We systematically explore a simple class of global attractors, called *Sturm* due to nodal properties, for the semilinear scalar parabolic partial differential equation (PDE) $ u t= u x x+f(x,u, u x)$ on the unit interval $0<x<1$, under Neumann boundary conditions. This models the interplay of reaction, advection, and diffusion. Our classification is based on the Sturm meanders, which arise from a shooting approach to the ordinary differential equation boundary value problem of equilibrium solutions $ u t=0$. Specifically, we address meanders with only three “noses,” each of which is innermost to a nested family of upper or lower meander arcs. The Chafee–Infante paradigm, with cubic nonlinearity $f=f(u)$, features just two noses. Our results on the gradient-like global PDE dynamics include a precise description of the connection graphs. The edges denote PDE heteroclinic orbits $ v 1\u21dd v 2$ between equilibrium vertices $ v 1, v 2$ of adjacent Morse index. The global attractor turns out to be a ball of dimension $d$, given as the closure of the unstable manifold $ W u( O)$ of the unique equilibrium with maximal Morse index $d$. Surprisingly, for parabolic PDEs based on irreversible diffusion, the connection graph indicates time reversibility on the ( $d\u22121$)-sphere boundary of the global attractor.

The global dynamics of the nonlinear interplay among diffusion, reaction, and advection is little understood. This holds true even for a single equation on finite intervals, where a decreasing energy functional and nonlinear nodal properties of Sturm type considerably simplify the dynamics. Part of this predicament is caused by an undue focus on the particular dynamics of particular nonlinearities: spatial chaos, for example, may lead to large numbers of globally competing stable and unstable equilibria. Instead, we explore a rich class of nonlinearities with prescribed meandric equilibrium configurations of three-nose type. The global attractors, in that case, turn out to be balls with an attracting boundary sphere of potentially arbitrarily large dimension. For the first time, in that class, we provide a detailed dynamic description via the global graph structure of heteroclinic orbits between equilibria. Much to our surprise, we encountered signs of *time reversibility* within the attracting boundary sphere. This contradicts common “knowledge” of diffusion as *the* paradigm of irreversibility.

## I. INTRODUCTION

*Fisher equation*$f=\lambda u(1\u2212u)$ of genetic selection,

^{53}and the slightly more general

*Kolmogorov–Petrovsky–Piskunov*(KPP) variant of population growth.

^{68}See also the stochastic branching processes addressed in Ref. 11. Cubic $f$ arise in the Allen–Cahn description of interface motion in binary alloys

^{1}and, as a singular limit, in the Nagumo equation of nerve conduction. The famous Chafee–Infante cubic $f= \lambda 2u(1\u2212 u 2)$ falls into that class, and inspired much of the PDE analysis in the area;

^{15}see also (4.1) and Sec. IV. The prefactor $ \lambda 2$, which we now omit, arises from scaling a spatial interval $0<x<\lambda $ to unit length. The

*Zeldovich–Frank–Kamenetskii equation*(ZFK) $f=u(1\u2212u)exp\u2061(\u2212\beta (1\u2212u))$ models combustion and, with the proper Arrhenius exponential instead, non-isothermal catalysis. Chemical reactions in permeable catalysis or tubular reactors provide examples, where reaction, advection, and diffusion arise under their proper name.

^{6}Quasilinear variants of (1.2) arise, for example, in curve shortening and interface flows.

^{4,31,32}Many applications involve singular limits. For applications to viscous hyperbolic balance laws, see, for example, Ref. 61. A few of the formidable complications of $x$-dependent nonlinearities $f$ have been tackled with in Ref. 5; see also Refs. 47 and 62. The PDE (1.2) also appears as a parabolic limit in problems of, both, elliptic and hyperbolic type

^{51,82,84,95}A spatially discrete variant models a coupled chain of overdamped pendula;

^{27}see also Sec. VIII A. Conley index theory, as a homotopy-invariant, global topological tool, has extended the Chafee–Infante paradigm further to include applications to certain beam equations, and settings like FitzHugh–Nagumo, Cahn–Hilliard, and certain phase field equations.

^{63,83}See Ref. 52 for a broad earlier survey on phase field equations. More recently, and mostly for systems of equations in biological context, see Ref. 85. See also the survey

^{50}for further mathematical and applied aspects. In the spirit of (1.2), very interesting global results for Ginzburg–Landau patterns on two-spheres, and other compact surfaces of revolution, have recently been obtained by Refs. 17 and 18. Meanwhile, the mathematical literature on reaction-diffusion equations alone, as referred in Zentralblatt under MSC 35K57, has grown to more than 15 000 entries.

