Boundary Orders and Geometry of the Signed Thom–Smale Complex for Sturm Global Attractors

We embark on a detailed analysis of the close relations between combinatorial and geometric aspects of the scalar parabolic PDE *ut=uxx+f(x,u,ux)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u_t = u_{xx} + f(x,u,u_x) \end{aligned}$$\end{document}on the unit interval 0<x<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0< x<1$$\end{document} with Neumann boundary conditions. We assume f to be dissipative with N hyperbolic equilibria v∈E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v\in {\mathcal {E}}$$\end{document}. The global attractor A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}$$\end{document} of (*), also called Sturm global attractor, consists of the unstable manifolds of all equilibria v. As cells, these form the Thom–Smale complexC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}$$\end{document}. Based on the fast unstable manifolds of v, we introduce a refinement Cs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}^s$$\end{document} of the regular cell complex C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}$$\end{document}, which we call the signed Thom–Smale complex. Given the signed cell complex Cs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}^s$$\end{document} and its underlying partial order, only, we derive the two total boundary orders hι:{1,…,N}→E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_\iota :\{1,\ldots , N\}\rightarrow {\mathcal {E}}$$\end{document} of the equilibrium values v(x) at the two Neumann boundaries ι=x=0,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\iota =x=0,1$$\end{document}. In previous work we have already established how the resulting Sturm permutation σ:=h0-1∘h1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sigma :=h_{0}^{-1} \circ h_1, \end{aligned}$$\end{document}conversely, determines the global attractor A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}$$\end{document} uniquely, up to topological conjugacy.


Introduction
For our general introduction we first follow [22][23][24] and the references there. Sturm global attractors A f are the global attractors of scalar parabolic equations on the unit interval 0 < x < 1. Just to be specific we consider Neumann boundary conditions u x = 0 at x ∈ {0, 1}. Standard theory of strongly continuous semigroups provides local solutions u(t, x) in suitable Sobolev spaces u(t, ·) ∈ X ⊆ C 1 ([0, 1], R), for t ≥ 0 and given initial data u = u 0 (x) at time t = 0. We assume the solution semigroup u(t, ·) generated by the nonlinearity f ∈ C 2 to be dissipative: there exists some large constant C, independent of initial conditions, such that any solution u(t, ·) satisfies u(t, ·) X ≤ C for all large enough times t ≥ t 0 (u 0 ). In other words, any solution u(t, ·) exists globally in forward time t ≥ 0, and eventually enters a fixed large ball in X . Explicit sufficient, but by no means necessary, conditions on f = f (x, u, p) which guarantee dissipativeness are sign conditions f (x, u, 0) · u < 0, for large |u|, together with subquadratic growth in | p|.
For large times t → ∞, the large attracting ball of radius C in X limits onto the maximal compact and invariant subset A = A f of X which is called the global attractor. Invariance refers to, both, forward and backward time. In general, the global attractor A consists of all solutions u(t, ·) which exist globally, for all positive and negative times t ∈ R, and remain bounded in X . See [34,47,53] for a general PDE background, and [3,10,11,32,33,38,48,51,54] for global attractors in general.
For Geneviève Raugel, in particular, global attractors were a main focus of interest. Her beautiful survey [48], for example, puts our past and present work on scalar one-dimensional parabolic equations in a much broader perspective.
For the convenience of the reader, we provide a rather complete background on our current understanding of the global attractors of (1.1). It is not required, and would in fact be pedantic, to read all technical references given. Rather, the present paper is elementary, although nontrivial, given the background facts which we will now summarize.
Equilibria u(t, x) = v(x) are time-independent solutions, of course, and hence satisfy the for 0 ≤ x ≤ 1, again with Neumann boundary. Here and below we assume that all equilibria v of (1.1), (1.2) are hyperbolic, i.e. without eigenvalues λ = 0 of their Sturm-Liouville linearization λu = u x x + f p (x, v(x), v x (x))u x + f u (x, v(x), v x (x))u (1.3) under Neumann boundary conditions. We recall here that all eigenvalues λ 0 > λ 1 > · · · are algebraically simple and real. The Morse index i (v) of v counts the number of unstable eigenvalues λ j > 0. In other words, the Morse index i(v) is the dimension of the unstable manifold W u (v) of v. Let E = E f ⊆ A f denote the set of equilibria. Our generic hyperbolicity assumption and dissipativeness of f imply that N := |E f | is odd. It is known that (1.1) possesses a Lyapunov function, alias a variational or gradient-like structure, under separated boundary conditions; see [26,36,39,40,42,56]. In particular, the time invariant global attractor consists of equilibria and of solutions u(t, ·), t ∈ R, with forward and backward limits, i.e. In other words, the αand ω-limit sets of u(t, ·) are two distinct equilibria v and w. We call u(t, ·) a heteroclinic or connecting orbit, or instanton, and write v w for such heteroclinically connected equilibria. See Fig. 1a for a modest 3-ball example with N = 9 equilibria. We attach the name of Sturm to the PDE (1.1), and to its global attractor A f . This refers to a crucial Sturm nodal property of its solutions, which we express by the zero number z. Let 0 ≤ z(ϕ) ≤ ∞ count the number of (strict) sign changes of continuous spatial profiles ϕ : [0, 1] → R, ϕ ≡ 0. For any two distinct solutions u 1 , u 2 of (1.1), the zero number t −→ z(u 1 (t, ·) − u 2 (t, ·)) (1.5) is then nonincreasing with time t, for t ≥ 0, and finite for t > 0. Moreover z drops strictly with increasing t > 0, at any multiple zero of the spatial profile x → u 1 (t 0 , x) − u 2 (t 0 , x); see [2]. See Sturm [52] for the linear autonomous variant (1.7) below.
