Chaotic Magnetic Field Lines and Fractal Structures in a Tokamak With

Chaos, Solitons and Fractals 104 (2017) 588–598

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena

journal homepage: www.elsevier.com/locate/chaos

Chaotic magnetic ﬁeld lines and fractal structures in a tokamak with magnetic limiter

∗ A.C. Mathias a, T. Kroetz b, I.L. Caldas c, R.L. Viana a, a Department of Physics, Federal University of Paraná, Curitiba, Paraná, Brazil b Department of Physics, Federal Technological University of Paraná, Pato Branco, Paraná, Brazil c Institute of Physics, University of São Paulo, São Paulo, Brazil

a r t i c l e i n f o a b s t r a c t

Article history: Open hamiltonian systems have typically fractal structures underlying chaotic dynamics with a number Received 12 April 2017 of physical consequences on transport. We consider such fractal structures related to the formation of

Revised 12 September 2017 a chaotic magnetic ﬁeld line region near the tokamak wall and the corresponding ﬁeld line dynamics, Accepted 15 September 2017 described by a two-dimensional area-preserving map. We focus on the exit basins, which are sets of points which originate orbits escaping through some exit and are typically fractal, also exhibiting the Keywords: so-called Wada property. We show qualitative as well as quantitative evidences of these properties. Fractal basin boundaries

1. Introduction be highly non-uniform and thus the risk of localized loadings can only be diminished, if not completely avoided [11–16] . An outstanding problem in the physics of magnetically confined This non-uniformity of chaotic regions is ultimately the con- fusion plasmas is the control of the interaction between plasma sequence of the underlying mathematical structure of chaotic or- particles and the inner wall in which they are supposed to be bits in area-preserving mappings, since they are structured on a contained (e.g. in a tokamak) [1] . Due to the so-called anomalous complicated homoclinic tangle formed by intersections of invariant transport, plasma particles escape towards the wall and are lost, manifolds stemming from unstable periodic orbits embedded in reducing the plasma density and thus the quality of confinement the chaotic orbits [17,18] . Hence it is important to use such knowl- [2,3] . Moreover, since many of these particles can be highly en- edge to understand how the escape is structured near the tokamak ergetic they interact with the metallic wall causing the release of wall, locating possible “hot spots” (i.e. high loading parts of the contaminants through sputtering and other processes [4] . One of wall) that should be protected so as not to cause extremely local- the proposed ways to control plasma-wall interactions is the use of ized particle fluxes from the plasma, for example by using divertor chaotic magnetic fields near the tokamak wall, spanning both the configurations [19–21] . plasma outer edge and the scrape-off layer between the plasma The main goal of this paper is to show the presence of frac- and wall [5–9] . tal structures in the chaotic region of escaping magnetic field lines Such a chaotic field line region can be obtained by applying caused by an ergodic limiter, by using a simple area-preserving suitable magnetic perturbations that break the integrability of the mapping proposed by Martin and Taylor [22] . We observe the pres- magnetic field line flow, causing the appearance of area-filling ence of fractal basins, in the case of two exits [17] , and the so- stochastic lines (ergodic magnetic limiter) [10] . The initial claim called Wada property in the case of three (or more) exits [23] . We was that such chaotic field would uniformize heat and particle emphasize the physical consequences of those mathematical prop- loadings, so diminishing localized attacks and their undesirable erties in terms of escape times and magnetic footprints caused by consequences [5–7] . However, both theoretical and experimental the self-similarity of the fractal structures [24] . evidences suggest that this is not true, i.e. the chaotic region can In previous papers dealing with exit basins due to ergodic limiter fields we have considered the fractality of exit basins from the direct calculation of the box counting dimension of the exit basin boundaries, and characterized the Wada property by a qualitative ∗ Corresponding author. argument [11,12] . This may be an insufficient characterization of E-mail addresses: viana@fisica.ufpr.br , [email protected] (R.L. Viana). http://dx.doi.org/10.1016/j.chaos.2017.09.017 0960-0779/© 2017 Elsevier Ltd. All rights reserved. A.C. Mathias et al. / Chaos, Solitons and Fractals 104 (2017) 588–598 589

Fig. 1. Schematic figure showing the basic tokamak geometry in (a) local and (b) rectangular coordinates. the exit basin structure because we would like to know the an- plified rectangular geometry to describe the situation therein, so swer to questions like: (i) to which degree are the fractal basins avoiding the use of unnecessarily complicated coordinate system. mixed together; or (ii) to what extent is the Wada property ful- In this slab geometry description we use three coordinates ( x, y, z ) filled? In this paper we provide answers to these questions by de- to describe a field line point, where x = bθ stands for the rectified veloping further the analysis by presenting quantitative techniques arc along the wall, and y = b − r is the radial distance measured introduced recently by Sanjuán and coworkers [25–27] . from the wall (which is located at y = 0 ) [ Fig. 1 (b)]. The coordi- This paper is organized as follows: in Section 2 we outline the nate z = Rφ measures the rectified arc along the toroidal direction. physical model used to describe the ergodic magnetic limiter as The magnetic field line equations in these coordinates are writ- well as the mapping equations governing the dynamics of the cor- ten as responding magnetic field line flow. Section 3 considers the frac- dx dy dz tal nature of the escape basin boundary, both qualitatively (us- = = (1) ing the invariant manifold structure as a guide) and quantitatively Bx By Bz (by computing the box-counting dimension of the boundary). In where the equilibrium magnetic field is B 0 = B T + B P . In the Section 4 we use the concepts of basin entropy and basin boundary lowest-order approximation we set B T = B 0 eˆ z and the field line entropy to characterize the fractality of the exit basins. Section 5 is equations can be written as [30] devoted to a discussion of the Wada property for three exit basins and to the application of a technique (grid approach) to verify in dx 1 = [ α + sy + o(y 2 )] , (2) what extent the Wada property is fulfilled by the system. The last dz 2 πR 0 Section is devoted to our Conclusions.

