fat boundaries ∂B i . We can start with r = 1,and if the result of the Wada test in step (e) is negative, we can start over the step(d) with higher values of r until we reach a stopping condition r = r max .(e) We check if all the slim boundaries fit into all the fat boundaries ∂B i ⊂ ∂B j ∀ i, j =1 , . . . , N a . If this is the case, we say that the basins have the Wada property. Of course,10his verification is reliable up to a resolution determined by the fattening parameter r .In case that the system is not Wada, the algorithm provides a list of non-Wada pointsof the basin. B. Analysis of the merging method
The whole method described above relies on a single parameter: the fattening parameter r . This parameter determines the confidence that we have in the result of the algorithm,since we will be able to say that the basin is Wada up to the resolution defined by thefat pixels that we use. Therefore, it is natural to analyze the behavior of the method fordifferent values of r in dynamical systems with different features. This is exactly our aim inthis section.In order to examine the behavior of the procedure with respect to the fattening parameter r , we can apply it to different Wada boundaries. We can characterize fractal boundaries bytheir fractal dimension and by the number of basins that they separate at the same time.Here, we examine two dynamical systems with Wada boundaries where we can easily varythese two quantities. Namely, the two paradigmatic dynamical systems under study are theH´enon-Heiles Hamiltonian [10] and the Newton method to find complex roots [8].The H´enon-Heiles Hamiltonian is defined by H = ( ˙ x + ˙ y ) + ( x + y ) + x y − y .For values of the energy above the critical one, the escape basins of this open Hamiltoniansystem show the Wada property [10]. The fractal dimension of the boundaries diminishesas the energy increases, but the Wada property is preserved, as reported in [22, 23]. Wehave used the merging method described in the previous section with different values ofthe energy E and of fattening parameter r . The results are plotted in Fig. 4-(a). We haveplotted only three different values of the energy for clarity, but we can observe that thealgorithm correctly determines that the basins are Wada for r ≥ r needed to correctly predict the Wada property and thefractal dimension of the boundaries. 11 % n o n - W a d a p o i n t s r % n o n - W a d a p o i n t s E= 0.27E= 0.47E= 0.77 (a) Hénon-Heiles (b) Newton method
Figure 4.
The role of the fattening parameter r . (a) The number of non-Wada points decreasesas the fattening parameter r increases for all the values of the energy studied in the H´enon-HeilesHamiltonian. Only three values of the energy are plotted for clarity, but we have checked thatthere is no relation between the percentage of non-Wada points and the value of the energy E ,i.e., there is no relation between the percentage of non-Wada points and the fractal dimension ofthe boundary. (b) The merging method converges for values of the fattening parameter r ≤ r > The second system where we have tested the merging method described before is theNewton method to find complex roots. This method can be described by the discrete complexvariable map z n +1 = z n − ( z R − / ( R · z R − ), where the parameter R determines the numberof roots and therefore the number of attractors N a = R . It has been reported that the basinsproduced by this complex variable map show the disconnected Wada property (the basins aredisconnected and also Wada) no matter the number of attractors determined by R [8, 24, 25].Thus, we ran the merging algorithm for an increasing number of roots R , and consequentlyof attractors N a . As shown in Fig. 4-(b), the merging method correctly classifies the basinsas Wada for all r > N a , which seldom appears intypical dynamical systems. Moreover, we have found no trivial relation between the numberof attractors N a and the percentage of non-Wada points. Again, we have performed thecomputations at different resolutions (up to 5000 × r needed for a correct classification ofthe basins does not directly depend on the fractal dimension of the boundaries nor on thenumber of attractors. This also proves that the method works correctly for disconnectedWada basins.Finally, we would like to mention another adjustment that could be added to the mergingalgorithm in case of need. Just as described before, the merging algorithm is an all or nothingtest. If there is a single pixel of a slim boundary that does not fit into a fat boundary, thenthe basin is labeled as non-Wada. However, it is clear that this can be too restrictivein some cases. For instance, if the basin is obtained by experimental procedures, it isvery likely to have some wrong pixels. In these cases, we could complement the mergingalgorithm with the measure of the fractal dimension of the non-Wada boundary, using abox-counting algorithm on the resulting image of the non-Wada points, for example. If thefractal dimension of these non-Wada boundary points is close to zero, then we can admitthat the basins have the Wada property, despite the misbehaved pixels. In any case, themerging method is able to determine whether a basin is Wada or not up to a given resolutionusing minimum requirements. IV. CONCLUSIONS
In the study of the asymptotic behavior of nonlinear dynamical systems, Wada basinsappear frequently. Initial conditions lying in the boundary of Wada basins can suffer anarbitrarily small perturbation leading the trajectory to any of the possible attractors of thesystem. This supposes a special kind of unpredictability different not only from basins withsmooth boundaries, but also from other fractal basins [6, 17].In this paper, we have seen how the action of merging the basins reveals a new aspectof Wada basins. Actually, Wada boundaries are those that remain unaltered under theaction of merging the basins. This perspective provides a new way to test Wada basins,that is faster than previous methods by orders of magnitude, and also much easier to use.Given a basin with three attractors with 1000 × ACKNOWLEDGMENTS
We acknowledge some enlightening discussions with Jim Yorke. This work has beensupported by the Spanish State Research Agency (AEI) and the European Regional Devel-opment Fund (FEDER) under Project No. FIS2016-76883-P. M.A.F.S. acknowledges thejointly sponsored financial support by the Fulbright Program and the Spanish Ministry ofEducation (Program No. FMECD-ST-2016).