For example, let ϕ j denote the j-th Sturm-Liouville eigenfunction ϕ j of the linearization (1.3) at any equilibrium v. Sturm not only observed that z(ϕ j ) = j. Already in 1836 he proved the much more general statement 0 = ϕ ∈ span {ϕ j , ϕ j+1 . . . , ϕ k } ⇒ j ≤ z(ϕ) ≤ k. (1.6) His proof was based on the solution ψ(t, ·) of the associated linear parabolic equation under Neumann boundary conditions and with initial condition ψ(0, ·) = ϕ. He then invoked nonincrease of the zero number t → z(ψ(t, ·)). Since the rescaled limits of ψ for t → ±∞ are eigenfunctions, this proved his claim (1.6). As a convenient notational variant of the zero number z, we will also write z(ϕ) = j ± (1. 8) to indicate j strict sign changes of ϕ, by j, and the sign ±ϕ(0) > 0, by the index ±.
For example, we may fix the sign of any j-th Sturm-Liouville eigenfunction ϕ j such that ϕ j (0) > 0, i.e. z(ϕ j ) = j + . The consequences of the Sturm nodal property (1.5) for the nonlinear dynamics of (1.1) are enormous. For an introduction see [7,8,25,29,31,41,43,49] and the many references there. Let us also mention Morse-Smale transversality, a prominent concept in [44][45][46]. The Sturm property (1.5) automatically implies Morse-Smale transversality, for hyperbolic equilibria. More precisely, intersections of unstable and stable manifolds W u (v) and W s (w) along heteroclinic orbits v w are automatically transverse: W u (v) − W s (w). See [1,35]. In the Morse-Smale setting, Henry [35] In a series of papers, based on the zero number, we have given a purely combinatorial description of Sturm global attractors A f ; see [14][15][16]. Define the two boundary orders h 0 , h 1 : {1, . . . , N } → E of the equilibria such that Fig. 1d for an illustration with N = 9 equilibrium profiles, E = {1, . . . , 9}, h 0 = id, h 1 = (1 8 3 4 7 6 5 2 9). The general combinatorial description of Sturm global attractors A is based on the Sturm permutation σ ∈ S N which was introduced by Fusco and Rocha in [30] and is defined as Already in [30], the following explicit recursions have been derived for the Morse indices i k := i(h 0 (k)): Similarly, the (unsigned) zero numbers z jk := z(v j − v k ) are given recursively, for j = k = j + 1, as Using a shooting approach to the ODE boundary value problem (1.2), the Sturm permutations σ ∈ S N have been characterized, purely combinatorially, as dissipative Morse meanders in [15]. Here the dissipativeness property requires fixed σ (1) = 1 and σ (N ) = N . The Morse property requires nonnegative Morse indices i k ≥ 0 in (1.12), for all k. The meander property, finally, requires the formal path M of alternating upper and lower half-circles defined by the permutation σ , as in Fig. 1c, to be Jordan, i.e. non-selfintersecting. See the beautifully illustrated book [37] for ample material on many additional aspects of meanders.
In [14] we have shown how to determine which equilibria v, w possess a heteroclinic orbit connection (1.4), explicitly and purely combinatorially from dissipative Morse meanders σ . This was based, in particular, on the results (1.12) and (1.13) of [30].
More geometrically, global Sturm attractors A f and A g of nonlinearities f , g with the same Sturm permutation σ f = σ g are C 0 orbit-equivalent [16]. Only locally, i.e. for C 1 -close dissipative nonlinearities f and g, this global rigidity result is based on the Morse-Smale transversality property mentioned above. See for example [44][45][46], for such local aspects.
More recently, we have pursued a more explicitly geometric approach. Let us consider finite regular cell complexes for the m-cell c v , by restriction of the characteristic map. The continuous map (1.15) is called the attaching (or gluing) map. For regular cell complexes, more strongly, the characteristic mapsB v →c v are required to be homeomorphisms, up to and including the attaching (or gluing) homeomorphism (1.15) on the boundary ∂ B v . The (m −1)-sphere ∂c v is also required to be a sub-complex of C m−1 . See [28] for some further background on this terminology. The disjoint dynamic decomposition  [21]; see [20] for a summary. We can therefore define the Sturm complex C f to be the regular Thom-Smale complex C of the Sturm global attractor Fig. 1b for the Sturm complex C f of the Sturm global attractor A f sketched in Fig. 1a, which is the closurec O of a single 3-cell. With this identification we may henceforth omit the explicit subscripts f , when the context is clear.
We can now formulate the main task of the present paper: Let the Thom-Smale complex C = C f of a Sturm global attractor A = A f be given, as an abstract regular cell complex. Derive the possible orders h ι : Once again: in the example of Fig. 1, this task requires to derive the red and blue boundary orders in (d) from the given complex (b)-of course without any previous knowledge of the red and blue path cheats in (b) which indicate precisely those orders.
For dim A = 1, the answer is almost trivial: any heteroclinic orbit is monotone, and therefore h 0 = h 1 . We have also solved this task for Sturm global attractors A of dimension (1.17) equal to two; see the planar trilogy [17][18][19]. For Sturm 3-balls A =c O , which are the closure of the unstable manifold cell c O of a single equilibrium O of maximal Morse index i(O) = 3, our solution has been presented in the 3-ball trilogy [22][23][24]. See (1.22)-(1.24) below and Sect. 5 for further discussion. The present paper settles the general case. It has turned out that the Thom-Smale complex C = C f does not determine the boundary orders h ι uniquely-not even when the trivial equivalences (1.38) below are taken into account. See [12,14]. Our unique construction of h ι therefore involves a refinement of the Thom-Smale complex C = C f which we introduce next: the signed Thom-Smale complex C s = C s f . The examples in [12,14,24] show that the same abstract regular cell complex C may possess one or several such refinements, as a signed Thom-Smale complex C s = C s f of Sturm type-or no such refinement at all.
