Transport barriers in symplectic maps
R. L. Viana, I. L. Caldas, J. D. Szezech Jr., A. M. Batista, C. V. Abud, A. B. Schelin, M. Mugnaine, M. S. Santos, B. B. Leal, B. Bartoloni, A. C. Mathias, J. V. Gomes, P. J. Morrison
TTransport barriers in symplectic maps
R. L. Viana , I. L. Caldas , J. D. Szezech Jr. , A. M. Batista , C.V. Abud , A. B. Schelin , M. Mugnaine , M. S. Santos , B. B. Leal , B. Bartoloni , A. C. Mathias , J. V. Gomes , and P. J. Morrison Departamento de F´ısica, Universidade Federal do Paran´a,Curitiba, Paran´a, Brazil; Departamento de F´ısica Aplicada,Instituto de F´ısica, Universidade de S˜ao Paulo, S˜ao Paulo,S˜ao Paulo, Brazil; Departamento de Matem´atica e Estat´ıstica,Universidade Estadual de Ponta Grossa, Ponta Grossa,Paran´a, Brazil; Universidade Federal de Goi´as, Goi´as,Brazil; Instituto de F´ısica, Universidade de Bras´ılia,Bras´ılia, Distrito Federal, Brazil; Department of Physics,The University of Texas at Austin, Texas, United States (Dated: February 8, 2021)
Abstract
Chaotic transport is a subject of paramount importance in a variety of problems in plasmaphysics, specially those related to anomalous transport and turbulence. On the other hand, agreat deal of information on chaotic transport can be obtained from simple dynamical systemslike two-dimensional area-preserving (symplectic) maps, where powerful mathematical results likeKAM theory are available. In this work we review recent works on transport barriers in area-preserving maps, focusing on systems which do not obey the so-called twist property. For suchsystems KAM theory no longer holds everywhere and novel dynamical features show up as non-resistive reconnection, shearless curves and shearless bifurcations. After presenting some generalfeatures using a standard nontwist mapping, we consider magnetic field line maps for magneticallyconfined plasmas in tokamaks. a Corresponding author. e-mail: viana@fisica.ufpr.br a r X i v : . [ phy s i c s . p l a s m - ph ] F e b . INTRODUCTION The main goal of the study of transport in Hamiltonian systems is to characterize themotion of groups of trajectories from one region of phase space to another [1]. Whendealing with non-integrable Hamiltonian systems, the study of transport is complicated bythe coexistence of periodic, quasi-periodic, and chaotic orbits [2]. In particular, chaotictransport is an issue of major importance in plasma physics, since plasma turbulence is theultimate cause of anomalous transport in magnetically confined plasmas [3].Fortunately many features of chaotic transport observed in real plasmas are also presentin low-dimensional systems like area-preserving symplectic maps [4]. If the latter satisfy theso-called twist condition, the celebrated KAM theorem warrants the existence of invarianttori with sufficiently irrational rotation numbers, provided the perturbation strength is smallenough [5]. KAM tori, or invariant curves, act as dikes preventing transport in a large scale inphase space. As the perturbation strength is increased, however, these tori are progressivelydestroyed, leaving there cantori as their remnants [6].If, however, the twist condition does not hold everywhere in the phase space region ofinterest, KAM theory no longer applies everywhere in phase space. As a consequence,novel features show up that influence transport in a dramatic way. For example, there areshearless tori for which the rotation number has a local extreme. These shearless tori, evenafter their breakup, if the perturbation is strong enough, decreases transport in such a waythat it becomes an effective transport barrier [7]. One of the observable consequences ofthese barriers is a ratchet current, when there is a symmetry breaking [8]. We have recentlyinvestigated the effect of a weak dissipation in nontwist systems, with the formation ofshearless attractors [9].Nontwist systems appear in several problems of plasma physical interest, like magneticfield line structure in Tokamaks with reversed magnetic shear [10–12], the E × B drift motionof charged particles in a magnetic field [13, 14], transport by traveling waves in shear flowswith non-monotonic velocity profiles [15], laser-plasma coupling [16], and magnetic fieldstructure in double tearing modes [17], among others.The so-called standard nontwist map (SNTM), proposed by Morrison and Del Castillo-Negrete in 1993, is considered a paradigm symplectic map for theoretical and computationalinvestigations of nontwist systems [15]. Drift trajectories in Tokamaks with reversed electric2hear can be