^{104}

It is, therefore, not our intention, in the present paper, to contribute just another analysis or simulation, for this or that particular nonlinearity $f$, arising in one or the other highly specialized applied context. For general $x$-dependent nonlinearities, on the other hand, the chaotic complexities of even the ODE equilibrium problem (1.1) seem to frustrate any all-out attack on the PDE dynamics of (1.2), *a priori*. Or, do they?

In fact, it is possible to characterize the class of all ODE equilibrium “configurations,” qualitatively, by certain permutations $\sigma $. See the following Sec. II. The permutations $\sigma $ themselves, as introduced by Fusco and Rocha,^{56} are based on the discrepancies between the orderings of the equilibria at the boundaries $x=0$ and $x=1$, respectively; see (2.4) and (2.5). Although each of the permutations will be represented by an open class of nonlinearities $f$, in principle, we will provide specific nonlinearities only in exceptional cases; but see (4.1) and Sec. IV for cubic $f$. In general,

it will therefore be the qualitative configuration of ODE equilibria (1.1), which we assume to be given, rather than some particular nonlinearity $f$.

In the present paper, we describe the global dynamics of the full PDE (1.2), for a certain subclass of permutations $\sigma $. This allows us to design certain time asymptotic global attractors of (1.2), with three competing attracting sinks. A plethora of other equilibria, of arbitrarily high unstable dimension, may be involved in the boundaries of their domains of attraction. The resulting PDE dynamics turns out to be gradient-like, by a general energy functional. In particular, the PDE dynamics on the global attractor will consist of equilibria and their heteroclinic orbits (2.1), only. Still, we will encounter at least some of the intricacies, which are caused by the competition among large numbers of highly unstable equilibria.

Some mathematical generalities are easily settled. For continuously differentiable nonlinearities $f\u2208 C 1$, standard theory of strongly continuous semigroups provides local solutions $u(t,x)$ of (1.2) in suitable Sobolev spaces $u(t,\u22c5)\u2208X\u2286 C 1([0,1], R)$, for $t\u22650$ and given initial data $u= u 0(x)$ at time $t=0$. See Refs. 64, 89, and 99 for a general PDE background.

We assume the solution semigroup $u(t,\u22c5)$ generated by the nonlinearity $f$ to be *dissipative*: any solution $u(t,\u22c5)$ exists globally in forward time $t\u22650$ 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 $f(x,u,0)\u22c5u<0$, for large $ |u |$, together with subquadratic growth in $ |p |$. For large times $t\u2192\u221e$, any large ball in $X$ then limits onto the same maximal compact and invariant subset $ A= A f$ of $X$ which is called the *global attractor*. In general, the global attractor $ A$ consists of all solutions $u(t,\u22c5)$ which exist globally, for all positive and negative times $t\u2208 R$, and remain bounded in $X$. Of course, $ A$, therefore, contains any equilibria, heteroclinic orbits, basin boundaries, or more complicated recurrence which might arise, in general. See Refs. 7, 16, 22, 24, 59, 60, 69, 92, 96, and 100 for global attractors in general.

## II. BACKGROUND AND OUTLINE

Admittedly, the above information on Sturm attractors $ A$ is quite general. However, it provides practically no information concerning the specific dynamics on $ A$. Rather than complacently pontificate a few pretty vague generalities, here, we aim to elucidate at least some of that very rich inner dynamics. Already the chaotic intricacies of the mere equilibrium ODE (1.1) may hint at the scope of our quest. In particular, after decades of dedication and quite a few unexpected results, we hope to convince our readers that the purportedly “trivial” dynamics of (1.2) is still poorly understood. That is why we proceed by examples.

*Lyapunov function*, alias a variational or gradient-like structure, under separated boundary conditions; see Refs. 30, 66, 73, 74, 79, 81, and 103. Therefore, the time invariant global attractor consists of equilibria and of solutions $u(t,\u22c5)$, $t\u2208 R$, with forward and backward limits, i.e.,

*heteroclinic*or

*connecting*orbit, or

*instanton*, and write $ v 1\u21dd v 2$ for such heteroclinically connected equilibria. See Figs. 1(c) and 1(d) for the modest three-ball example with $N=11$ equilibria. Although the variational structure persists for other separated boundary conditions, the possibility of rotating waves shows that it may fail under periodic boundary conditions. See, however, Refs. 30 and 49.