dy 2. Magnetic field line map = 0 , (3) dz A tokamak is a toroidal device for magnetic confinement of fu- where the parameters α and s are related to the equilibrium sion plasmas, in which the plasma is generated by the ionization magnetic field components, by expanding the safety factor q ( r ) of an injected low-pressure gas (hydrogen) through an induced emf in a power series around the plasma boundary r = b, such that caused by the discharge of a capacitor bank [28] . There are two ba- α = π / = π ( ) / 2 2 b qb and s 2 bq b qb [30]. Let us take, for example, a sic confining magnetic fields: a toroidal field B generated by coils ( ) = 2 / 2 = π/ T quadratic profile q r qb r b . This yields s 4 qb . Hence our mounted externally to the toroidal chamber and a poloidal field B = π = , P choice s 2 would correspond to qb 2 which is a value in generated by the plasma itself. In the axisymmetric case the result- agreement with Tokamak experiments. ing magnetic field lines are helicoidal, winding on nested toroidal The equations (2) and (3) can be integrated, using z as the in- surfaces (called magnetic surfaces) [29] . dependent variable, such that a Poincaré map is obtained by sam- A general non-axisymmetric perturbation will break up a cer- pling the values of x and y just after the n th crossing of a surface of tain number of these magnetic surfaces. As a result, some of the section at z = 0 (the z -direction has a well-defined periodicity cor- magnetic field lines will no longer lie on surfaces, ergodically fill- responding to the long way around the torus). In this way we de- ing a bounded volume inside the toroidal device. Such magnetic fine discrete-time variables ( x n , y n ) which are known as functions field lines will be called chaotic but, since we are dealing with of the same variables at the previous crossing, namely (x n −1 , y n −1 ) . strictly time-independent configurations, the word chaos must be The axisymmetric configuration (closed, toroidal magnetic sur- intended in a Lagrangian sense: two field lines, originated from faces) can be described in such a framework by a simple twist map very close points, separate spatially at an exponential rate as wind- (after a convenient rescaling of variables, such that 0 ≤ x < 2 π and ing around the torus. Such description can be made rigorous by y ≥ 0) considering a hamiltonian description of magnetic field lines [10] . The basic tokamak geometry is depicted in Fig. 1 (a) using cylin- x + x x + s (y ) y n 1 = n = n n n , φ T1 (4) drical coordinates (R, , Z): the tokamak vessel is a torus of major y n +1 y n y n radius R = R 0 with respect to the major axis parameterized by coordinate Z ). The major radius stands for the minor axis, such that The equality of y for consecutive iterations of the map reflects the the position of a field line point can be assigned to the local co- fact that a field line always lies on a magnetic surface y = const. . ordinates ( r, θ, φ), where R = R 0 + r cos θ and Z = r sin θ. The vari- In this paper we consider a particular kind of symmetry- ables θ and φ are also called poloidal and toroidal angles, respec- breaking perturbation on the above equilibrium configuration, the tively. The tokamak wall is located at r = b. so-called ergodic limiter, which is a ring-shaped coil at r = b with In this paper we will focus on the production of chaotic field m pairs of straight wire segments of length , aligned with the lines near the wall of the toroidal device. Hence we can use a sim- toroidal direction and conducting a current I [ Fig. 2 ]. The ergodic 590 A.C. Mathias et al. / Chaos, Solitons and Fractals 104 (2017) 588–598

[31] . On the other hand, the separation between two adjacent periodic islands is δy m ≈ 2 π/ s and does not depend on p . A consequence of the above properties is that, if the limiter strength p is increased, only the islands closer to the tokamak wall at y = 0 are significantly affected. Let us thus consider only the islands with m = 1 and m = 2 . The so-called stochasticity parameter is defined as √ + y1 y2 ps −π/s −π/s S 12 = = e 1 + e . (13) δy 1 π For the case s = 2 π (this value will be kept constant from now on) the exponential factors add up practically to the unity and S 12 ≈ Fig. 2. Schematic figure showing the ergodic magnetic limiter. 2 p/π. A rough but useful estimate for the threshold of global chaos is given by the well-known Chirikov criterion S 12 = 1 [31] , which indicates that after p ≈ 1.6 there is an area-filling chaotic or- magnetic limiter generates a spatially localized magnetic field bit encompassing the region near the tokamak wall, which was the