reduced to the SNTM [13]. The SNTM is also obtained for the magnetic fieldline behavior in tokamak, when the safety factor radial profile of the magnetic flux surfacesis non-monotonic, having local extrema [18].Effective transport barriers observed in nontwist symplectic maps can help to understandthe formation of internal transport barriers in tokamak plasmas. The latter are produced bymodifications of the current, safety factor, or electric field profiles by using external heatingand current drive [19] or voltage biasing [20]. Internal transport barriers can provide hightokamak confinement at modest plasma current values [21].In this work we aim to review recent theoretical and computational works aiming tounderstand the formation of effective transport barriers in nontwist symplectic maps, havingin mind applications in plasma physics problems such as the magnetic field line structurewith reversed shear. The basic dynamical mechanism underlying the formation of suchbarriers is the formation of dimerized magnetic island chains in both sides of the shearlesscurve, due to the non-monotonicity of the winding number profile. As the non-integrableperturbation is strong enough, when these islands overlap they are progressively destroyed,leaving in their places internal transport barriers that reduce (while not blocking at all)chaotic transport through them.The rest of the paper is organized as follows: in Section II we review some basic propertiesof the standard nontwist map, focusing on the existence and destruction of the shearless curveand the formation of internal transport barriers. Section III considers two magnetic fieldline maps in tokamaks with nontwist properties and the consequent formation of transportbarriers. Section IV presents the newly discovered phenomenon of shearless bifurcation, andalso the effect of reversed current. The last Section contains our Conclusions. II. STANDARD NONTWIST MAP
Let us consider a two-dimensional area-preserving map of the general form y n +1 = y n − f ( x n ) , (1) x n +1 = x n + ω ( y n +1 ) , (mod 1) , (2)where y n ∈ R and x n ∈ [0 ,
1) are canonical variables. We require that the function f beperiodic with period-1. If the latter vanishes everywhere in the cylindrical phase space we3 IG. 1. (color online) (a) Phase space of the SNTM for a = 0 .
631 and b = 0 . S : x = 0 .
5. (c) a zoom of a region of (b). The red star indicatesthe y -position of the shearless curve along S . have simply x n +1 = x n + ω ( y n ) and the system is integrable. Hence the orbits lie on curves y n = const. , along which ω ( y n ) is the so-called rotation number, defined more generally as ω = lim n →∞ x n − x n . (3)The system (1)-(2) is a twist map, provided the condition (cid:12)(cid:12)(cid:12)(cid:12) ∂x n +1 ∂y n (cid:12)(cid:12)(cid:12)(cid:12) = | ω (cid:48) ( y n +1 ) | ≥ c > , (4)holds for every value of ( y n , x n ) along a curve in the phase plane. A nontwist system is suchthat this condition is not fulfilled somewhere, for example when the derivative above (alsocalled shear) crosses zero.A two-dimensional paradigm for the study of nontwist systems is the so-called standardnontwist map (SNTM), for which f ( x ) = − b sin(2 πx ) and ω ( y ) = a (1 − y )[15]: y n +1 = y n − b sin (2 πx n ) , (5) x n +1 = x n + a (1 − y n +1 ) , (mod 1) , (6)4 IG. 2. (color online) Phase space of the SNTM for b = 0 .
475 and (a) a = 0 . a = 0 . a = 0 . a = 0 .
71. The shearless curves are indicated by the red curves. where a ∈ [0 ,
1) and b ∈ R . The parameter b is a measure of the non-integrability of thesystem, and a is proportional to the shear along ( x, y ) curves. This map has for symmetrylines, namely S = { ( x, y ) | x = 1 / } , S = { ( x, y ) | x = 0 } , S = { ( x, y ) | x = a (1 − y ) / } , S = { ( x, y ) | x = a (1 − y ) / / } , that are useful to find periodic orbits of any period[22]. For example, orbits with odd period n on S are obtained by searching for points( x = 0 , y ) on S that are mapped to S or S after ( n + 1) / b = 0 the SNTM is integrable and the shear is simply ω (cid:48) = 2 ay , changing sign at5 IG. 3. (color online) Phase space of the SNTM for b = 0 . a = 0 . a = 0 . y = 0, for which the map is nontwist. At each side of y = 0 we have two invariant curves y = y and y = − y with the same rotation number for a given value of y . Switching onthe perturbation ( b (cid:54) = 0) there will appear twin chains of periodic islands with the samerotation number, as illustrated by Fig. 1(a), where a phase portrait of the SNTM is shownfor a = 0 .
