Sturm 3-ball global attractors 3: Examples of Thom-Smale complexes

Figure 1.1.  Example of a Sturm 3-ball global attractor $\mathcal{A}_f = clos W^u(\mathcal{O})$; see attractor $9.3^2$, alias case 2, of figs. 6.3, 6.4, and case 2, $5.2|7.2^2$ of table 6.5. Equilibria are labeled as $\mathcal{E}_f=\{1,\ldots,9\}$. The previous papers [23,24] established the equivalence of the viewpoints (a)-(d). (a) The Sturm global attractor $\mathcal{A}_f$, 3d view, including the location of the poles $\mathit{\boldsymbol{N}}$, $\mathit{\boldsymbol{S}}$, the (green) meridians $\mathit{\boldsymbol{WE}}$, $\mathit{\boldsymbol{EW}}$, the central equilibrium $\mathcal{O}$ and the hemispheres $\mathit{\boldsymbol{W}}$ (green), $\mathit{\boldsymbol{E}}$. (b) The dynamic Thom-Smale complex $\mathcal{C}_f$ of the boundary sphere $\Sigma^2 = \partial c_\mathcal{O}$, including the Hamiltonian SZS-pair of paths $(h_0,h_1)$, (red/blue), induced by the bipolar orientation of the 1-skeleton $\mathcal{C}_f^1$. The right and left boundaries denote the same $\mathit{\boldsymbol{EW}}$ meridian and have to be identified. See fig. 1.2 for the general case. (c) The Sturm meander $\mathcal{M}_f$ of the global attractor $\mathcal{A}_f$. The meander $\mathcal{M}_f$ is the curve $a \mapsto (v, v_x)$, at $x = 1$, which results from Neumann initial conditions $(v, v_x) = (a, 0)$ at $x = 0$ by shooting via the equilibrium ODE (1.2). Intersections of the meander with the horizontal $v$-axis indicate equilibria. See fig. 1.3 for the general case. (d) Spatial profiles $x\mapsto v(x)$ of the equilibria $v \in \mathcal{E}_f$. Note the different orderings of $v(x)$, by $h_0 = \mathrm{id}$ at the left boundary $x = 0$, and by the Sturm permutation $\sigma_f = h_1 = (1\ 8\ 3\ 4\ 7\ 6\ 5\ 2\ 9)$ at the right boundary $x = 1$. The same orderings define the meander in (c) and the Hamiltonian SZS-pair $(h_0, h_1)$ in the Thom-Smale complex (b)

Figure 1.2.  The 3-cell template, generalizing fig. 1.1(b). Shown is the 2-sphere boundary of the single 3-cell $c_\mathcal{O}$ with poles $\mathbf{N}$, $\mathbf{S}$, hemispheres $\mathbf{W}$ (green), $\mathbf{E}$, and separating meridians $\mathbf{EW}$, $\mathbf{WE}$ (both green). The right and the left boundaries denote the same $\mathbf{EW}$ meridian and have to be identified. Dots $\bullet$ are sinks, and small circles $\circ$ are sources. (a) Note the hemisphere decomposition (ii), the edge orientations (iii) at meridian boundaries, and the meridian overlaps (iv) of the $\mathbf{N}$-adjacent meridian faces $\otimes = w_-^\iota$ with their $\mathbf{S}$-adjacent counterparts $\odot = w_+^\iota$; see also (1.31). (b) The SZS-pair $(h_0, h_1)$ in a 3-cell template $\mathcal{C}$, with poles $\mathbf{N}, \mathbf{S}$, hemispheres $\mathbf{W}, \mathbf{E}$ and meridians $\mathbf{EW}, \mathbf{WE}$. Dashed lines indicate the $h_\iota$-ordering of vertices in the closed hemisphere, when $\mathcal{O}$ and the other hemisphere are ignored, according to definition 2.4(i). The actual paths $h_\iota$ tunnel, from $w_ -^\iota \in \mathbf{W}$ through the 3-cell barycenter $\mathcal{O}$, and re-emerge at $w_+^\iota \in \mathbf{E}$, respectively. Note the boundary overlap of the faces $\mathbf{NW}, \mathbf{SE}$ of $w_-^1, w_+^0$ from $v_-^{\mu-1}$ to $v_-^{\mu ' +1}$ on the $\mathbf{EW}$ meridian. Similarly, the boundaries of the faces $\mathbf{NE}, \mathbf{SW}$ of $w_-^0, w_+^1$ overlap from $v_+^{\nu -1}$ to $v_+^{\nu ' +1}$ along $\mathbf{WE}$. For many additional examples see sections 5 to 7. See also fig. 2.3