The refinement is crucially based on the disjoint signed hemisphere decomposition with the nonempty equilibrium sets as barycenters, for 0 ≤ j < i(v). Equivalently, we may define the hemisphere decompositions, inductively, via the topological boundary j-spheres j (v) of the fast unstable manifolds W j+1 (v), as Here W j+1 (v) is tangent to the eigenvectors ϕ 0 , . . . , ϕ j of the first j + 1 unstable Sturm-Liouville eigenvalues λ 0 > · · · > λ j > 0 of the linearization (1.3) at the equilibrium v. In fact j−1 (v) becomes an equator in j (v), recursively, defining the two remaining hemispheres j ± (v) in j (v). See [22] for further details. We call the resulting refined Thom-Smale regular cell complex C = C f of a Sturm global attractor A = A f , together with the above hemisphere decompositions of all cell boundaries, the signed Thom-Smale complex C s = C s f . Abstractly, we define a signed hemisphere complex C s via a regular refinement of a given regular cell complex C as follows. We recursively bisect each closed n-cellc =c n , equatorially, by closed cellsc j of successively lower dimensions j = n − 1, . . . , 1. (The cell interiors c j are an abstraction of the fast unstable manifolds W j of the barycenter of v with Morse index i(v) = n.) On each boundary sphere j = ∂c j+1 , this induces a decomposition of j \ j−1 into two hemispheres. For bookkeeping, we may assign signs ± to these hemispheres and denote them as (1. 24) In this case, a complete characterization of the signed Thom-Smale complexes C s = C s f has been achieved. See Sect. 5, Definition 5.1 and Theorem 5.2, for further discussion.
To return to our main task, let us now fix any unstable equilibrium O ∈ A of Morse index n := i(O) ≥ 1. It is our task to identify the predecessors and successors of O, along the boundary orders h ι at x = ι = 0, 1. As input information, we will only use the geometric information encoded in the signed Thom-Smale complex C s = C s f , of Sturm type. Specifically, the hemisphere refinements are given by the hemisphere decompositions j ± (v) of the Sturm complex. We will illustrate all this for the example of Fig. 1 at the end of the present section. Suffice it here to say that it remains a highly nontrivial task to pass from the partial order, defined by the collection of all signed hemisphere decompositions, to the two total orders required by h 0 and h 1 .
To determine the ι-neighbors w ι ± of O geometrically, in case i(w ι ± ) = i(O) − 1, we develop the notion of descendants next. See [23] for the special case n = 3.  For any given O, the descendant v j (s) only depends on s j , . . . , s 0 . In Sect. 2 we show that the descendants v j (s) are indeed defined uniquely. We also determine the Morse indices i(v j (s)) = j and show that the descendants define a staircase sequence of heteroclinic orbits between equilibria of descending adjacent Morse indices: Clearly, the notion (1.27)-(1.29) of descendants is purely geometric: it is based on the signed hemisphere decomposition j ± (O) = ∂c j O,± in the abstract signed hemisphere complex C s , Beyond the partial order implicit in the hemisphere signs, we do not require any further explicit data on the total boundary orders h ι which we derive. In Sect. 5 we will illustrate this viewpoint, based on specific examples.

Theorem 1.2 Consider any unstable equilibrium
Assume that any one of the boundary successors w ι Then that w ι ± of O is given by the leading descendant v n−1 (s) of O, according to the following list: for odd n; (1.35) for even n; v n−1 (+ + · · · ), for odd n.
Next, consider all barycenters of cell dimension dim A − 1. Unless their h ι -neighbors possess higher Morse index, and those adjacencies have already been taken care of, we may apply theorem 1.2 again to determine their remaining h ι -neighbors, this time of Morse index i = dim A − 2. Iterating this procedure we eventually determine all h ι -adjacencies and our main task is complete.
In Sects. 2 and 3 we prove Theorem 1.2 by the following strategy. First we reduce the four cases (1.33)-(1.36) to the single case by four trivial equivalences. Indeed, the class of Sturm attractors A remains invariant under the transformations  (1.34) under the combination of both. We henceforth restrict to the case s = (+ + · · · ) of (1.36).

. In Sects. 3 and 4 we study the additional elements
. As a corollary, for k = n − 1, this proves theorem (1.2) and completes our task. In Sect. 4 we show, in addition to v k = v k , that v k =v k for all k; see Theorem 4.3. For an alternative proof of the minimax Theorem 4.3 see also the companion paper [50]. That more elementary proof is based on a more direct and quite detailed ODE analysis of the Sturm meander. Strictly speaking, Theorem 4.3 is not required for the identification task to derive the h ι -neighbors w ι ± from O. However, it much facilitates the task to identify the equilibria E Therefore the alternating and constant descendants v j (s) of O = 4 are given by (1.28), and therefore v 2 = 7. Since v 2 = v 2 , by theorem 3.1, we conclude that the successor w 1 This agrees with the equilibrium profiles in Fig. 1d. The companion paper [50] presents a meander based proof of Theorem 4.3. The property v n−1 = v n−1 of Theorem 3.1, which holds independently of Theorem 4.3, then allows us to identify, conversely, the geometric location of predecessors, successors, and signed hemispheres in the associated Thom-Smale complex. These results combined, can therefore be viewed as first steps towards the still elusive goal of a complete geometric characterization of the signed Thom-Smale complexes which arise as Sturm global attractors.