*Sturm nodal property*, which we express by the

*zero number*$z$. Let $0\u2264z(\phi )\u2264\u221e$ count the number of (strict) sign changes of continuous spatial profiles $\phi :[0,1]\u2192 R,\phi \u22620$. For any two distinct solutions $ u 1$, $ u 2$ of (1.2), the zero number

^{98}for the linear autonomous variant.

The consequences of the Sturm nodal property (2.2) for the nonlinear dynamics of (1.2) are enormous. For an introduction, see Refs. 12, 13, 50, 55, 57, 76, 80, and 93 and the many references there. Already Sturm observed that all eigenvalues $ \mu 0> \mu 1>\u2026$ of the PDE linearization of (1.2) at any equilibrium $v$ are algebraically simple and real. In fact $z( \phi j)=j$, for the eigenfunction $ \phi j$ of $ \mu j$. *We assume all equilibria are hyperbolic, i.e., all eigenvalues are nonzero.* The *Morse index* $i(v)$ of $v$ then counts the number of unstable eigenvalues $ \mu j>0$. In other words, the Morse index $i(v)$ is the dimension of the unstable manifold $ W u(v)$ of $v$. Let $ E\u2286 A$ denote the set of equilibria. Our generic hyperbolicity assumption and dissipativeness of $f$ imply that $N$:= $ | E |$ is odd; see also (5.5).

^{2,65}More precisely, intersections of unstable and stable manifolds $ W u( v 1)$ and $ W s( v 2)$ along heteroclinic orbits $ v 1\u21dd v 2$ are automatically transverse: $ W u( v 1)\u22d4 W s( v 2)$. In the Morse–Smale setting, Henry

^{65}also observed

*boundary orders*$ h 0, h 1$: ${1,\u2026,N}\u2192 E$ of the equilibria such that

*Sturm permutation*$\sigma \u2208 S N$, defined by Fusco and Rocha in Ref. 56 as

Using a shooting approach to the ODE boundary value problem (1.1), the Sturm permutations $\sigma \u2208 S N$ have been characterized, purely combinatorially, as *dissipative Morse meanders* in Ref. 35. Here, the *dissipativeness* property, abstractly, requires fixed $\sigma (1)=1$ and $\sigma (N)=N$. In fact, the shooting meander emanates upwards, toward $ v x>0$, from the leftmost (or lowest) equilibrium at $\sigma (1)=1$, and terminates from below, $ v x<0$, at x=1. The *meander* property requires the formal path $ M$ of alternating upper and lower half-circle arcs defined by the permutation $\sigma $, as in Fig. 1(c), to be Jordan, i.e. non-selfintersecting. For dissipative meanders, the recursion in (2.6), and $ i 1=0$, define all Morse numbers $ i j$. Note how $j$ and $ i j$ are always of opposite parity, $ mod2$. In particular, $N$ is odd, and $ i N$=0 follows automatically. The *Morse* property, finally, requires non-negative Morse indices $ i j\u22650$ in the formal recursion (2.6), for all $j$. For brevity, we also use the term *Sturm meanders*, for dissipative Morse meanders.

For a simple recipe to determine the Morse property of a meander, *the Morse number increases by 1, along any right turning meander arc, but decreases by 1 along left turns.* This holds, independently, for upper and lower meander arcs, and remains valid even when the proper orientation of the arc is reversed; see (2.6). For example, see Figs. 1 and 3–5. The beautifully illustrated book, Ref. 67, contains ample material on many additional aspects of meanders. Even “just” counting meanders, with a prescribed number of “noses” (2.11), is a deep and fascinating subject.^{19,20} The results for Morse meanders are much less explicit, so far.^{58}^{,}^{102}

In the present paper, we address Sturm meanders. We will return to the intriguing issue of non-Morse dissipative meanders with some negative “Morse indices” $ i j<0$, briefly, in Proposition 3.1 and Sec. VI. See also our sequel.^{46}

More geometrically, global Sturm attractors $ A f$ and $ A g$ of dissipative nonlinearities $f,g$ with the same Sturm permutation $ \sigma f= \sigma g$ are $ C 0$ orbit-equivalent.^{36} Only locally, i.e., for $ C 1$-close nonlinearities $f$ and $g$, this global rigidity result is based on the Morse–Smale transversality property mentioned above. See, for example,^{86–88} for such local aspects. Section III discusses some “trivial equivalences” between Sturm attractors $ A f$ and $ A g$ with different Sturm permutations $ \sigma f\u2260 \sigma g$.