Figure 1.3.  The 3-meander template. Note the $\mathbf{N}$-polar $h_1$-serpent $\mathbf{N} = v_+^{2n} \ldots v_+^\nu$ terminated at $v_+^\nu$ by the subsequent source $w_-^0$ which is, both, $h_1$-extreme minimal and the lower $h_0$-neighbor of $\mathcal{O}$. This serpent overlaps the anti-polar, i.e. $\mathbf{S}$-polar, $h_0$-serpent $v_+^{\nu '} \ldots v_+^\nu \ldots v_+^0 = \mathbf{S}$, from $v_+^{\nu '}$ to $v_+^\nu$. Similarly, the $\mathbf{N}$-polar $h_0$-serpent $\mathbf{N} = v_-^0 \ldots v_-^{\mu '}$ overlaps the anti-polar, i.e. $\mathbf{S}$-polar, $h_1$-serpent $v_-^\mu \ldots v_-^{\mu '} \ldots v_-^{2n} = \mathbf{S}$, from $v_-^\mu$ to $v_-^{\mu '}$. The $h_1$-neighbors $w_\pm^1$ of $\mathcal{O}$ are the $h_0$-extreme sources, by the two polar $h_0$-serpents. Similarly, the $h_0$-neighbors $w_\pm^0$ of $\mathcal{O}$ define the $h_1$-extreme sources. See also sections 6, 7 for specific examples

Figure 2.1.  Traversing a face vertex $\mathcal{O}$ by a ZS-pair $h_0, h_1$. Note the resulting shapes "Z" of $h_0$ (red) and "S" of $h_1$ (blue). The paths $h_\iota$ may also continue into adjacent neighboring faces, beyond $w_\pm^\iota$, without turning into the face boundary $\partial c_{\mathcal{O}}$

Figure 2.2.  The Sturm $(m, n)$-gon disk with source $\mathcal{O}$, $m+n$ sinks, $m+n$ saddles, and hemisphere decomposition $\Sigma_\pm^j$, $j \in \{0, 1\}$, of $\mathcal{A} = \text{clos } W^u(\mathcal{O})$. Saddles and sinks are enumerated by $v_\pm^k$ with odd and even exponents $k$, respectively. (a) The associated Thom-Smale dynamic complex $\mathcal{C}$. Arrows on the circular boundary indicate the bipolar orientation of the edges of the 1-skeleton. Edges are the whole one-dimensional unstable manifolds of the saddles; the orientation of the edge runs against the time direction on half of each edge. The poles $\mathbf{N}$, $\mathbf{S}$ are the extrema of the bipolar orientation. Geometrically, we obtain an $(m+n)$-gon, with $m$ edges to the right of the poles and $n$ edges to the left. The resulting bipolar orientation determines the ZS-pair $(h_0, h_1)$, by definition 2.1. Colors $h_0$ (red), $h_1$ (blue). (b) The meander $\mathcal{M}$ defined by the ZS-pair $(h_0, h_1)$ of the $(m, n)$-gon (a). Along the horizontal axis, equilibria $v \in \mathcal{E}$ are ordered according to the directed path $h_1$ (blue). The directed path $h_0$ (red) defines the arcs of the meander $\mathcal{M}$. Note the two full polar $h_0$-serpents $v_-^0v_-^1 \ldots v_-^{2m-1}$ and $v_+^0v_+^1 \ldots v_+^{2n-1}$. The two full polar $h_1$-serpents are $ v_+^{2n} \ldots v_+^1$ and $v_-^{1} \ldots v_-^{2m}$. Also note how the $h_\iota$-neighboring saddles $w_\pm^\iota$ to the source $\mathcal{O}$, at $x = \iota$, become the $h_{1-\iota}$-extreme saddles at the opposite boundary