Descendants
In this section we fix any unstable hyperbolic equilibrium O of positive Morse index n := i(O) > 0, in a Sturm global attractor. Let Concerning the descendants v j = v j (s) of O, according to Definition 1.1, we also fix any sequence s = s n−1 . . . s 0 of n signs s j = ±, for 0 ≤ j < n. We first explain why the descendants v j are well-defined. After a pigeon hole Proposition 2.1, we collect some elementary properties of descendants in Lemma 2.2. Except for that last lemma, we do not require the sign sequence s to be constant or alternating. We do not require assumption (1.32) of Theorem 1.2 to hold anywhere, in the present section. In particular, the descendant v n−1 here need not coincide with any immediate successor or predecessor w ι ± of O on any boundary x = ι = 0, 1. Let us examine the recursive Definition 1.1 first. For s 0 = ±, the equilibrium {v 0 } := 0 s 0 (O) is defined uniquely by (1.29). Now consider 1 ≤ j < n and assume v 0 , . . . , v j−1 have been well-defined, already. By the Schoenflies result [21] on the j-sphere boundary we have the disjoint decomposition clos We claim that there exists a unique cell This follows again from [21], which asserts the following. Let ϕ j denote the j-th Sturm-Liouville eigenfunction of the linearization at O, with sign chosen such that z(ϕ j ) = j + . The eigenprojection P j projects the closed j-dimensional hemisphere clos j s j into the tan- The projection P j is homeomorphic onto a topological j-dimensional ball with Schoenflies ( j − 1)-sphere boundary. This homeomorphic projection preserves the regular Thom-Smale cell decomposition of clos j s j . In particular, any ( j − 1)-cell in the j-dimensional interior hemisphere j s j possesses precisely two j-cell neighbors in j s j , separating them as a shared boundary.
This proves that (2.5) defines v j uniquely, and explains why all descendants v j are well-defined, in Definition 1.1. Since for all 0 ≤ j < n. As we have mentioned in (1.9) already, (2. [35] and the Sturm transversality property. This proves the unique staircase (1.30) of heteroclinic orbits, i.e.
. This heteroclinic staircase with Morse indices descending by 1, stepwise, motivates the name "descendants" for the equilibria v j . Any heteroclinic orbit v j v j−1 in the staircase of descendants, from Morse index j to adjacent Morse index j − 1, is also known to be unique; see [8,Lemma 3.5]. Note that Sturm transversality of stable and unstable manifolds implies transitivity of the relation " ". In particular, not only does O connect to We briefly sketch an alternative possibility to construct the heteroclinic staircase (2.8), directly. Our construction is based on the y-map, first constructed in [7] by a topological argument. The y-map allows us to identify at least one solution u(t, x), with initial condition u(0, ·) in any small sphere around O in W u (O), such that the signed zero numbers Here t n−1 := −∞, t −1 := +∞, and the remaining dropping times t = t j of the zero number z can be chosen arbitrarily. Consider sequences of t j such that the length of each finite interval (t j , t j−1 ) tends to infinity. Separately for each 1 ≤ j < n, we consider the suitably time-shifted sequences u j (t, ·) := u(t + τ j , ·).
Passing to locally uniformly convergent subsequences, the u j will then converge to the desired heteroclinic orbits Here dropping of Morse indices along any heteroclinic orbit v w implies that the v j constructed from adjacent t-intervals in fact coincide.
Uniqueness of the heteroclinic staircase, however, is not obtained by the above topological construction.
Before we collect more specific properties of +descendants, in Lemma 2.2, we record a useful pigeon hole triviality which we invoke repeatedly below. In the following we call v j with j even the even descendants. Odd descendants v j refer to odd j. We occasionally use the abbreviations (2.14) to indicate that v 1 (x) < v 2 (x) holds at x = 0, and at x = 1, respectively.

Lemma 2.2 Consider the +descendants
with constant sequence s = + + · · · . Then the following statements hold true for any 0 ≤ j, k < n = i(O): (iii) for even k and even descendants, Therefore v j is between O and v k , at x = 0. For the heteroclinic orbit u(t, ·) from O to v k this implies the strict dropping On the other hand, the z-inequalities were already observed in [6]; see also [21] for a more recent account. Hence the heteroclinic orbit This contradiction proves claim (iii) on even j, k.
The case (iv) of odd j, k is analogous. We just argue indirectly, for odd j < k and for those j, we invoke the pigeon hole Proposition 2.1. Assumption (2.12) holds, for m := k, To show that the sequence ζ j increases strictly, with j, we compare ζ j−1 and ζ j for 1 ≤ j < k. Since j − 1 and j are of opposite even/odd parity, v j−1 and v j lie on opposite sides of v k , at x = 1; see (iii), (iv). Therefore v j v j−1 implies strict dropping of z To show that the unique +descendantsṽ j ∈ j + (v k ) of v k , for 0 ≤ j < k, coincide with the +descendants v j of O, it only remains to show (2.24) This proves (2.24), claim (vi), and the lemma.
We conclude this section with an illustration of the action, on lemma 2.2 (ii)-(iv), of the four trivial equivalences generated by u → −u and x → 1 − x from (1.38). First note that (ii) refers to a monotone order of all descendants v k , whereas (iii) and (iv) address the alternating order, depending on the even/odd parity of k, at the opposite end of the x-interval. The trivial equivalence u → −u flips the monotone order (ii) into the opposite monotone order O > v n−1 > . . . > v 0 , at x = 0, which corresponds to constant s j = −. The trivial equivalence x → 1 − x, in contrast, makes the monotone order (ii) and its opposite appear at x = 1, respectively. Therefore, the four trivial equivalences are characterized by the unique one of the four half axes of u, at x = 0 and x = 1, where all descendants are ordered monotonically. The alternating orders appear on the x-opposite u-axis. In summary, x → 1 − x swaps constant with alternating sign sequences s, whereas u → −u reverses the sign of s.