^{101},

*adjacent*, if there does not exist any

*blocking equilibrium*$w$ strictly between $ v 1$ and $ v 2$, at $x=0$ (or, equivalently, at $x=1$) such that

Clearly, any heteroclinic orbit $u(t,.): v 1\u21dd v 2$ implies adjacency: by (2.2), any blocking equilibrium $w$ would force $z(u(t,.)\u2212w)$ to drop strictly at the Neumann boundary $x=0$, for some $t= t 0$. This contradicts the equal values of $z$ at the limiting equilibria $ v 1, v 2$ of $u$, for $t\u2192\xb1\u221e$.

As a trivial corollary, for example, we conclude $ v 1\u21dd v 2$, for neighbors $ v 1, v 2$ on any boundary order $ h \iota $. Here, we label $ v 1, v 2$ such that $i( v 1)=i( v 2)+1$; see (2.6). For an in-depth analysis and many more examples, see Ref. 94.

We encode the above heteroclinic structure in the directed *connection graph* $ C$. See Fig. 1(d) for an example. The connection graph is graded by the Morse index $i$ of its equilibrium vertices. Directed edges are the heteroclinic orbits $ v 1\u21dd v 2$ running downwards between equilibria of adjacent Morse index. Uniqueness of such heteroclinic orbits, given $ v 1, v 2$, had already been observed in Lemma 3.5 of Ref. 13; see also Ref. 56.

Directed paths in the connection graph in fact encode all heteroclinic orbits. Indeed, the heteroclinic relation $\u21dd$ on $ E$ is transitive, by Morse–Smale transversality and the $\lambda $-Lemma.^{87} Therefore, any directed path from $ v 1$ to $ v 2$ also defines a direct heteroclinic orbit $ v 1\u21dd v 2$. Given $ v 1\u21dd v 2$, conversely, the cascading principle first described in Ref. 13 asserts an interpolating sequence of heteroclinic orbits between equilibria of adjacent Morse indices, from $ v 1$ to $ v 2$.

The *basin of attraction* of an $i=0$ sink vertex $v$ in $ A$, for example, consists of $v$ itself, and all heteroclinic orbits $ v 1\u21ddv$. The *basin boundary* consists of just those other equilibria $ v 1$, and all heteroclinic orbits among them. The connection graph $ C$ readily identifies all those equilibria. See our discussion in reversibility Subsection VIII D for nontrivial geometric examples (8.7) and (8.8) based on the connection graph of Fig. 8(c).

*Thom–Smale complex*or

*dynamic complex*; see, for example, Refs. 54, 10, and 9. In our Sturm setting (1.2) with hyperbolic equilibria $v\u2208 E$, the Thom–Smale complex is a finite regular cell complex, in the terminology of algebraic topology: the boundaries $ clos W u\u2216 W u$ of the open $i(v)$-cells $ W u(v)$ are homeomorphic to spheres of dimension $i(v)\u22121$. The proof follows from the Schoenflies property.

^{40,41}We, therefore, call the regular cell decompositions (2.10) of the Sturm global attractor $ A$ the

*Sturm complex*$ S$.

We call $d=dim\u2061 A:= max v \u2208 Ei(v)$ the *dimension* of $ A$, or of the complex $ S$. Then, at least one equilibrium $ O$ has maximal Morse index $i( O)=d$, i.e., $i(v)\u2264d$ for all other Morse indices. If $ A= clos W u( O)$ is the closure of a single $d$-cell, then the Sturm complex turns out to be a closed $d$-ball.^{41} We call this case a *Sturm $d$-ball*.

A three-dimensional Sturm complex $ C$, for example, is the regular Thom–Smale complex of a three-dimensional Sturm global attractor $ A$. See Fig. 1(c) for the Sturm complex $ S$ of the Sturm three-ball $ A$ associated with the meander in Fig. 1(a).

In the Sturm-ball trilogy,^{42–44} we have characterized all Sturm 3-balls $ S$. Earlier, the trilogy^{37–39} had characterized all planar Sturm complexes $ S$, i.e., the case $dim\u2061 A=2$. The case $dim\u2061 A=1$, i.e., $\sigma = id N$ with odd $N\u22653$, is a trivial line with alternating $i=0$ sinks and $i=1$ saddles. Global asymptotic stability of a unique sink equilibrium is the case $N=1$ of $dim\u2061 A=0$.