˜ ( , ) , ≤ ≤ intended function of the magnetic limiter. ( , , ) = B1 x y if 0 z B 1 x y z (5) An example is provided by the phase portrait depicted in 0 , if < z < 2 πR 0 , Fig. 3 (a), obtained for p = 2 . 0 . For this value of the perturbation where = strength the two first periodic islands (with m 0 and 1) have al- μ ready been destroyed, whereas remnants of the island with m = 2 0 mI −my/b mx B˜ 1 x (x, y ) = − e cos , (6) = π bπ b can still be observed centered about x . This remnant island is also distorted in shape thanks to the nonzero shear s . For a larger μ value of p [Fig. 3(b)] even that remnant is further destroyed. In 0 mI −my/b mx B˜ (x, y ) = e sin , (7) both cases a large chaotic orbit near the tokamak wall has been 1y bπ b developed. The chaotic region displayed in Fig. 3 (a) and (b) was obtained = × = B˜ 1 z (x, y ) = 0 (8) from a mesh of N 8 20 160 initial conditions distributed along both x and y directions, in the following way: The limiter field is superimposed to the equilibrium field, i.e. ( , ) = ( π/ , ( − ) / ) , ≤ ≤ , ≤ ≤ , we make B = B 0 + B 1 , leading to the destruction of a number of x0 k y0 k 4 ymax ymin 20 0 k 7 0 19 magnetic surfaces. Martin and Taylor have, by an ingenious inte- such that, for every initial condition, we computed the 10 0 0 first gration of the field line equations (1) , obtained an analytical map map iterations. T 2 representing the limiter field action on the magnetic field lines twisting according to the map (4) [22] . The composition of the two 3. Fractal escape basin boundaries mappings, T = T 1 ◦ T 2 yields A closer look at Fig. 3 (a) and (b) shows that the area-filling x + x x + s y + g(x , y ) n 1 = T n = n n n n , (9) chaotic orbits therein are far from being uniform, i.e. the distribu- + ( , ) yn +1 yn yn h xn yn tion of iterates (corresponding to intersections of field lines with a where fixed surface of section) is non-uniform. In some cases, it appears that “voids” develop within these chaotic orbits, what is obviously ( − −y ) cos x p e cos x h (x, y ) = ln , (10) only an artifact of the finite number of iterations used to gener- cos x ate those plots. If we were to plot an infinitely large number of iterates then the whole available region would be filled. However, − g(x, y ) = −p e y cos x + s h (x, y ) , (11) this would not add to the clarity of the picture, hence we use a moderately large number of map iterates to illustrate the chaotic = μ 2 / 2 π in which p 0 m I B0 b is a positive parameter representing orbits. the strength of the limiter magnetic field. It depends on the limiter Nevertheless, those non-uniformities in the chaotic orbits of current and length [30] . If p = 0 (no limiter field) we recover the Fig. 3 teach us some lessons on the type of behavior we expect simple twist map (4) . to see for charged particles belonging to the confined plasma. The A direct calculation of the corresponding jacobian determinant most basic motion for them, when in a magnetic field, is a gyra- shows that the above map is area-preserving. This is a neces- tion with finite Larmor radius and whose guiding center follows, sary requirement due to the divergenceless property of the mag- in a first approximation, the field lines (in this case we would ne- netic fields involved. Being a perturbed twist mapping, we expect glect any kinds of particle drifts due, for example, to curvature and a rich dynamical behavior from its iteration. The general aspects non-uniformities of the magnetic field) [32] . In the realm of this of the dynamics are well-known [22,30] . The fixed points of the approximation it is possible to describe the transport of charged map (9) are roughly located at (0, 2 πm / s ) and ( π, 2 πm / s ), where plasma particles by considering the topology of the magnetic field m = 0 , 1 , 2 , . . .. The former are saddles and the latter, centers of pe- lines, in such a way that particles will closely follow those lines. riodic islands with half-widths This approximation is justified if the characteristic drift times are

larger than the typical time used to describe a given plasma phe- p −πm/s y m ≈ 2 e , (12) nomenon. s Since the chaotic orbits generated by limiter action intercept which decrease with y , thanks to the fact that the limiter mag- the tokamak wall, a given particle following the corresponding netic ﬁeld strength decays exponentially as we go inside the ﬁeld lines will eventually hit the tokamak wall and be lost (phys- toroidal vessel. The non-equality of the island widhts and the non- ically it corresponds to a metallic wall which absorbs particles, periodicity of the y -direction make the map (9) in many ways very perhaps with the production of undesirable quantities of contami- different from the widely studied Chirikov–Taylor standard map nants by sputtering and other processes). The original design of the A.C. Mathias et al. / Chaos, Solitons and Fractals 104 (2017) 588–598 591