First Descendants and Nearest Neighbors
In this section we prove our main result, Theorem 1.2. As explained in the introduction, the trivial equivalences (1.38) reduce the four cases (1.33)-(1.36) to the single case s = + + · · · of +descendants v k = v k (+ + · · · ) with k = n − 1, n := i(O); see (1.37), (1.39). We also recall the notation Invoking Theorem 3.1 for the special case k = n − 1, we will then prove Theorem 1.2.
of the descending heteroclinic staircase (1.30) of the O descendants implies v j v j−1 . Lemma 2.2(iii), (iv) and (3.2) imply that v k is strictly between v j and v j−1 at x = 1, due to the opposite parity of j −1 and j, mod 2. Strict dropping of the zero number t → z(u(t, ·)−v k ) along the heteroclinic orbit u(t, ·) from v j to v j−1 , at the boundary x = 1, therefore implies We have already used dropping arguments of this type in our proof of Lemma 2.2, repeatedly. In [21, Lemma 3.6(i)], on the other hand, we have already observed that for any two elements u 1 , u 2 of the same closed hemisphere clos k + (O). In particular for the distinct equilibria v k , v k ∈ k + (O). By the standard pigeon hole argument of Proposition 2.1, however, the k + 1 distinct integers 0 ≤ ζ 0 < · · · < ζ k of (3.4) cannot fit into the k available slots 0, . . . , k − 1 of (3.6). This contradiction proves the theorem.
in the notation of the present section. In particular we may consider +descendants s = + + · · · . We first claim O w 1 − . To prove that claim we recall Wolfrum's Lemma; see [55] and [23,Appendix] , and only if, there does not exist any equilibrium w with boundary values strictly between v 1 and v 2 , at x = 0 or x = 1 alike, such that as claimed in (3.7), for even n. For odd n and even n − 1, we can repeat the exact same steps for the successor w 1 + of O at x = 1, instead of the predecessor w 1 − . This proves Theorem 1.2 for s = + + · · · . The remaining cases of constant or alternating s follow by the trivial equivalences (1.38), as already indicated in the introduction.

Minimax: The Range of Hemispheres
shows how, within + k (O), minimal distance from O, along the meander axis h 1 of x = 1, coincides with maximal distance from O, along the meander h 0 of x = 0.
Throughout this section we fix k. In Lemma 4.1 we show in correspondence to i(v k ) = k. We then study the +descendants w j ∈ j + (v k ) ofv k , for 0 ≤ j < k. In Lemma 4.2, in particular, we show (4.5) Combining (4.4) and (4.5) will then prove the claim (4.2) of Theorem 4.3.
We conclude the section, in Corollary 4.4, with a summary of our results for all four cases of constant and alternating descendants.

Lemma 4.1 Claim (4.4) holds true, i.e., i(v k ) = k for any
To prove i(v k ) = k, indirectly, we may therefore suppose i(v k ) < k. We will then reach a contradiction in two steps, below. In step 1 we show that there exists an initial condition Since u 0 is chosen in the invariant open hemisphere k + , we can follow the nonlinear backwards trajectory u(t, ·) from u 0 , for t ≤ 0, and define a second equilibriumṽ as the α-limit set of u(t, ·). In step 2 we then show thatṽ satisfies But (4.8) contradicts maximality ofv k in k + (O), at x = 0; see Definition (1.41) ofv k . This contradiction will prove the lemma.

It remains to showṽ
Our construction ofṽ as the α-limit set of u(t, ·) ∈ W u (ṽ), and (2.16), (3.5) imply (4.12) In the first equality here, we used that u 0 can be chosen arbitrarily close tov k , for small ε > 0. This contradiction toṽ ∈ W u (ṽ) ⊂ k−1 (O) completes the proof of claim (4.8), and of the lemma.
We consider the +descendants w j ∈ j (4.13) By construction, z(w j −v k ) = j + . However, this does not yet determine z(w k−1 − O) to be k − 1, as claimed in (4.5).

Lemma 4.2 For any
, the first +descendant w k−1 ofv k satisfies claim (4.5). In particular (4.14) Proof By construction of the +descendant w k−1 ofv k , we have by (4.13), we therefore obtain . This contradicts the maximality ofv k in k + (O), at x = 0, and proves z(w k−1 − O) = k − 1, as claimed in (4.5). Since O < 0v k < 0 w k−1 still holds, we also obtain z(w k−1 − O) = (k − 1) + . Moreover we recall O w k−1 . Together this establishes w k−1 ∈ k−1 + (O), as claimed in (4.14), and the lemma is proved. Proof For k = 0, where 0 + (O) = {v 0 } consists of a single equilibrium anyway, there is nothing to prove. Therefore consider 1 ≤ k < n. We proceed indirectly and supposev k = v k . Let w j ∈ j + (v k ) denote the +descendants ofv k , as in (4.13) and in Lemma 4.2. To reach a contradiction we prove the following three contradictory claims, separately: is strictly between O andv k , both, at x = 0 and x = 1, by our indirect assumption v k =v k . From (3.5) and (4.21).
So far, we have only considered descendants v k = v k =v k based on the constant sign sequence s = + + · · · of (1.37). The four trivial equivalences (1.38) provide the following extension of minimax Theorem 4.3.

Discussion
In this final section we explore and illustrate what our main Theorem 1.2 does, and does not, say. We first review the most celebrated Sturm global attractor, the n-dimensional Chafee-Infante attractor CI n [9]. Contrary to the common approach, which starts from a symmetric cubic nonlinearity, an ODE discussion of equilibria, and the time map of their pendulum boundary value problem, we start from an abstract description of CI n as the n-dimensional Sturm attractor with the minimal number N = 2n + 1 of equilibria. We derive the associated signed Thom-Smale complex next, in this much more general context. Invoking Lemma 2.2 will then provide the well-known shooting meander, and the Sturm permutation, of the Chafee-Infante attractor CI n .
In the second part of our discussion, we present examples of abstract signed regular complexes which are 3-balls. We first adapt the general recipe of Theorem 1.2 for the construction of the associated boundary orders h 0 , h 1 to the special case of Sturmian 3-balls, in the spirit of [22,23]. See Theorem 5.2 and Definitions 5.1, 5.3, 5.4. Going beyond the introductory example of Fig. 1, we address three further examples for illustration.
Our first new example, in Fig. 4, constructs h 0 , h 1 and the permutation σ = h −1 0 • h 1 for the unique Sturm solid tetrahedron with two faces in each hemisphere. The locations of the poles N, S along the 4-edge meridian circle turn out to be edge-adjacent, necessarily.