Figure 2.3.  Western $(\mathbf{W})$ and Eastern $(\mathbf{E})$ planar topological disk complexes. In $\mathbf{W}$, (a), all edges of the 1-skeleton $\mathbf{W}^1$ with a vertex $v \neq \mathbf{N}$ on the disk boundary are directed outward, i.e. towards $v$. In $\mathbf{E}$, (b), all 1-skeleton edges with a vertex $v\neq \mathbf{S}$ on the disk boundary are directed inward, i.e. away from $v$. Note the respective full $\mathbf{S}$-polar $h_0, h_1$-serpents $v_+^{2n-1} \ldots v_+^0 = \mathbf{S}$, $v_-^1 \ldots v_-^{2m} = \mathbf{S}$, dashed red/blue in (a), and the full $\mathbf{N}$-polar $h_0, h_1$-serpents $\mathbf{N} = v_-^0 \ldots v_-^{2m-1}$, $\mathbf{N} = v_+^{2n} v_+^{2n-1} \ldots v_+^1$, dashed red/blue in (b). Here we use ZS-pairs $(h_0, h_1)$ in $ \mathbf{E}$, but SZ-pairs $(h_0, h_1)$ in $\mathbf{W}$

Figure 3.1.  The effects of the trivial equivalences $\kappa, \rho$, and $\kappa\rho$ on a 3-cell template $\mathcal{C}$ with 19 equilibria. The cell complex $\mathcal{C}$ is drawn as the boundary sphere $S^2 = \partial c_{\mathcal{O}}$, in the style of figs. 1.1(b) and 1.2. (a) The original 3-cell template $\mathcal{C}$. (b)-(d) The 3-cell templates $\gamma \mathcal{C}, \ \gamma \in \lbrace \kappa, \rho, \kappa\rho \rbrace$. Annotations refer to the resulting template with hemisphere decomposition given by (1.21). See also the summary in table 3.1, at the end of this section. All entries of table 3.1 can be recovered from the figure, in principle, because corresponding cells are easily identified by their shape and connectivity. Note how the Klein 4-group $\langle \kappa, \rho \rangle$ acts transitively on the four elements $w_\pm^\iota$

Figure 4.1.  (a) Schematics of an EastWest complex $\mathcal{C}_0$. Poles are $\mathbf{N}, \ \mathbf{S}$. Arrows indicate the bipolar orientation. Green: pole-to-pole boundary paths. The paths are contained in the boundaries $\partial c_{w^\iota}$ of the boundary faces $c_{w^\iota}$ with barycenters $w^\iota$, respectively, for $\iota \in \{0, 1\}$. (b) The meander $\mathcal{M}$ resulting from the SZ-pair $(h_0, h_1)$ in $\mathcal{C}$. Note the full $\mathbf{N}$-and $\mathbf{S}$-polar serpents

Figure 4.2.  Three EastWest complexes $\mathcal{C}_0$ as Western, left, disks in 3-cell templates. See also fig. 1.1 for an example, and fig. 1.2 for general notation. (a) A single-face $(2, 3)$-gon $\mathcal{C}_0$, with barycenter $w_-^0 = w_-^1$. See 2.2 for a general description of $(m, n)$-gons. (b) A double face lift with two faces. Each face takes care of one meridian. (c) An eye lift where $\mathcal{C}_0$ possesses one interior closed face, the eye, which is detached from the two meridians

Figure 4.3.  The 3-meander template of a West lift of an Eastern disk $\text{clos } \mathbf{E} = \mathcal{C}_+$ by a (Western) EastWest disk $\text{clos } \mathbf{W} = \mathcal{C}_0$. Note the full $\mathbf{N}$-polar serpents, inherited from the Eastern disk $\mathbf{E}$. The $\mathbf{S}$-polar serpents are not full, in general, but are overlapped completely by their full $\mathbf{N}$-polar counterparts. This leads to a subtle simplification of the general 3-meander template, fig. 1.3