Fig. 3. Phase portraits of the map (9) for s = 2 π and p = 2 . 0 (a) and 2.5 (b). ergodic limiter was based on the assumption that a chaotic orbit gion the basins of A and B are intermixed, sometimes in a very in- would uniformize particle fluxes near the tokamak wall, so avoid- tricate way. The fraction of initial conditions escaping through the ing localized particle loadings [5–7] . However, it turns out that the exits A and B with respect to the total number of escaping initial chaotic orbit itself is highly non-uniform, hence it is somewhat un- conditions are 46.71% and 53.29%, respectively. avoidable to have some degree of non-uniform particle loading on It is worthwhile to note that there are initial conditions the wall. (painted in green) very close to the exit A that nevertheless be- One numerical characterization of the non-uniformity inherent long to the basin of B . This suggests the existence of escape chan- to the chaotic orbit is the computation of the so-called escape nels that in some way provide a preferential escape through B . On basins. Such a basin is the set of points in the Poincaré phase increasing the perturbation strength [ Fig. 4 (b)] it is clearly visible plane that generate orbits which escape through a specified exit an increase of the red points belonging to the basin of A . Indeed, [17] . Hence the application of this concept requires an open sys- the fraction of initial conditions escaping through A and B is re- tem. Since a field line (and the corresponding particle that follows spectively 29.46% and 70.54%. In a certain sense the intermixing of it) is considered lost once it hits the tokamak wall at y = 0 we di- the basins has been enhanced with increasing perturbation. vide this line into two segments, or exits: A : (0 ≤ x < π, y = 0) , From a mathematical point of view, the complicated structure and B : (π ≤ x < 2 π, y = 0) . of the escape basins revealed by Fig. 4 can be understood by con- We divide the phase plane into a 10 3 × 10 3 mesh of points, each sidering the invariant chaotic sets underlying the large chaotic or- of them being the initial condition (x0 , y0 ) for an orbit of the map bit near the tokamak wall. We begin by considering an unstable (9) . Just after this orbit crosses the wall at y = 0 we depict the periodic orbit embedded in the chaotic region, with its stable and initial condition pixed in red if the escape occurs through exit A unstable manifolds. They are invariant sets of points which asymp- and green if through exit B . The union of the red (green) regions tote to the periodic orbit under forward and backward iterations of is a numerical approximation for the basin of the exit A ( B ). Since the map, respectively [33] . The intersection of the stable and un- the map gives us the field lines coordinates at the Poincaré sec- stable manifolds, called chaotic saddle , is a non-attracting in- tion, the collision with the wall usually happens between two suc- variant chaotic set containing a dense orbit [34] . cessive iterations of the map. Therefore, the escape ( y e = 0 ) occurs The boundary of the escape basins coincides with the stable when y n < 0 < y n +1 . The polar coordinate x e related to the escape manifold of the chaotic saddle (actually the boundary is the clo- is obtained by linear interpolation: x e = x n − [(x n +1 − x n ) / (y n +1 − sure of the stable manifold). The convoluted aspect of the escape y n )] y n . It must be emphasized that, since x is defined in the in- boundary reveals a smooth component along the stable manifold terval [0, 2 π), if the escape point x e is outside [0, 2 π) we first and a fractal Cantor-like set transversal to it. It is thus necessary reduce to this interval to decide through which exit it is escaping. that the escape boundary crosses the stable manifold in order to Each initial condition is iterated by a maximum of 10 4 times. If have a fractal structure [35] . the initial condition has not escaped after this limit, we consider The basic dynamical behavior in the phase space region R con- it as non-escaping. In fact this does not imply that the initial containing the chaotic saddle is the following: the orbits of all ini- dition cannot escape at all (since it might escape after, say, 10 5 tial conditions in R will eventually leave R through either one of iterations). However, the number of this false-positive points is ex- the two exits A or B , except for a measure zero set of initial con- pected to be relatively small and does not affect appreciably our ditions lying on the stable manifold of . Those initial conditions main results. that are off but close to the stable manifold are first attracted to In Fig. 4 (a) we show the exit basins for the same parameters and stay close to its neighborhood for a number of map itera- used to generate the Poincaré phase plane depicted in Fig. 3 (a). tions, before they are repelled, following the unstable manifold of The white regions correspond to initial conditions which do not [36] . The closer the initial condition is to the stable manifold of escape (within 10 4 iterations), corresponding to orbits inside peri- the longer the number of map iterations it takes before leaving odic islands as well as chaotic orbits separated by KAM tori from the phase space region R . the large chaotic sea near the wall. Within most of the chaotic re- 592 A.C. Mathias et al. / Chaos, Solitons and Fractals 104 (2017) 588–598

Fig. 4. Escape basins of exits A : (0 ≤ x < π, y = 0) (red) and B : (π ≤ x < 2 π, y = 0) (green) for s = 2 π and p = 2 . 0 (a) and 2.5 (b). White pixels stand for initial conditions which do not escape at all. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

initial conditions constitute an approximation to the stable manifold of the chaotic saddle [ Fig. 6 (a)]. On the other hand, the end points of the trajectories related to the recorded initial conditions (i.e., the orbit points for those trajectories after n c = 100 map iterations), are an approximation to the unstable manifold of the saddle [ Fig. 6 (b)]. The points from the middle of these trajectories ( n¯ = 0 . 5 n c = 50 ) are thus representations for the chaotic saddle [ Fig. 6 (c)]. On comparing Fig. 6 (a) and 4 (b), both obtained with the same perturbation strength p = 2 . 5 , it is apparent that the part shown of the stable manifold traces out the boundary between the exit basins of A and B . Moreover, the shape of the unstable manifold shown in Fig. 6 (b) helps us to understand the asymmetry between the basins of exits A and B , as illustrated by Fig. 4 . Points belonging to the green basin escape through exit B and points nearby it are mostly green. On the other hand, many green points can be found very close to the exit A (whose basin points are painted red). This is because the unstable manifold intersects the tokamak wall mostly at exit B , as clearly seen in Fig. 6 (b). An important physical consequence of the invariant manifold Fig. 5. Connection lengths (measured in number of map iterations) for initial conditions in the phase portrait of the Martin-Taylor map for s = 2 π and p = 2 . 5 . structure revealed by Fig. 6 is that the spatial distribution of connection lengths is highly non-uniform, as shown by Fig. 5 . By comparing it with Fig. 6 (a) there follows that the regions of higher connection lengths tend to follow the stable manifold of the boundary.