The second example, Fig. 5, deviates from the above tetrahedral signed Thom-Smale complex, by misplacing the South pole and reversing the orientation of a single edge, along the same meridian circle as before. The new location of the South pole is not edge-adjacent to the North pole, along the meridian, but diametrically opposite, instead. Still, our recipe succeeds to construct a Sturm permutation σ := h −1 0 • h 1 . The associated Sturm global attractor, however, fails to be the regular solid tetrahedron.
The third example, Fig. 6, starts from a signed regular solid octahedron complex with antipodal pole locations. It was first observed in [20] that such antipodal octahedra cannot be of Sturm type. Nevertheless, our construction of the total orders h 0 and h 1 still succeeds, at first sight. The permutation σ := h −1 0 • h 1 , however, fails to define a meander. We conclude with comments on the still elusive goal of a geometric characterization of all Sturmian Thom-Smale complexes C s f as abstract signed regular cell complexes C s . The n-dimensional Chafee-Infante attractor CI n is usually presented as the Sturm global attractor of on the unit interval 0 < x < 1, with parameter 0 < (n − 1)π < λ < nπ, cubic nonlinearity, and for Neumann boundary conditions. See [9] for the closely related original Dirichlet setting. Geometrically, CI 0 can be thought of as the single trivial equilibrium O ≡ 0, and CI n is the one-dimensionally unstable double cone suspension of CI n−1 , inductively for n > 0. See [12,35]. The double cone suspension is a generalization of the passage to a sphere n from its equator n−1 , of course. The Chafee-Infante attractor CI n can also be characterized as the Sturm attractor of maximal dimension n = (N − 1)/2, for any (necessarily odd) number N of equilibria. Equivalently, CI n is the n-dimensional Sturm attractor with the minimal number N = 2n + 1 of equilibria. See [12].
Let us start from that latter characterization, abstractly, without any ODE or PDE recurrence to the explicit equation (5.1) whatsoever. We will first derive the relevant part of the signed Thom-Smale complex C s for the n-dimensional Chafee-Infante attractor CI n with N = 2n + 1 equilibria. We will then determine the two boundary orders h ι , and arrive at the meander permutation σ = h −1 0 • h 1 of the Chafee-Infante attractor CI n , without any ODE shooting analysis of the equilibrium boundary value problem (1.2).
Fix any dimension n ≥ 1. By assumption, C s contains at least one cell c O of dimension n. The relevant part of the signed Thom-Smale complex C s is the signed hemisphere decom- For the s descendants v k (s) ∈ k s k , defined in (1.27), (1.28), the hemisphere characteriza- In particular, Theorems 3.1 and 4.3 become trivial. Invoking Lemma 2.2(ii) for constant s = + + · · · and s = − − · · · , however, implies the total order v 0 of all equilibria, at the boundary x = 0. Here we have applied the trivial equivalence u → −u of (1.38), as in the end of Sect. 2, to also derive the ordering of v k − at x = 0. This determines h 0 .
To determine the total order h 1 of all equilibria at the opposite boundary x = 1, we invoke the trivial equivalence x → 1 − x of (1.38). Since this equivalence swaps s j with (−1) j s j , and h 0 with h 1 , we can apply (5.3) and (5.4) Indeed the equilibria v k + appear below O for odd k, in increasing order, and above O for even k, in decreasing order. For the intercalating equilibria v k − the parities of k are swapped. See Fig. 2 for the resulting meander h 0 , based on the orders (5.4), (5.5). Notably the meander consists of two collections of nested arcs, one above and one below the horizontal h 1 axis. The two outermost arcs join the two poles v 0 ± with v 1 ± , respectively. The two innermost arcs join the inflection point O with v n−1 ± . In cycle notation, the Chafee-Infante Sturm permutation σ = h −1 0 • h 1 of the N = 2n + 1 equilibria follows easily from (5.4), (5.5) to be the involution σ = (2 2n) (4 2n − 2) . . .
for all 0 ≤ j < k < n. See (1.11) and (1.12). Conversely, a priori knowledge of all signed zero numbers determines the Sturm permutation σ = h −1 0 •h 1 , in any Thom-Smale complex. Indeed, the signs of z immediately determine the total order h 0 of all equilibria v k , at x = 0. Keeping the even/odd parity of k in mind, the same signs determine the total order h 1 of all equilibria v k , at x = 1.
For general abstract signed regular complexes, however, matters are not that simple. The prescribed hemisphere signs do not keep track of the relative boundary orders of all barycenter pairs v j , v k , explicitly. This information is only available for those pairs v j , v k for which one barycenter is in the cell boundary of the other (a posteriori, in other words, these are the heteroclinic pairs v k v j in the resulting Sturm attractor). How to extend this partial order to the two total orders h 0 , h 1 , uniquely for each of them, was the main result of the present paper. See Theorem 1.2.
To illustrate this point further, we turn to 3-ball Sturm attractors A f next. A purely geometric characterization of their signed hemisphere decompositions (1.18)-(1.21) has been achieved in [22,23]; see also [24] for many examples. Dropping all Sturmian PDE interpretations, we defined 3-cell templates, abstractly, in the class of signed regular cell complexes C s and without any reference to PDE terminology or to dynamics. Recall Fig. 1b for a first illustration. We recall here that an edge orientation of the 1-skeleton (C s ) 1 is called bipolar if it is without directed cycles, and with a single "source" vertex N and a single "sink" vertex S on the boundary of C s . Here "source" and "sink" are understood, not dynamically but, with respect to edge direction. The edge orientation of any 1-cell c v runs from 0 − (v) to 0 + (v). The most elementary hemi-"sphere" decomposition of 1-cells, in other words, can simply be viewed as an edge orientation. Bipolarity is a local and global compatibility condition for these orientations which, in particular, forbids directed cycles.
By Definition 1.1 of descendants, the 2-cells NE of w 0 − and SW of w 1 + denote the unique faces in W, E which contain the first, last edge of the meridian WE in their boundary, respectively. In Definition 5.1(iv), the boundaries of NE and SW are required to overlap in at least one shared edge along that meridian WE.