631 and b = 0 . S : { x = 0 . , − ≤ y ≤ } , iterated until n = 10 [Fig. 1(b)]. The rotation number has a local maximum when the shear changes sign[see also the zoon in Fig. 1(c)], what occurs at points along the so-called shearless curve,represented as the red curve in Fig. 1(a), where the twist condition is also violated.As the system parameters change many of the invariant curves are destroyed and chaoticdynamics sets in. In nontwist systems, however, the shearless curve is remarkably resilientand survives even when neighboring curves have disappeared. Figure 2 shows a representa-tive example of this phenomenon, with phase portraits of the SNTM obtained for constant b and varying the parameter a . The twin period-5 islands at both sides of the shearless curveare “dephased”, i.e. the elliptic point of one corresponds to a hyperbolic point of the other[Fig. 2(a)].These island chains approach mutually and their separatrix reconnect, as the parameter6 IG. 4. (color online) b = 0 . a = 0 . a = 0 . y = 1 and −
1, respectively. Each initial condition wasiterated until n = 400. a changes (actually the reconnection affects the respective chaotic layers, since the system isno longer integrable). A further change in a leaves each hyperbolic point with a homoclinicand a heteroclinic manifold and, in the region between the chains, new invariant curvesappear which are not graphs over the x -axis and are called meanders [Fig. 2(b)]. Changing a again makes the elliptic and hyperbolic points to collide and chaotic regions are formed,survived by the meander [Fig. 2(c)]. For increasing a even the meander is destroyed, leavinga large chaotic region with remnants of the period-5 islands [Fig. 2(d)].While the survival of the shearless curves (or meanders) is a barrier for large scale chaotictransport, even though they disappear there is still an effective barrier, as illustrated by Fig.3. In Fig. 3(a) and (b) we show phase portraits of the SNTM for b = 0 .
619 and a = 0 .
642 and0 . a = 0 .
642 the two chaotic orbitsdo not show signs of mixing, suggesting the presence of an internal transport barrier [Fig.3(a)], whereas for a = 0 .
645 the two colors are mixed, signaling a higher degree of chaotictransport [Fig. 3(b)]. 7
IG. 5. (color online) Parameter plane ( a vs. b ) obtained from the barrier transmissivity for theSNTM. We considered 10 initial conditions randomly placed along the line y = − .
0, iterateduntil n = 5000. The colorbar indicates how many of those initial conditions reach the line y = 5 . We have found that the difference between these cases is the different configuration ofthe unstable manifolds stemming from the Poincar´e-Birkhoff fixed points associated withthe periodic island chains in both sides of the shearless curve [23]. These unstable manifoldsintercept at heteroclinic points and, if the system is nontwist, these heteroclinic points canconnect the twin island chains, increasing the transport [24]. Slight variations in the systemparameters, however, can alter qualitatively the geometry of the invariant manifolds anddecreasing the transport. This change occurs due to the formation of structures calledturnstiles [6, 25]Another way to regard the sudden increase of transport as the parameters are varied isto consider the SNTM as an open dynamical system, and consider that the orbits in phasespace can escape to plus or minus infinity if they cross the lines { ( x, y ) | < x < , y = 1 . } or { ( x, y ) | < x < , y = − . } , respectively. We call the escape basin the set of initialconditions which produce map orbits escaping the system through a given exit. Figure 4,8btained for the same parameter values as Fig. 3, shows the escape basins of these twoexits. The extent of chaotic transport is given by the degree in which these escape basinsmix together. For small chaotic transport [Fig. 3(a)] this mixing is limited to the regionneighboring the twin island chains, whereas for large chaotic transport this mixing occursthrough an unbounded region of the phase space, thanks to the incursive fractal fingers [Fig.3(b)].The sensitive dependence of the chaotic transport on the system parameters can bequantitatively described by the transmissivity, which is the fraction of map orbits thatcross the region between the twin island chain. A numerical estimate of this