Figure 4.4.  The 3-meander template resulting from the lift of a general (Western) EastWest disk $\mathcal{C}'_0$ by a general (Eastern) EastWest disk $\mathcal{C}_0$, welded at the shared $(m+n)$-gon meridian boundary. Interchanging the Eastern and Western roles of $\mathcal{C}_0$ and $\mathcal{C}'_0$ interchanges $m$ and $n$

Figure 4.5.  Two examples of 3-meander templates involving $(m+n)$-gons. (a) The single face lift of a Western $(n, m)$-gon by an Eastern $(m, n)$-gon, called pitchfork lift. Note the modification of the planar $(m, n)$-gon meander of fig. 2.2(b) by a pitchfork bifurcation of the face center $\mathcal{O}$. (b) The lift of a Western $n$-striped disk by an Eastern $m$-striped disk. Note the resulting unstable double cone suspension, of the planar $(m, n)$-gon meander of fig. 2.2(b), by a 180$\, ^\circ$ rotation and the addition of two new polar arcs $\mathbf{N} v_-^1$ and $v_+^1 \mathbf{S}$

Figure 4.6.  Two mirror-symmetric 3-ball attractors $\mathcal{A}^+$ (left) and $\mathcal{A}^-$ (right). Note the degenerate eye lifts. The attractors are not trivially equivalent, but result from lifts of the same EastWest disks $\mathcal{C}_0, \ \mathcal{C}'_0$ in swapped Eastern and Western roles

Figure 5.1.  The dual complex $\mathcal{C}^*$ of an example 3-cell complex $\mathcal{C}$. The seven face sources $\circ$ of $\mathcal{C}$ become vertices of the dual, $\mathcal{C}^*$. The eight sources $\bullet$ of faces in $\mathcal{C}^*$ are the sink vertices of $\mathcal{C}$. The orientation of $S^2 \subseteq \mathbb{R}^3$ is taken to be standard planar, when viewed from outside. (a) The 1-skeleton $\mathcal{C}^{1}$ for the 3-cell template $\mathcal{C}$; see also fig. 1.2(a). The boundaries $\mathbf{EW}$ are identified. The thirteen edges $e$ of $\mathcal{C}$ (solid) are crossed by dual edges $e^*$ (dashed) from left to right. (b) Schematics of the dual complex $\mathcal{C}^{*, 1}$. Edges $e^*$ of $\mathcal{C}^*$ are solid, and $e$ of $\mathcal{C}$ are dashed. The poles $\mathbf{N}, \mathbf{S}$ become the barycenters of the exterior face $\mathbf{N}^*$ and some interior face $\mathbf{S}^*$, respectively. Note the annulus bounded by the dual polar circles $\partial \mathbf{N}^*, \partial \mathbf{S}^*$ (solid blue), which surround the pole $\mathbf{N}$ clockwise(!), and $\mathbf{S}$ counter-clockwise. Bipolarity holds within each dual hemisphere core, $\mathbf{W}^* = \mathcal{C}_-^{2, *}$ and $\mathbf{E}^* = \mathcal{C}_+^{2, *}$ (both gray), with North poles $w_\pm^0$ and South poles $w_\pm^1$, and with dual meridians (orange). The duals to overlap edges form single-edge bridges between the polar circles, directed from the dual South poles $w_\pm^1$ in one hemisphere core to the dual North poles $w_\mp^0$ in the opposite hemisphere core

Figure 5.2.  Minimal construction of an EastWest disk by the dual of any planar Sturm attractor $\mathcal{A}^2$. Note how the poles $w_+^0$ and $w_+^1$ of $\mathcal{A}^2$ become faces adjacent to the extra edges of A and B which constitute the respective meridian boundaries. Without these edges we obtain the standard construction of the planar dual Sturm attractor $\mathcal{A}^{2, *}$; see [18,20]

Figure 6.1.  The eight Eastern dual cores $\mathbf{E}^*$, up to trivial equivalences, written as planar Sturm attractors with $N^* \leq 7$ equilibria. See (6.8) for the classification scheme. Circles "$\circ$" indicate vertices of $\mathbf{E}^*$ and Morse index $i = 2$ face barycenters of $\mathbf{E}$. Dots "$\bullet$" are face barycenters of $\mathbf{E}^*$ and Morse stable $i = 0$ vertices of $\mathbf{E}$. The bipolar orientation of $\mathbf{E}^*$ runs from $w^0$ (red) to $w^1$ (blue), in each case