Similarly, the 2-cells NW of w 1 − and SE of w 0 + denote the unique faces in W, E, respectively, which contain the first, last edge of the other meridian EW in their boundary, respectively. The boundaries of NW and SE are required to overlap in at least one shared edge along that meridian EW.
The main result of [22,23], in our language of descendants, reads as follows.

of a 3-ball Sturm attractor A f if, and only if, C s is a 3-cell template.
In [22,  The ordering is uniform for 0 ≤ x ≤ 1, and holds at x ∈ {0, 1}, in particular. The poles N and S indicate the lowest and highest equilibrium, respectively, in that order. Again we refer to Fig. 1 for an illustrative example.
In our present language, properties (i) and (ii) thus describe the 3-cell template as a hemisphere decomposition of the boundary 2 = ∂c O of the single 3-cell O. The meridian cycle is To prepare our construction, we first consider planar regular cell complexes C, abstractly, with a bipolar orientation of the 1-skeleton C 1 . Here bipolarity requires that the unique poles N and S of the orientation are located at the boundary of the embedded regular complex C ⊆ R 2 .
To traverse the vertices v ∈ E of a planar complex C, in two different ways, we construct a pair of directed Hamiltonian paths Of course, this choice, together with the bipolar orientation and the boundary extrema on each cell, identifies the abstract planar regular cell complex C as a signed regular cell complex C s , with certain global rules on the cell signatures. Indeed, bipolarity serves as an additional global constraint, necessarily satisfied by planar Thom-Smale complexes of Sturm type. In addition, note how shared edges between adjacent faces receive opposite signatures from either face. For an SZ-pair, in contrast, we just have to swap the signature roles of 1 ± (O), in every 2-cell. The planar trilogy [17][18][19] contains ample material and examples on the planar case.
After these preparations we can now return to the general 3-cell templates C s of Definition 5.1 and define the SZS-pair (h 0 , h 1 ) associated to C s . Given the Sturm signed Thom-Smale complex of Fig. 1b, with the orientation of the 1-skeleton determined by the poles N = 1 and S = 9, we thus arrive at the SZS-pair (h 0 , h 1 ) indicated there. This determines the boundary orders of the equilibria in Fig. 1d. The meander in Fig. 1c is then based on the Sturm permutation σ = h −1 0 • h 1 , as usual. In summary, Theorem 1.2 and, for 3-cell templates equivalently, Definition 5.4 reconstruct the same generating Hamiltonian paths h 0 , h 1 , and hence the same generating Sturm permutation, of any 3-cell template.
In the general case, not restricted to 3-balls, we have assumed that the signed regular complex C s = C s f is presented as a signed Thom-Smale complex of Sturm type, from the start. In particular, all hemisphere signs were determined by the signed zero numbers. We have then described the precise relation between that signed complex C s = C s f and the boundary orders, at x = ι = 0, 1, of the paths h ι traversing the complex. In particular we have proved that the signed Thom-Smale complex C s = C s f determines the Sturm permutation σ = σ f , uniquely. Conversely, abstract Sturm permutations determine their signed Thom-Smale complex, uniquely. See [14,16,21,22]. This provides a 1-1 correspondence between Sturm permutations and their signed Thom-Smale complexes.
In general, however, we are still lacking a purely geometric characterization of those signed regular cell complexes C s which arise as signed Thom-Smale complexes C s = C s f of Sturm type. Indeed, the characterization by Theorem 5.2 covers 3-cell templates O, only. Three difficulties may arise in an attempt to realize a given signed regular cell complex C s as a Sturm complex C s = C s f . First, the recipe of Theorem 1.2 might fail to provide Hamiltonian paths h 0 , h 1 . For example, the same barycenter w of an (n − 1)-cell may be identified as the successor w = w ι + of the barycenters O and O of two different n-cells, for the same directed path h ι . Or that "path" might turn out to contain cyclic connected components, instead of defining a single Hamiltonian path which visits all barycenters. Second, even if both paths turn out to be Hamiltonian, from "source" N to "sink" S, the resulting permutation σ = h −1 0 • h 1 may fail to define a Morse meander-precluding any realization in the Sturm PDE setting (1.1). Third, and even if we prevail against both obstacles, the lucky original signed regular complex C s may fail to coincide, isomorphically, with the signed Thom-Smale complex C s f associated to the thus constructed Sturm permutation σ = σ f .
Let us bolster those nagging abstract doubts with three specific examples. Our first example, Fig. 4, recalls the unique Sturm tetrahedron 3-ball with 2+2 faces in the hemispheres 2 ± = 2 ± (O), alias W and E; see the detailed discussion in [24]. For such a hemisphere decomposition of the 3-ball tetrahedron, there essentially exists only one signed Thom-Smale complex which complies with all requirements of Definition 5.1; see Fig. 4a. In particular, both, the edge-adjacent location, along the meridian circle, of the poles 0 ± = 0 ± (O), alias N and S, and the bipolar orientation are then determined uniquely, up to geometric automorphisms of the tetrahedral complex and trivial equivalences.
In Fig. 4b we construct the resulting SZS-pair of Hamiltonian paths h 0 , h 1 . We follow the practical recipes of Definition 5.4(ii), for the h ι predecessors and successors w ι ± of O, and of Definition 5. (5.10) Our second example, Fig. 5, starts from a minuscule variation (a) of the same signed tetrahedral 3-ball. We only move the South pole S away from the position 4, which is edgeadjacent to N = 1 along the meridian circle. The new, more "symmetric" location 3 of S is not edge-adjacent to N along the meridian circle. We keep the 2+2 hemisphere decomposition unchanged, and only adjust the bipolarity of the 1-skeleton accordingly. Only the orientation of edge 9, from 3 to 4, has to be reversed to accommodate the misplaced South pole as an orientation sink. By tetrahedral symmetry, we may keep the orientation of edge 10, from 2 to 4, without loss of generality. Note however that any orientation of edge 10 now violates the orientation condition (iii) of Definition 5.1 in the hemisphere E = 2 + . All other requirements of Definition 5.1, including the overlap condition (iv), remain satisfied.