AUTHOR CONTRIBUTIONS STATEMENT
A.D., A.W., and M.A.F.S. devised the research. A.D. and A.W. performed the numericalsimulations. A.D., A.W., and M.A.F.S. analyzed the results and wrote the paper.
ADDITIONAL INFORMATION
Competing interests : The authors declare no competing interests. [1] K. Yoneyama, Tokohu Math. J. , 43 (1917).[2] J. G. Hocking and G. S. Young, Topology (1988).[3] C. Kuratowski, Fundamenta Mathematicae , 130 (1924).[4] J. Kennedy and J. A. Yorke, Physica D , 213 (1991).[5] H. E. Nusse, E. Ott, and J. A. Yorke, Phys. Rev. Lett. , 2482 (1995).
6] A. Daza, A. Wagemakers, B. Georgeot, D. Gu´ery-Odelin, and M. A. F. Sanju´an,Sci. Rep-UK , srep31416 (2016).[7] L. Poon, J. Campos, E. Ott, and C. Grebogi, Int. J. Bifurcation Chaos , 251 (1996).[8] B. Epureanu and H. Greenside, SIAM Rev , 102 (1998).[9] D. Sweet, E. Ott, and J. A. Yorke, Nature , 315 (1999).[10] J. Aguirre, J. C. Vallejo, and M. A. F. Sanju´an, Phys. Rev. E , 066208 (2001).[11] J. Vandermeer, Ecol. Model. , 65 (2004).[12] A. Daza, A. Wagemakers, and M. A. F. Sanju´an, Commun. Nonlinear Sci. , 220 (2017).[13] Z. Toroczkai, G. K´arolyi, A. P´entek, T. T´el, C. Grebogi, and J. A. Yorke,Physica A , 235 (1997).[14] J. Aguirre and M. A. F. Sanju´an, Physica D , 41 (2002).[15] Y. Zhang and G. Luo, Phys. Lett. A , 3060 (2012).[16] Y. Zhang and G. Luo, Phys. Lett. A , 1274 (2013).[17] J. Aguirre, R. L. Viana, and M. A. F. Sanju´an, Rev. Mod. Phys. , 333 (2009).[18] H. E. Nusse and J. A. Yorke, Physica D , 242 (1996).[19] H. E. Nusse and J. A. Yorke, Phys. Rev. Lett. , 626 (2000).[20] J. S. E. Portela, I. L. Caldas, R. L. Viana, and M. A. F. Sanju´an,Int. J. Bifurcation Chaos , 4067 (2007).[21] A. Daza, A. Wagemakers, M. A. F. Sanju´an, and J. A. Yorke,Sci. Rep-UK , srep16579 (2015).[22] F. Blesa, J. M. Seoane, R. Barrio, and M. A. F. Sanju´an,Int. J. Bifurcation Chaos , 1230010 (2012).[23] E. E. Zotos, Meccanica , 2615 (2017).[24] M. Frame and N. Neger, Coll. Math. J. , 192 (2007).[25] P. Ziaukas and M. Ragulskis, Nonlinear Dyn. , 871 (2017)., 871 (2017).