quantitycan be obtained by placing a large number ( N = 10 ) of initial conditions on the line { ( x, y ) | < x < , y = − . } and iterating each of them by 5 × times. The transmissivityis the fraction of the orbits which reach the line { ( x, y ) | < x < , y = +5 . } . If thistransmissivity is zero, there exists a transport barrier between the island chains, otherwisethere is some degree of chaotic transport.In Figure 5 we show (in a colorbar) the transmissivity of the trajectories for the SNTMas a function of its parameters a and b . The zero transmissivity regions are painted black,indicating the existence of a robust transport barrier, which we can identify as the shearlesscurve (and perhaps other remaining tori in both sides of it). The boundary of the no-transmissivity region has been investigated from the point of view of a fractal curve [26].Low transmissivity, on the other hand, can be identified with an internal transport barrierrelated to the presence of turnstiles, like in Figs. 3(a) and 4(a). Larger values of thetransmissivity are thus characteristic of the absence of any barrier. III. MAGNETIC FIELD LINE MAPS
Nontwist maps appear naturally in some problems of interest in Plasma Physics, like themagnetic field line structure in toroidal devices like Tokamaks and Stellarators. In this workwe will focus in a Tokamak whose vessel has minor radius b and major radius R [Fig. 6].The aspect ratio R /b is supposed to be large enough that we can approximate the tokamakby a periodic cylinder of length 2 πR , in which the plasma column has a radius a < b .A field line point in this geometry can be identified by its cylindrical coordinates ( r, θ, z ),where 0 ≤ r < b , 0 ≤ θ < π , and 0 ≤ z < πR .9 IG. 6. Schematic figure of a large aspect ratio Tokamak with chaotic limiter.
In the tokamak, magnetic field lines can be modeled through the following hamiltonianstructure: dψdϕ = − ∂H∂θ , dθdϕ = ∂H∂ψ . (7)Here, H is the poloidal flux, θ and ψ correspond to the canonical coordinate and momentum[27, 28] and the toroidal angle ϕ = z/R acts as a timelike variable.We can divide the Hamiltonian into two parts: H , which corresponds to the non-perturbed flux and H , the perturbed flux. Together they form H = H + (cid:15)H where: H ( ψ ) = (cid:90) dψq ( ψ ) , (8)with q ( ψ ) as the safety profile. The conditions for MHD equilibrium imply that the magneticfield lines lie on flux surfaces ψ = const. , describing helical trajectories whose pitch isdetermined by the rotational transform ι ( r ) = 1 /q ( r ).10he strength of the perturbations is given by (cid:15) and H is written in terms of the followingFourier series: H ( ψ, θ, ϕ ) = (cid:88) m,n H m,n ( ψ ) cos( mθ − nϕ + χ m,n ) , (9)with m and n as the poloidal and toroidal mode numbers and χ m,n as their phases.This magnetic perturbation is periodic, which allows for the creation of stroboscopic mapswith sections at ϕ = ϕ n = (2 π/s ) n , with ( n = 0 , ± , ±
2) and s ≥
1. Taking ( ψ n , θ n ) as theintersection points, we can write the field line map as( ψ n +1 , θ n +1 ) = F ( ψ n , θ n ) . (10)Using ψ = r / H is a two-dimensional symplectic map F , derived from the magnetic field line equations, ofthe form r n +1 = r n − a sin( θ n ) , (11) θ n +1 = θ n + 2 πq ( r n +1 ) + a cos( θ n ) , (mod 2 π ) , (12)where the parameter a gives the toroidal correction to the cylindrical approximation. Weshall use a = − .
04. A non-monotonic safety factor profile for the equilibrium plasma isgiven by the expression [18] q ( r ) = q a r a (cid:40) − (cid:18) β (cid:48) r a (cid:19) (cid:18) − r a (cid:19) µ +1 H ( a − r ) (cid:41) − , (13)where β , µ and β (cid:48) = β ( µ + 1) / ( β + µ + 2) are equilibrium parameters and H ( x ) is theHeaviside unit-step function.The conditions for the formation of a chaotic region in the Poincar´e surface of section( r, θ ) are fulfilled if a non-integrable magnetic perturbation sets in. One example is the so-called chaotic limiter, which consists in a grid of m pairs of wires with length (cid:96) [Fig. 6], sointroducing a “time”( z )-dependence which breaks the integrability of the equilibrium withtoroidal correction given by the map F . The magnetic field produced by a chaotic limiter11 IG. 7. (color online) Phase space for the Ullmann map F ◦ F with m = 3 and (a) (cid:15) = 0 .
03, (b)0 .