Figure 6.2.  The twelve regular 2-sphere complexes, with at most $N-1 \leq 12$ cells on the sphere $\partial c_\mathcal{O}$. The omitted 3-cell barycenter $\mathcal{O}$ contributes to the total count $N$ of cells, in the notation $N.n^{k_n}\ldots$ of (6.8). The 2-sphere is represented by one-point compactification of the plane, i.e. each 1-skeleton is drawn as a Schlegel graph. In other words, we also consider the exterior as a face. Left: $c_0 \leq c_2$, i.e. at least as many faces as zero-cell vertices. Right: $c_0 \geq c_2$, by standard duality. Note the two self-dual cases $7.2^2$ and $11.3^2 2$. The cases $13.3^2 2^2-1$ and $-2$ differ by degrees 433 and 442 at their vertices, respectively. See fig. 6.3 for the associated Sturmian Thom-Smale complexes, which turn out to be nonunique quite frequently

Figure 6.3.  The 31 3-cell templates of 3-ball Sturm attractors with at most 13 equilibria, up to trivial equivalences. See tables 6.5, 6.6 for case numbers 1-31, hemisphere notation, and Sturm permutations. Cases are arranged in rows, on right, according to the twelve regular Thom-Smale $S^2$-complexes of fig. 6.2, listed left. The two self-dual $S^2$-complexes are marked by $*$. On the left, $S^2$ is the compactified plane. On the right, with meridians in green, the right and left $\mathbf{EW}$ meridian have to be identified. All omitted bipolar orientations result from the poles $\mathbf{N}$, top, versus, $\mathbf{S}$, bottom, and the hemisphere assignments $\mathbf{W}$, left, versus $\mathbf{E}$, right

Figure 6.4.  The 31 3-meander templates of 3-ball Sturm attractors with at most 13 equilibria, up to trivial equivalences. Horizontal $h_1$ axis omitted. See tables 6.5, 6.6, and fig. 6.3 for case numbers 1-31

Figure 7.1.  The five Platonic solids $c_\mathcal{O}\, $: tetrahedron $(\mathbb{T})$, octahedron $(\mathbb{O})$, cube or hexahedron $(\mathbb{H})$, dodecahedron $(\mathbb{D})$, and icosahedron $(\mathbb{I})$. In each case we depict the planar Schlegel graph of the 1-skeleton $\mathcal{C}^1$ for the regular cell complex $\mathcal{C}^2$ of the boundary sphere $S^2 = \partial c_\mathcal{O}$. Again, we consider the exterior as another face in the one-point compactification of the plane.

Figure 7.2.  Left: the unique Sturm tetrahedron $\mathbb{T}.1$ with a single Western face (exterior). The bipolar orientation on $S^2 = \partial \mathbb{T}$ is uniquely determined by the pole location and the hemisphere decomposition. Right: the Sturm meander $\mathcal{M}$ determined from the SZS-pair $(h_0, h_1)$ on the left

Figure 7.3.  The unique Sturm tetrahedron $\mathbb{T}.2$ with two Western faces, (b). The bipolar orientation on $S^2 = \partial \mathbb{T}$ is uniquely determined by the pole location and the hemisphere decomposition. Left, (a): the dual tetrahedron $\mathbb{T}^*$ with the one-dimensional dual core attractors $\mathbf{W}^*, \ \mathbf{E}^*$ (both shaded gray), dual poles $w_\pm^\iota$, polar circles $\partial \mathbf{N}^*, \ \partial \mathbf{S}^*$ (blue) and the predual meridian circle (green). All orientations follow from lemma 5.1. Right, (c): the Sturm meander $\mathcal{M}$ and Sturm permutation $\sigma = h_0^{-1} \circ h_1$ resulting from the (omitted) SZS-pair $(h_0, \ h_1)$ in the 3-cell template (b). Note the 1, 2, 4 nested lower, and 7 nested upper arcs