We obtained the invariant manifolds of the chaotic saddle using While points exactly on the stable manifold will not escape at all, the so-called sprinkler method, which consists in partitioning the points off but very close to the stable manifold will take more map R phase space region into a fine mesh of N0 points [37,38]. We iterations to escape. compute for each point (initial condition) the connection length, The fractal nature of the escape basin boundary can also be i.e. the number of map iterations it takes for the corresponding quantitatively investigated by computing its box-counting dimen- R orbit to leave [Fig. 5]. The initial conditions with a connection sion using the uncertain fraction method. We randomly choose a length larger than some specified value nc represent an approxi- large number of initial conditions (x0 , y0 ) in the chaotic region mation to the stable manifold of , and their last iterations before and iterate the map (9) until the corresponding orbit reaches the R leaving are an approximation of the unstable manifold [36,39]. tokamak wall at some exit ( A or B ). For each initial condition we = ξ , Those initial conditions with a given connection length n¯ nc choose another one in a random way within a ball of radius ε and < ξ < where 0 1 are an approximation to the chaotic saddle . repeat the iteration. If this second initial condition goes to a differ- ξ The values of nc and must be chosen after trial and error, ent exit the first one is dubbed ε-uncertain. The fraction of uncer- but the numerical results seem not to be substantially affected tain conditions f ( ε) is expected to increase with ε as a power-law α by them. nc must be large compared with the average connec- f ( ε) ∼ ε , where α is the so-called uncertainty exponent [35,41] . tion length, and ξ = 0 . 5 seems to be a good choice for most dy- Let D p be the phase space dimension. If D is the box-counting = π namical systems [40]. Fig. 3 shows our results for the case s 2 dimension of the basin boundary there results that the uncertainty and p = 2 . 5 , where we have used n c = 100 and ξ = 0 . 5 . Starting exponent satisfies the relation α = D p − D . A heuristic proof of this = 4 × 4 from a grid of N0 10 10 initial conditions, we record those result is the following [35] : let N ( δ) be the minimum number of initial conditions with a connection length larger than n c . Such A.C. Mathias et al. / Chaos, Solitons and Fractals 104 (2017) 588–598 593

Fig. 6. (a) Stable and (b) unstable manifold and (c) chaotic saddle for a chaotic orbit in the case s = 2 π and p = 2 . 5 . There was used the sprinkler method with n c = 100 iterations and ξ = 0 . 5 .

D p -dimensional hypercubes of side δ required to completely cover 4. Basin entropy the escape basin boundary. The box-counting dimension of the latter is Besides the calculation of the basin boundary box-counting dimension, as shown in the last section, we can compute the so- ln N(δ) D = lim , (14) called basin entropy, which quantifies the degree of uncertainty δ→ 0 ln (1 /δ) due to the fractality of the basin boundary [26,27] . Let us consider a phase space region which includes a part of the escape basin (δ) ∼ δ−D δ = ε, in such a way that N . If we set the volume of the boundary, and divide this region into boxes by using a fine mesh, ε uncertain region in the phase space will be N( ) times the volume assigning to each mesh point a random variable. The values taken ε D p of the Dp -dimensional cubes, which is . Since the initial condi- on by this random variable characterize each different escape, and tions are uniformly chosen over the phase space region, the uncer- the basin entropy is obtained by application of the information (ε ) ε D p = ε D p −D tain fraction is of the order of the total volume N . entropy definition to this set. In the general case, let the system α = − Hence Dp D. A rigorous proof of this statement can be found = consists of NA escapes (NA 2 in the present case), and the phase = , α = − , in Ref.[35]. In our case we have Dp 2 for which 2 D in space region is covered by a mesh of boxes with a given size. = α = , such a way that a smooth boundary with D 1 has 1 whereas We assign to each mesh point, or initial condition, a color labeled 0 < α < 1 characterizes a fractal basin boundary. from 1 to N A . ε ε Fig. 7(a) and (b) show log-log plots of f( ) vs. for two differ- The colors in each mesh point stand for an integer random vari- ent values of the perturbation strength. The solid lines represent able j defined in the interval [1, N A ], and the values are distributed least squares fits, and yield the following values of the dimensions according to a given probability. Let pi, j denote the probability that = . ± . = . of the fractal escape boundaries: D 1 10 0 02 for p 2 0 and the j th color is assigned to the i th box, where i = 1 , 2 , . . . , N, N be- = . ± . = . , D 1 25 0 02 for p 2 5 confirming our first impression that ing the number of mesh points, or boxes. On supposing statisti- the increase of perturbation leads to a more complex basin bound- cal independence, the information (Gibbs) entropy of the i th box is ary structure. Moreover, the fractality of the basin boundary leads [26] to important physical consequences on the predictability of the fi- m nal state the system will asymptote to. i 1 = , Si pi, j log (15) As an example, consider the escape basin structure depicted in , = pi j Fig. 4 (b), for which α ≈ 2 − 1 . 25 = 0 . 75 . Any basin point (initial j 1 ∈ condition for iterate the map) is known within a given uncertainty where mi [1, NA ] is the number of colors for the ith box. The total ε, such that the fraction of uncertain points scales as ε0.75 . Suppose entropy for the mesh covering the region is obtained by sum-

= N we can reduce this uncertainty by 50%. There follows that the frac- ming over the N boxes, or S i =1 Si . The basin entropy is then = / tion of uncertain point will be reduced by only a factor of 0.59. If the total entropy per box, or Sb S N. For a single escape the basin α decreases to zero this reduction is even shorter, hence a prac- entropy is zero, which means that there is no uncertainty at all. On tical consequence of the fractality of the escape boundary is that the other hand, if we have N A equiprobable escapes the basin en- = , our ability of improving the knowledge of the future state of the tropy reaches a maximum of Sb log NA so characterizing a com- system is impaired [41] . pletely randomized escape basin structure. 594 A.C. Mathias et al. / Chaos, Solitons and Fractals 104 (2017) 588–598

Fig. 7. Uncertainty exponent versus uncertainty radius for (a) p = 2 . 0 and (b) p = 2 . 5 .