08, (c) 0 .
30, (d) 0 .
40. The remaining parameters are listed in the Appendix. The rectangularcoordinates here are x = θ and y = ( b − r ) /b . with m pairs of wires yields a perturbation map F of the form r n +1 = r ∗ n +1 + mC(cid:15)bm − (cid:18) r ∗ n +1 b (cid:19) m − sin( mθ n +1 ) , (14) θ ∗ n +1 = θ n +1 − C(cid:15) (cid:18) r ∗ n +1 b (cid:19) m − cos( mθ n +1 ) , (15)where C = 2 m(cid:96)a /R q a b , and (cid:15) = I (cid:96) /I p , where I (cid:96) is the limiter current and I p is the plasmacurrent. The composed map F ◦ F was proposed by Ullmann and Caldas in 2000 [29]. The12 IG. 8. Non-monotonic safety factor profile of the Tokamap. parameter values used in the numerical simulations are b = 0 . m (major radius), a = 0 . m (minor radius), (cid:96) = 0 . m , β = 2 . µ = 1 . q a = 3 . m = 3 are shown in Fig. 7 for different values ofthe perturbation strength (cid:15) , which is proportional to the current applied at the limiter ring.For small values of the latter we have the formation of twin dimerized islands separated by ashearless curve (in red) [Fig. 7(a)]. Increasing the perturbation strength these island chainsapproach each other and the shearless curve meanders around them [Fig. 7(b)]. Even whenthe perturbation is stronger, forming chaotic regions in both sides of the shearless curve,it continues to act as a transport barrier [Fig. 7(c)]. Further increase in the perturba-tion strength breaks down this barrier and allows a larger chaotic region with some islandremnants [Fig. 7(d)].Using the original set of canonical coordinates, another symplectic field line map, thetokamap, was proposed by Balescu et al. [30]: ψ n +1 = ψ n − εψ n +1 ψ n +1 sin( θ n ) , (16) θ n +1 = ψ n + 2 πq ( ψ n +1 ) − ε cos( θ k )(1 + ψ k +1 ) . (17)The tokamap was not directly derived from the magnetic field line equations. However, itfits important characteristics for the system, namely: (i) there are no negative values of ψ ,such that ψ = 0 and ψ n ≥ n ; (ii) it follows a realistic safety factor profile q ( ψ ) [30].Here we use the non-monotonic profile shown in figure 8, given by: q ( ψ ) = q m − α ( ψ − ψ m ) , (18)13 IG. 9. Poincar´e section of the revtokamap (16)-(17) for ε = 0 .
35. In red we show the shearlessbarrier at ψ = 0 . with α = (1 − q m /q ) ψ − m , and ψ m is the minimum of q given by: ψ m = (cid:32) (cid:115) − q m /q − q m /q (cid:33) − , (19)where q = q (0) = 3 e q = q (1) = 6. With this profile, the mapping is also known as the revtokamap [30, 32].In Fig. 9 we show the Poincar´e section of field lines in ( ψ, θ )-plane for the revtokamap(16)-(17), in which the shearless curve is drawn in red color. The qualitative evolution issimilar to that exhibited by the previous maps here presented. IV. SHEARLESS BIFURCATION AND REVERSED CURRENT
In the previous Section we have seen examples of symplectic field line maps, for which theexistence of shearless barriers is due to the non-monotonicity of the safety factor profile ofthe plasma equilibrium. However, it is possible to obtain a shearless transport barrier evenwith monotonic safety factor profiles, provided we are close enough to some bifurcationsnear primary resonant islands.Dullin, Meiss, and Sterling showed, in 2000, the existence of a shearless torus in theneighborhood of the tripling point of an elliptic fixed point of a generic Hamiltonian system[33]. Further numerical investigations have shown the existence of shearless tori near aquadrupling bifurcation [34]. 14
IG. 10. (color online) Phase space of the Ullmann map with (cid:15) = 0 . m = 6. The insetshows a quadrupling bifurcation. In the context of the Ullmann map, tripling and quadrupling bifurcations of an ellipticfixed point show up over a wide range of the perturbation parameter (cid:15) [35]. In Figure 10we show a phase space of the Ullmann map for (cid:15) = 0 .