Figure 7.4.  The impossibility of pole distance $\delta = 2$ in the octahedron $\mathbb{O}$ with cube dual $\mathbb{O}^* = \mathbb{H}$. Note the orientations of the disjoint polar circles $\partial \mathbf{N}^*, \ \partial \mathbf{S}^*$, and the pairs of directed polar bridges $e_* = w_\pm^1w_\mp^0$, dual to meridian edges $e$. The remaining meridian edges are polar. The meridian circle separates the (impossible) dual tri-star core $\mathbf{W}^*$ of $A, \ w_-^0, \ w_-^1, \ B$ from the tri-star core $\mathbf{E}^*$ (both shaded gray). Polar bridges directed from $w_\pm^1$ to $w_\mp^0$ are indicated (orange)

Figure 7.5.  The impossibility of $\dim \mathbf{W}^* = 2$ in the octahedron with cube dual $\mathbb{O}^* = \mathbb{H}$. Note how the exterior square $\mathbf{W}^*$ (gray) forces the location of the meridian circle (green), with edge adjacent poles $w_-^\iota$ on the polar circle $\partial \mathbf{N}^* \, \cap \, \partial \mathbf{W}^*$. These force $w_+^{1-\iota} \in \partial \mathbf{N}^*$ to be adjacent on the inner square $\partial \mathbf{S}^*$. The resulting position of $\mathbf{S}$ in the central square is not on the meridian circle, and hence is impossible

Figure 7.6.  An orientation conflict arising from the (gray) Western core $\mathbf{W}^*$ with $\eta = 3$ vertices. Note how the location of the $\mathbf{E}^*$-poles $w_+^\iota$ forces the face $\mathbf{S}^*$ to possess an inconsistent orientation of its polar circle, $\partial \mathbf{S}^*$ from $A = w_+^0$ to $w_+^1$

Figure 7.7.  A viable dual core $\mathbf{W}^*$ (gray) with $\eta = 2$ vertices, (a). The locations $A, \ B$ are not viable for $\mathbf{S}$ because $\partial \mathbf{S}^*$ cannot accommodate $w_+^0$ and $w_+^1$ at edge distance 1 across the meridian from $w_-^1$ and $w_-^0$, respectively. We draw the only viable location for $\mathbf{S}$. The bipolar orientation of (gray) $\mathbf{E}^*$, from $w_+^0$ to $w_+^1$ determines all other edge orientations uniquely. See (b) for the resulting bipolar octahedron complex $\mathbb{O}.2$. See table 7.2 for the Sturm permutation $\sigma$

Figure 7.8.  The four possible bipolar orientations of the Sturm octahedron with a single face Western hemisphere $\mathbf{W}$ (exterior). The orientations only differ on the acyclic central triangle $ABC$ inside the Eastern hemisphere $\mathbf{E}$. The four possibilities on the right arise by the selection of a maximal and a minimal vertex among $A, \ B, \ C$. Bipolarity prevents $C$ from being minimal. The case $AB$, for example, chooses $A$ as maximal and $B$ as minimal

Figure 7.9.  The four 3-meander templates of single-face lift octahedra, $\eta = 1$. Note the identical locations of the four core poles $w_\pm^\iota$, and the different location configurations of the central triangle vertices $A, \ B, \ C$. See fig. 7.8 for bipolar orientations, and table 7.3 for Sturm permutations and case labels

Figure 7.10.  The unique cube $\mathbb{H}.3.3$ with pole distance $\delta = 3$. Note the resulting 3-gon, $\eta = 3$, Western and Eastern cores $\mathbf{W}^*, \ \mathbf{E}^*$ (both gray). (a) Disjoint polar circle triangles $\partial \mathbf{N}^*, \ \partial \mathbf{S}^*$, with poles $\otimes = w_-^\iota$ of $\mathbf{W}^*$ in $\partial \mathbf{N}^*$. The bridge options for poles $\odot = w_+^\iota$ of $\mathbf{E}^*$ in $\partial \mathbf{S}^*$, across the meridians, are dotted or solid. Only the solid option is compatible with the proper left orientation of $\partial \mathbf{S}^*$. (b) The resulting cube 3-cell template $\mathbb{H}$ with uniquely determined bipolar orientation. (c) The cube meander $\mathcal{M}$ generated by the SZS-pair $(h_0, h_1)$ of the cube 3-cell template (b). Note the nested $3^2, \ 3, \ 1$ upper arcs, and $1, \ 3, \ 3^2$ lower arcs of $\mathcal{M}$, from left to right. The meander exhibits full isotropy under all trivial equivalences $\langle \kappa, \rho \rangle$. Flip-isotropy $\kappa$ is in compliance with the requirements of corollary 3.2. Still, it is neither pitchforkable nor realizable by a pendulum type nonlinearity $f = f(u)$. See table 7.4, case 7, for the Sturm permutation $\sigma$