It is also possible to adapt this entropy calculation to evaluate the uncertainty related to the escape basin boundary, rather than the uncertainty of the escape basin itself. In order to accomplish this new task we repeat the above calculation, but restricting the computation of the total entropy to the number Nbb of boxes which contain more than one color. In this way we obtain the boundary = / basin entropy, or Sbb S Nbb [26]. In the case analyzed in the previous section, there are two exits, namely A : (0 ≤ x < π, y = 0) (red basin in Fig. 4 ) and B : (π ≤ x < 2 π, y = 0) (green basin in Fig. 4 ). Choosing a representative phase space region comprising both exit basins (and perhaps points that do not exit at all) we cover it with a grid of 10 0 0 × 10 0 0 points of each basin, distributed inside 4 × 10 4 boxes. For each grid point we computed a maximum number of 10 4 iterations of the map, and exclude from the statistics those initial conditions leading to orbits that do not escape during this maximum iteration Fig. 8. Basin entropy and basin boundary entropy for the case of two exits as a time. Naming n A and n B the number of points into each box corre- function of p. We also plot the fraction occupied by the red basin. (For interpre- sponding to the exits A and B , respectively, we compute the prob- tation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.) abilities n A n B p A = , p B = , (16) n A + n B n A + n B such that the entropy for each box is (we use base 2 logarithms)

S = −p log p − p log p . (17) A A B B that the red and green exit basins in Fig. 4 become progressively

The basin entropy Sb is obtained by summing up the entropy val- more mixed and involved, their fractality being most pronounced ues with respect to those boxes for which all the initial condi- at p ≈ 1.75 and decreases for higher values of p . tions escape and dividing the result by the number of boxes Nb for The behavior of the entropies can be qualitatively understood which all initial conditions escape. The boundary basin entropy is by comparing them with the fraction A occupied by the red basin obtained similarly, excluding from the summation those boxes for (corresponding to the exit through the ﬁrst half poloidal arc on = = which either pA 0 or pB 0. In other words, for the computing the tokamak wall), which is also depicted in Fig. 8. Both Sb and Sbb the boundary basin entropy we consider only those boxes which follow the increase of the red basin fraction. A cursory inspection have at least one boundary in their interior, such that Sbb equals of Fig. 4 shows a predominance of the green basin over the red the summation of the entropy S for all boxes that contain at least one. Since the basin entropies measure the degree of mixing of one boundary divided by the number Nbb of boxes having at least both basins, there follows that the entropies increase as the red = one boundary. Since NA 2 both entropies vary between zero and basin becomes larger and comparable to the green basin. Indeed, at log 2 = 1 . the maximum value of the basin entropy (for p ≈ 1.75) the fraction

We plot in Fig. 8 the values of the basin entropy Sb and the of the red basin is roughly 50%. As the perturbation p increases, basin boundary entropy Sbb relative to the case of the escape so the fraction of the red basin also increases, but this time the through A : (0 ≤ x < π, y = 0) (red points in Fig. 4 ) and B : (π ≤ fraction of the green basin will decrease, and the entropies become x < 2 π, y = 0) (green points in Fig. 4 ), as a function of the parame- smaller again. Hence, besides the exit basin boundary dimension, ter p . As a trend the entropies increase as p goes from zero to 1.75 it is important to compute the basin entropies to know at what and suffer a decrease afterwards. The increase of entropy means extent the basins are mixed together. A.C. Mathias et al. / Chaos, Solitons and Fractals 104 (2017) 588–598 595

Fig. 9. Escape basins of exits A : (0 ≤ x < 2 π/ 3 , y = 0) (blue), B : (2 π/ 3 ≤ x < 4 π/ 3 , y = 0) (red), and C : (4 π/ 3 ≤ x < 2 π, y = 0) (green), for s = 2 π and p = 2 . 0 (a) and 2.5 (b). White pixels stand for initial conditions which do not escape at all. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

5. Wada escape basin boundaries presence of strips of the three basins in ﬁner scales, suggesting that at least some of the boundary points have the Wada property, If we divide the tokamak wall into three or more partitions, since a small neighborhood of such point will intersect all basins. each of them can be considered a new exit with its own escape Although a compelling evidence, this criterion does not indi- basin. For example we can divide the tokamak wall into three cate which boundary points have the Wada property. If many but exits, namely A : (0 ≤ x < 2 π/ 3 , y = 0) , B : (2 π/ 3 ≤ x < 4 π/ 3 , y = not all boundary points do, then we have only partial fulﬁllment 0) , and C : (4 π/ 3 ≤ x < 2 π, y = 0) . In Fig. 9 (a) we plot the corre- of this property. This characterization is provided by the grid ap- sponding escape basins in different colors (blue for the basin of A , proach proposed in Ref. [25] , by considering a bounded phase space = . ≥ = , , , red for B and green for C) for p 2 0. The fraction of initial con- region containing NA 3 disjoint escapes Aj , j 1 2 NA the cor- B ditions escaping through the exits A, B and C with respect to the responding escape basins being denoted by j . The region is = { , , . . . , } , total number of escaping initial conditions are 35.58%, 36.46%, and covered by a set of grid boxes P b1 b2 bK where bi de-

27.96%, respectively. A similar situation is displayed in Fig. 9(b), ob- notes the ith box, and such that any pair of boxes bi and bj are tained for p = 2 . 5 , and for which the fraction of initial conditions non-overlapping. escaping through the exits A, B and C changed to 54.98%, 33.08%, By iterating the map we can know, for each point ( x, y ) ∈ to and 11.94%, respectively. which exit Aj the corresponding trajectory escapes through, i.e. to B Given that, for two basins the boundary is a fractal curve, which escape basin j it belongs to. We thus define an integer- one may wonder what this would mean for a higher number of valued function C(xi ; yi ) related to the initial condition (xi ; yi ), such ( ; ) = ( ; ) ∈ B ( ; ) = basins. The answer lies in the so-called called Wada property: if that C xi yi j if xi yi j and C xi yi 0 if (xi ; yi ) does not B the boundary between two basins is smooth, three of such basins belong to any of such basins j . For any rectangular box bj we de- ( ) = ( ; ) , have only one common boundary point. However, if the boundary fine C b j C xi yi where (xi ; yi ) is the point at the center of the is fractal (non-smooth) there may be an infinite number of such box taken as an initial condition. If (xi ; yi ) does not belong to any ( ) = common points [42]. of the escapes, then C b j 0.