189 and m = 6. The chaotic layerhas embedded remnants of an island chain. In the inset we exhibit a period-5 island chainwhose elliptic point bifurcated into a period-4 one.In order to understand how a local shearless barrier is formed near the island chainundergoing a quadrupling bifurcation, we show in Figure 11 the evolution of the phasespaces (left panels) and the corresponding rotation number profiles (right panels) in theneighborhood of the quadrupling bifurcation. Just before the latter [Fig. 11(a)] the rotationnumber profile is monotonic, with no local extrema. On increasing the (cid:15) parameter therehappens a quadrupling bifurcation, through which there is a local minimum and a localmaximum [Fig. 11(b)], and thus two shearless tori have been formed therein. As (cid:15) isfurther increased, the bump in the rotation number profile has increased its size until thelocal maximum achieves the value ω = 1 / IG. 11. (color online) Phase space (left) And rotation number profile (right) for the Ullmann mapwith m = 6 and (a) (cid:15) = 0 . . . current density profile J z with a reversed current given by [36]: J z ( r ) = I p R πa ( δ + 2)( δ + 1) δ + γ + 2 (cid:18) δ r a (cid:19) (cid:18) − r a (cid:19) γ , (20)where a = 0 .
18 m is the plasma radius, I p = 20 kA is the plasma current, δ = − . γ = 5. Such parameters are obtained from the TCABR tokamak [39]. The correspondingpoloidal magnetic field profile is depicted in Fig. 12, and the resulting safety profile is given16 IG. 12. Poloidal magnetic field radial profileFIG. 13. Numerical safety factor profile with x = 0 .
5. The inset shows two shearless points: amaximum at y = 0 . y = 0 . by: q ( r ) = q ( a ) r a (cid:40) − (cid:34)(cid:18) β (cid:48) r a (cid:19) (cid:18) − r a (cid:19) γ +1 (cid:35)(cid:41) (cid:20) − r R (cid:21) − / . (21)where q ( a ) = 5 . β (cid:48) = δ ( γ + 1) / ( δ + γ + 2).We apply this non-monotonic safety profile in the Ullmann map, with a = − .
04. Oncomputing the rotation number we obtain Figure 13 for a cross section at x = 0 .
5. There isa divergence at y = 0 . IG. 14. Shearless curve for the Ullmann map with reversed current at y and its correspondingisland chains. The inset in Fig. 13 shows the existence of two local extrema (one maximum at y =0 . y = 0 . y = y with its corresponding twin island chains. V. CONCLUSIONS
The existence of shearless barriers in tokamaks has deep consequences in terms oftransport properties and the quality of plasma confinement that can be achieved. Theshearless barriers are basically magnetic surfaces with some kind of robustness againstsymmetry-breaking perturbations. The presence of shearless barriers is usually related tonon-monotonic safety factor profiles.A paradigm of this behavior is provided by the standard nontwist map of Morrison anddel Castillo-Negrete. The shearless curve, in this case, is remarkably robust against theincrease of a non-integrable perturbation strength. The location of the shearless curve is alocal extremum of the rotation number profile, where the twist condition is violated for themap.Even after the shearless curve has been destroyed, however, transport is affected by theinvariant manifold structure in the region formerly occupied by the shearless curve. The18reakup of the shearless curve is extremely sensitive to the parameter values taken by thestandard nontwist map. The boundary (in parameter space) between the two situation iscomplicated (with fractal features).In this paper we show two magnetic field line maps in tokamaks with non-monotonicsafety factor profiles. One of them considers a tokamak with chaotic limiter, which is anexternal arrangement of current wires designed to create a peripheral region of chaotic fieldlines near the tokamak wall. For both cases the shearless tori are located at local extremaof the rotation number profile.We also shown that there are cases for which a field line map can exhibit shearless barrierseven when the safety factor profile is monotonic. This occurs if the map parameters areclose to a tripling or quadrupling bifurcation, so creating local extrema in the correspondingrotation number profiles.Finally, we consider explicitely a situation in which the non-monotonic safety factor profilehas a well-defined physical reason, namely the existence of a current density profile with asign reversal, a situation usually present in tokamak scenarios which partially explains whyshearless barriers are so often observed.
ACKNOWLEDGMENTS
This work was made possible by the partial financial support of the following Brazil-ian government agencies: CNPq (proc. 301019/2019-3, 428388/2018-3, 310124/2017-4),CAPES, Funda¸c˜ao Arauc´aria, and FAPESP (grant 2018/03211-6).
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