Figure 7.11.  Dual octahedron with a 3-gon Western core $\mathbf{W}^*$ (exterior, gray) and surrounding meridian. The direction of the dual edge $e_* = w_-^0w_-^1$ follows because $\mathbf{N}^*$, with barycenter $\mathbf{N}$ on the meridian, cannot coincide with the exterior face of $\mathbf{W}^*$. The left orientation of the Southern polar circle $\partial \mathbf{S}^*$ contradicts the direction $w_+^0w_+^1$, for $\mathbf{S} = \mathbf{S}_1$ or $\mathbf{S}_2$ and all (dotted) candidates $w_+^{1-\iota}$ paired with $w_-^\iota$. Therefore $\mathbf{S} = \mathbf{S}_3$ is the antipode of $\mathbf{N}$, as in fig. 7.10

Figure 7.12.  The unique cube $\mathbb{H}.2.2$ with $\eta = 2$ Western faces. Note the unique edge $e_* = w_-^0 w_-^1$ in the Western core $\mathbf{W}^*$ and the double 3-gon Eastern core $\mathbf{E}^*$ (both gray). (a) Exterior polar circle $\partial \mathbf{N}^*$ and meridian circle in the octahedral dual $\mathbb{O} = \mathbb{H}^*$. Only South poles $\mathbf{S} = \mathbf{S}_2$ and $\mathbf{S}_4$ are viable options, trivially equivalent under $\rho$. Lemma 5.1 forces the location $w_+^0 = A$, and hence $w_+^1 = B$. All remaining orientations of dual edges follow. (b) The resulting cube 3-cell template $\mathbb{H}$ with uniquely determined bipolar orientation. (c) The cube meander $\mathcal{M}$ generated by the SZS-pair $(h_0, h_1)$ of (b), without remaining isotropy. For the Sturm permutation $\sigma$ see table 7.4, case 6

Figure 7.13.  The three possible bipolar orientations, each, of the Sturm cube with single face Western hemisphere, exterior. The orientations only differ on the acyclic central square $ABCD$, respectively, for the case (a) of adjacent poles, $\delta = 1$, and for the case (b) of diagonally opposite poles, $\delta = 2$, on the Western face. Note how acyclicity of $ABCD$ makes $D$ a local minimum in (c). The cases $A, \ B, \ C$ refer to the location of the local maximum

Figure 7.14.  The five 3-meander templates of single face lift cubes, $\eta = 1$. Note the three identical locations of the four core poles $w_\pm^\iota$, for each pole distance $\delta = 1, 2$, and the different configurations of the central square $A, \ B, \ C, \ D$. See fig. 7.13 for bipolar orientations, and table 7.4 for Sturm permutations and case labels

Figure 7.15.  The dodecahedral dual $\mathbb{I}^* = \mathbb{D}$ of the icosahedron complex $\mathbb{I}$. (a) Exterior polar face dual $\mathbf{N}^*$ with oriented boundary $\partial \mathbf{N}^*$. Representative barycenters $\mathbf{S}_\delta$ of candidate face duals $\mathbf{S}_\delta^*$ denote South poles $\mathbf{S}$ at distances $\delta = 1, 2, 3$ from the exterior North pole $\mathbf{N}$. Note the single bridge $BE$ between the polar circle $\partial \mathbf{N}^*$ and $\partial \mathbf{S}_2^*$, as well as the absence of bridges between $\partial \mathbf{N}^*$ and $\partial \mathbf{S}_3^*$. (b) Viable placement of the four pole $w_\pm^\iota$ of the dual cores $\mathbf{W}^*, \ \mathbf{E}^*$ in case $\mathbf{S} = \mathbf{S}_1$. Only the locations $A, \ldots, F$ allow for single-edge directed bridges $w_\pm ^1w_\mp^0$. The bridges must lie in the solid boundary parts of the two adjacent pentagonal polar faces. The orientations of the polar circles $\partial \mathbf{N}^*, \ \partial \mathbf{S}_1^*$ are indicated, and result in the green meridian circle