Let us consider the escape basin of some exit, say A. It has a In addition we denote by b(bj ) the collection of grid boxes con- boundary point p if every open neighborhood of p intersects the sisting of bj and all the grid boxes having at least one point in basin of A and at least the basin of other exit, like B or C. We common with bj . For example, in a two-dimensional phase space × define the basin boundary as the set of all boundary points corre- region b(bj ) turns to be a 3 3 collection of boxes, with bj being sponding to that basin. the central box. The number of different colors in b(bj ) is denoted = The boundary point p is called a Wada point if every open M(bj ). Provided M(bj ) 1, NA we select the two closest boxes in neighborhood of it intersects at least three different basins, in our b(bj ) with different colors and draw a line segment between them, case A, B , and C [43] . If every boundary point of the basin of A is a computing the color of the midpoint for the line segment. If the ( ) = Wada point, then the boundary is said to possess the Wada prop- computed color completes all colors inside b(bj ) then M b j NA erty. A necessary (albeit not sufficient) condition for the latter to and we stop the procedure. Otherwise we iterate this procedure ( ) = , hold is that the unstable manifold of p must intersect the basin of by choosing intermediate points until M b j NA or the number every exit [44,45] . of points exceeds a limit. ( ) = In order to make a first check for the validity of this condi- Let Gm be set of all the original grid boxes such that M b j m. = tion, in Fig. 10(a) we exhibit a magnification of a portion of the If m 1 then the corresponding set (G1 ) contains points belonging exit basin structure, depicted as a rectangular box in Fig. 9 (b). The to the interior of the j th escape basin, because all the boxes inside = black lines in Fig. 10(a) represent a portion of the unstable man- the ball b(bj ) have the same color. For m 2 the set G2 belongs ifold of some periodic orbit embedded in the chaotic region, and to the boundary between two basins, since there are two different it is clearly seen that such manifold intersects all the three escape colors inside the ball b(bj ). Analogously, the set G3 contains points

q basins. Successive magniﬁcations [Fig. 10(b) and (c)] conﬁrm the satisfying the Wada property. We call Gn the set Gn at the qth pro- 596 A.C. Mathias et al. / Chaos, Solitons and Fractals 104 (2017) 588–598

Fig. 10. Successive magniﬁcations of the escape basins of exits A : (0 ≤ x < 2 π/ 3 , y = 0) (blue), B : (2 π/ 3 ≤ x < 4 π/ 3 , y = 0) (red), and C : (4 π/ 3 ≤ x < 2 π, y = 0) (green), for s = 2 π and p = 2 . 5 . The black points in (a) indicate a portion of the unstable manifold of some periodic orbit embedded in the chaotic region. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

cedure step. Calling N (G ) the number of points of the set G we compute q N (G ) = m , Wm lim (18) q →∞ NA N ( q ) j=2 G j

where m ∈ [2, N A ]. If W m = 0 the system has no grid boxes that belong to the boundary separating the m escape basins. Conversely, if W m = 1 then all the boxes in the boundary separate m escape basins. If W = 1 the system is said to have the Wada property, since it will NA always be possible to ﬁnd a third color between the two other colors. If 0 < W m < 1, with m ≥ 3, the basins partially satisfy the Wada property. = In the case depicted in Fig. 9(b) we have NA 3 exits and thus we use (18) to compute, for q procedure steps of computing colors at the intermediate points between adjacent boxes, the following ratios Fig. 11. Basin structure of Fig. 9 (b), showing in black points belonging to G 1 set

N (G 2 ) (internal points), in red points of the G2 set (boundary points between two basins), W 2 = , (19) and in blue points of the G set (boundary points between three basins), after N (G ) + N (G ) 3 2 3 q = 18 refinement steps. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) N (G 3 ) W 3 = , (20) N (G 2 ) + N (G 3 ) = and check, for each qth iteration of the procedure, whether or not points), G2 (red points) and G3 (blue points), after q 18 proce- ( − ) points of G2 may belong to G3 by testing 2 q 2 initial conditions dure of refinement steps. intermediate between the central box and a neighbor box of differ- Once we have tracked a number of points with the Wada prop- ( ent color. If some of those 2 q − 2) intermediate initial conditions erty we can quantify at what extent this property is fulfilled in show the missing third color, the central box will be reclassified the system. Fig. 12 (a) shows a histogram for the number of points

as G3 and so on. We depict in Fig. 11 points classiﬁed as G1 (black initially classiﬁed as belonging to the set G2 but which are found A.C. Mathias et al. / Chaos, Solitons and Fractals 104 (2017) 588–598 597

Fig. 12. (a) Number of reclassified points in each refinement step, using the grid approach for the basin structure depicted in Fig. 9 (b). (b) Values of W 2 and W 3 as a function of the refinement step.