In Fig. 5b we construct the resulting paths h 0 , h 1 from the practical recipes of Definitions 5.3 and 5.4 , as before, with the usual labels of equilibria. This time, we obtain  Fig. 4. Only the edge-adjacent poles have been replaced by N = 1, S = 3, which are not edge-adjacent along the meridian circle (green). a Adapted bipolar orientation of the 1-skeleton, and hemisphere decomposition. Only the orientation condition (iii) of Definition 5.1 is violated, necessarily, by the orientation of edge 10 in the hemisphere E. b The SZS-pair h 0 (red) and h 1 (blue), constructed according to Definitions 5.3 and 5.4 , still provides Hamiltonian paths (5.11). c The dissipative Morse meander defined by the label-independent Sturm permutation σ = h −1 0 • h 1 ; see also (5.12). Violating Definition 5.1(iii), the original signed regular tetrahedron 3-ball complex is not Sturm. Therefore the Sturm permutation σ necessarily fails to describe the original non-Sturm signed complex (a). Instead, σ describes a Sturm signed Thom-Smale complex which is not a 3-ball (Color figure online) Alternatively to this indirect proof we could also have shown blocking of any heteroclinic orbit from O = 15 to the face equilibrium 14, based on zero numbers.
In fact we should have expected such failure: our construction of h 0 , h 1 in Theorem 1.2 is based on a signed cell complex C s which is assumed to be the signed Thom-Smale complex C s f of a Sturm global attractor A f . In the example of Fig. 5, we kind of started from a Sturm 3-ball A f , albeit with a misplaced South pole S and an incorrect signature orientation of the cell c 9 in its signed complex. Cooked up from the recipes of Theorem 1.2 and Definition 5.4 by sheer luck, the Sturm permutation (5.12) still described a Sturm global attractor A g , with an associated signed Thom-Smale complex C s g . However, A g = A f turns out to be a geometrically different global attractor, and C s g = C s f is not even a 3-ball.  Fig. 6 A signed regular octahedron 3-ball complex with antipodal poles N = 1 and S = 6. By [20,22,24], there does not exist any Sturm octahedron complex with antipodal poles. See Figs. 4 and 5 for our general setting and notation. a Equilibria E = {1, . . . , 27}, bipolar orientation of the 1-skeleton, and hemisphere decomposition W, E into 4 + 4 faces, one exterior. Only the overlap condition (iv) of Definition 5.1 is violated by the faces of the two pairs w ι − , w 1−ι + , respectively. b The SZS-pair h 0 (red) and h 1 (blue), constructed according to Definitions 5.3 and 5.4 , provides Hamiltonian paths. See also (5.13) for the resulting paths h 0 , h 1 . c The involutive permutation σ = h −1 0 • h 1 of (5.14) is dissipative and Morse, but fails to define a meander. There are 16 self-crossings. Therefore the signed regular octahedron 3-ball complex (a) with antipodal poles fails to define a Sturm signed Thom-Smale complex (Color figure online) That failure should caution us against another premature temptation. It is true that each closed hemisphere 2 ± (O) of any 3-ball Sturm attractor is a planar Sturm attractor, itself. See Definition 5.1(i), (ii). However it is not true, conversely, that we may glue any pair of planar Sturm disks, with matching poles and boundaries, along the shared meridian boundary to form the 2-sphere boundary 2 (O) of a Sturm 3-ball attractor. In fact, the two disks also have to satisfy the mandatory compatibility constraints of Definition 5.1(iii), (iv), on edge orientations and overlap, to define a Sturm 3-ball attractor. In Fig. 5, the two quadrangular closed disks clos W and clos E, each with a single diagonal, are such a non-matching pair. Indeed, the Eastern disk clos E violates Definition 5.1(iii).
Our third and final example, Fig. 6, applies our path construction to an octahedral 3-ball. Figure 6a prescribes a signed octahedron complex with diagonally opposite poles N = 1 and S = 6. In [22,24], however, we have already shown that there does not exist any signed Thom-Smale octahedral complex of Sturm type with diagonally opposite poles. See also [20] for this surprising phenomenon. So our construction is asking for trouble, again. To be specific we consider a symmetric decomposition into hemispheres W, E with 4+4 faces, as indicated in Fig. 6a. The hemisphere splitting avoids direct edges between the meridians EW and WE, like edge 10 in Fig. 5, which would contradict Definition 5.1(iii). All edge orientations in the bipolar 1-skeleton are then determined uniquely by conditions (i)-(iii) of Definition 5.1. Only the overlap condition (iv) is violated, this time. See Fig. 6b.
In conclusion we see how the recipe of Theorem 1.2, for the construction of the unique Hamiltonian boundary orders h 0 , h 1 and the unique associated Sturm permutation σ = h −1 0 • h 1 , works well for signed regular complexes C s -provided that these complexes are the signed Thom-Smale complex C s = C s f of a Sturm global attractor A f , already. In other words, there is a 1-1 correspondence between Sturm permutations and Sturm signed Thom-Smale complexes. For non-Sturm signed regular complexes, however, the construction recipe for h 0 , h 1 may fail to provide a Sturm permutation σ = h −1 0 • h 1 . This was the case for the octahedral example of Fig. 6. But even if the construction of a Sturm permutation σ succeeds, by our recipe, the result will-and must-fail to produce the naively intended Sturm realization of a prescribed non-Sturm signed regular complex C s . This was the case for the second tetrahedral example of Fig. 5. The goal of a complete geometric description of all Sturm signed Thom-Smale complexes C s f , as abstract signed regular complexes C s , therefore requires a precise geometric characterization of the fast unstable manifolds and the Sturmian signatures, on the cell level. Only for planar cell complexes and for 3-balls has that elusive goal been reached, so far. and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.