Figure 7.16.  The icosahedral dual $\mathbb{D}^* = \mathbb{I}$ and the dodecahedral complex $\mathbb{D}$. (a) Exterior polar face dual $\mathbf{N}^*$ with oriented boundary $\partial \mathbf{N}^*$. Representative candidate face duals $\mathbf{S}_\delta^*$ indicate South poles $\mathbf{S}$ at distances $\delta = 1, \ldots, 5$ from the North pole $\mathbf{N}$. Bridges between $\partial \mathbf{S}_5^*$ and $\partial \mathbf{N}^*$ are absent. Bridges are unique between $\partial \mathbf{S}_4^*$ and $\partial \mathbf{N}^*$. (b) Placement of the directed 4-cycle $BCDEB$ of $w_\pm^\iota$ in (a), for the case $\mathbf{S} = \mathbf{S}_2$ of pole distance $\delta = 2$. See also table 7.5. (c) The resulting meridian segments (green, solid) in $\mathbb{I} = \mathbb{D}^*$ for the configuration (b) of $w_\pm^\iota$. For the closure of the meridian circle (green, dashed) see text. (d) The resulting hemisphere decomposition with pole distance $\delta = 2$ and $\eta = 2$ Western faces $w_-^0 = B, \ w_-^1 = C$, in the original dodecahedron 3-cell template $\mathbb{D}$. Only mandatory parts of the bipolar orientation are indicated

Figure 7.17.  A sample Sturm icosahedron Thom-Smale complex $\mathbb{I}$ with pole distance $\delta = 1$ and with $\eta = 2$ Western faces $\otimes$. Note the required orientation arrows emanating from the North pole $\mathbf{N}$, directed away from the meridians into the Eastern hemisphere, and terminating at the South pole $\mathbf{S}$. The SZS-pair $(h_0, h_1)$ results from the bipolar orientation: $h_0$ (red), $h_1$ (blue), $h_0 \& h_1$ (purple). See fig. 7.19 and table 7.6 for the Sturm meander $\mathcal{M}$ of the resulting Sturm permutation $\sigma = h_0^{-1}\circ h_1$

Figure 7.18.  A sample Sturm dodecahedron Thom-Smale complex $\mathbb{D}$ with maximal pole distance $\delta = 2$ and with $\eta = 2$ Western faces. The SZS pair $(h_0, h_1)$ results from the bipolar orientation; see also fig. 7.16(d). See fig. 7.20 and table 7.6 for the Sturm meander $\mathcal{M}$ of the Sturm permutation $\sigma = h_0^{-1} \circ h_1$

Figure 7.19.  The Sturm meander $\mathcal{M}$ for the icosahedron $\mathbb{I}$ of fig. 7.17. The marked sources $A, \ldots, F$ correspond to fig. 7.15(b) and to the icosahedral Thom-Smale complex $\mathbb{I}$. Note the extreme positions of the poles $w_\pm^\iota$ of the dual cores $\mathbf{W}^*, \ \mathbf{E}^*$

Figure 7.20.  The Sturm meander $\mathcal{M}$ for the dodecahedron $\mathbb{D}$ of fig. 7.18. The marked sources $A, \ldots, F$ correspond to figs. 7.16 and 7.18. For further comments on $w_\pm^\iota$; see fig. 7. 19

Figure 8.1.  (a) The Snoopy bun 3-ball Sturm attractor with $N = 13$ equilibria. See fig. 6.3 and table 6.6, cases 13, 14, 19, 24 for inequivalent realizations. (b) The Snoopy burger with an additional 3-cell bun $c_{\mathcal{O}}$, and hemisphere $\mathbf{H}$ packed on top. This regular cell complex of dimension 3, with two adjacent 3-balls sharing 3 faces, is not a Sturm dynamic complex