to belong to the set G3 in each refinement step q. We see that, the exit basin boundary were smooth (with an integer dimension) after three refinement steps the number of reclassified points de- an improvement in the precision to which we know a given ini- creases to zero as q increases. For q = 18 the method converges in tial condition causes a corresponding decrease in the uncertainty a very satisfactory way. The values of the quantities W2 and W3 , as of the final state. On the other hand, for fractal basin boundaries given by Eq. (18) are shown in Fig. 12 (b) as a function of q , show- the scaling is unfavorable, such that even a huge improvement in ≈ ing good convergence after less than 10 steps. Since W2 0.112 and the precision of the initial condition might cause only a slight de- ≈ W3 0.888 for a large number of refinement steps we say that the crease of the final-state uncertainty. basin structure in Fig. 9 is partially Wada. Moreover the fractality of the basin structure was evidenced by computing the basin and basin boundary entropies. The latter vary 6. Conclusions between zero (in the case of just one exit, meaning no uncertainty at all) and unity (for two exit basins), when the basins are ex- Open hamiltonian systems may present an arbitrarily large tremely intertwined and we have maximum final-state uncertainty. number of exits through which a given trajectory can escape. The The entropies were found to increase with the symmetry-breaking set of initial conditions leading to trajectories leaving a phase space perturbation p in the same way as the fraction occupied by one ≈ area through an exit is called its exit basin. When there is a chaotic of the basins, until a critical perturbation p c 1.75 is achieved. For > (area-filling) orbit in a part of the phase space region it turns out p pc , although the fraction occupied by one basin continues to that the exit basins are fractal as well as their common boundary. increase, the entropies show some decrease. This fractality comes from structures such as invariant (unstable This behavior is ultimately due to the meaning of the basin en- and stable) manifolds underlying the chaotic orbit. tropy in terms of the degree of mixing of the exit basins. When In this paper we investigated the fractality of exit basins in the one exit basin has its area increased (by augmenting p ) the other context of magnetic field line flow in magnetically confined fusion exit basin has its area decreased, and the entropies achieve a max- plasmas, such as those produced in a tokamak. Although the equi- imum (slightly less than unity) when the areas have comparable librium magnetic fields yield an integrable hamiltonian system in fractions (around 0.5 each). As p is further increased, the exit basin an axisymmetric system, the presence of symmetry-breaking per- areas become distinct again and the entropies decrease accord- turbations causes the formation of chaotic orbits coexisting with ingly. < islands and KAM tori. In particular, we investigate the chaotic mag- However the behavior of the entropies for p pc is differ- > netic field region appearing near the tokamak wall due to an er- ent from the case p pc , since the entropy increase observed for godic limiter, as described by a two-dimensional area-preserving smaller p has a larger rate than the entropy decrease for larger p . map. The reason is in the qualitative dynamics of the system in the two We divided the tokamak wall into two regions through which cases. For small p the chaotic region is restricted to the vicinity the magnetic field lines can escape and considered the basins cor- of the separatrices of the islands nearer the tokamak wall (since responding to each exit. The basin structure exhibits an infinite the perturbation strength is higher therein). As p increases the is- number of incursive fingers which follow the intersection of a lands also augment their widths, while their mutual distance re- basin boundary segment with the stable manifold of an unsta- main practically unchanged. ble periodic orbit embedded in the chaotic orbit. If we make a For p large enough the local chaotic regions of adjacent islands transversal cut of this fingering structure the result is the product merge together forming a single connected region of global chaos. of a smooth curve (dimension 1) and a Cantor-like fractal set. In- We used a simple overlapping criterion to estimate the value of ≈ deed, we obtained directly the box-counting dimension of the exit p c 1.6, which is close to the value (1.75) which maximizes the < basin boundary by using the uncertainty exponent method. The entropy. Hence the entropies increases with p (for p pc ) because basin boundary dimension was found to increase with the inten- the available region for the chaotic orbit also increases. However, > sity of the symmetry-breaking perturbation. for p pc the chaotic region increases at a lower rate, essentially by It is worthwhile to point out that the fractal dimension of the engulfing the remnant islands still embedded in the chaotic region. exit basin has important consequences on the predictability of the Another fractal signature we have found in the basin bound- final state of the system, i.e. to which exit will escape the orbit ary structure, in the case of three exits (we divided the tokamak (magnetic field line, in our case) from a given initial condition. If wall into three instead of two escape regions) is the Wada prop- 598 A.C. Mathias et al. / Chaos, Solitons and Fractals 104 (2017) 588–598 erty. 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ﬁrst characterization of this property by considering the intersec- [11] Silva ECd , Caldas IL , Viana RL , Sanjuán MAF . Phys Plasmas 2002;9:4917 . tions of the stable manifold with all three basins. However, since [12] Portela JSE , Caldas IL , Viana RL , Sanjuán MAF . Int J Bifurcat Chaos this does not lead to quantitative information we used a recently 2007;17:4067 .

[13] Jakubowski MW, Schmitz O, Abdullaev SS, Brezinsek S, Finken KH, developed technique to evaluate the fraction of points of the basin Krämer-Flecken A , et al. (TEXTOR Team), Phys Rev Lett 20 06;96:0350 04 . boundary possessing the Wada property. In a representative case [14] Jakubowski MW , Evans TE , Fenstermacher ME , Groth M , Lasnier CJ , we found that approximately 11.2% of the boundary points sepa- Leonard AW , et al. Nucl Fusion 2009;49:095013 . rate two basins, and 88.8% of them separate three basins, i.e. have [15] Evans TE, Roeder RKW, Carter JA, Rapoport BI. Contrib Plasma Phys 2004;44:235 . the Wada property. Since these results are consistently obtained [16] Evans TE , Roeder RKW , Carter JA , Rapoport BI , Fenstermacher ME , Lasnier CJ . J with other values of the perturbation we conclude that the basins Phys: Conf Ser 2005;7:174 . are only partially Wada. [17] Bleher S, Grebogi C, Ott E, Brown R. Phys Rev A 1988;38:930. [18] Aguirre J , Viana RL , Sanjuán MAF . Rev Mod Phys 2009;81:333 .

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