Conversely, we have described in Refs. 45 and 94 how the boundary orders $ h \iota $ of (2.4), and therefore the Sturm permutation $\sigma $ of (2.5), are determined uniquely by the *signed hemisphere decomposition*. This is a slight refinement of the Sturm complex $ S$, which we do not pursue in further detail here. In Fig. 1, for example, the signed hemisphere complex (c) determines how the boundary orders $ h 0$ [red in (a)] and $ h 1$ (blue) traverse the equilibrium vertices, from the North pole $ A 0$ to the South pole $ D 0$. The predecessors and successors, on $ h \iota $, of the repelling sphere barycenter $ O$ are marked by small annotated red and blue circles, everywhere in Fig. 1.

*noses*[called “pimples,” in Ref. 20; see also the (2,1)-lieanders in Ref. 19]. Here, noses are subscripts $j\u2208{1,\u2026,N\u22121}$ such that

The simplest case, of just two noses, is called the *Chafee–Infante attractor*. This has been well-studied, ever since it first arose for cubic nonlinearities $f= \lambda 2u(1\u2212 u 2)$ in Ref. 15. As a warm-up on terminology, and as a simple illustration of our approach, we review this case in Sec. IV. For a three-nose meander, see Fig. 1.

Section V then presents our main results on the general case of primitive three-nose meanders $ M p q$ with two nose arcs above the horizontal axis, each as the innermost of $p$ and $q$ nested upper arcs, respectively. Below the horizontal axis, the only remaining nose is centered as the innermost of the complementing $p+q$ lower arcs. Since all lower arcs are nested, we also call that configuration a (lower) *rainbow*. It turns out that the resulting curves are meanders if, and only if, $p\u22121$ and $q+1$ are co-prime, i.e., they do not share any nontrivial integer factor. See Theorems 5.1 and 5.2, where it is also established that the dissipative meander $ M p q$ is Sturm if, and only if, $p=r(q+1)$, for some $r,q\u22651$. Let $ \sigma r q$ denote the associated Sturm permutations. The resulting global attractors $ A r q$ are all distinct—except for the not immediately obvious “trivial” linear flow equivalence upon interchange of $r$ and $q$; see Corollary 5.3. In Theorem 5.4, the Sturm complex $ S r q$ turns out to be a Sturm ball of dimension $r+q$. The three-ball attractor of Fig. 1, for example, is trivially equivalent to the simple case $r=2,q=1$, in the sense of Sec. III.

*the*paradigm of irreversibility.

*Time reversibility*in its strongest form means the existence of an involutive

*reversor*$ R:\Sigma \u2192\Sigma $ which reverses the time direction of PDE orbits of (1.2), on a “large” invariant subset $\Sigma \u2282 A$. In particular, with any two equilibria $ v 1, v 2\u2208\Sigma $ such that $ v 1\u21dd v 2$ in $ A$, the subset $\Sigma $ should also contain some of those heteroclinic orbits. Restricted to equilibria $ v 1, v 2\u2208 E\u2229\Sigma $, strong reversibility implies the weaker statement

We prove Theorem 5.1 in Sec. VI. To circumvent tiresome mathematical pedantry, we only provide proofs for the simplest interesting case $r=1,p=q+1$ of our remaining results, in Sec. VII. This includes the explicit connection graphs $ C 1 q$ for $q\u22652$; see Theorem 7.6.

Section VIII touches the general case $ \sigma r q$, which will be addressed in our sequel.^{46} We also discuss some non-dissipative PDE aspects, and a spatially discrete ODE variant of (1.2). We conclude with more geometric ODE models of the connection graphs $ C 1 q$ and their time reversibility.

## III. ROTATIONS, INVERSES, AND SUSPENSIONS

To reduce the sheer number of cases, a proper consideration of symmetries is mandatory. In this section, we recall the notion of trivial equivalence for Sturm attractors $ A$, meanders $ M$, permutations $\sigma $, and connection graphs $ C$, as introduced in Ref. 34; see also Sec. 3 in Ref. 44. As a prelude to induction over the number of arcs in three-nose meanders, we also discuss double cone suspensions $ A ~, M ~, \sigma ~, C ~$ of the entourage $ A, M,\sigma , C$. See also previous accounts in Refs. 36, 67, and 94.

*Trivial equivalences*are generated as the Klein four-group $\u27e8\kappa ,\rho \u27e9$ with commuting involutive generators

A small subtlety arises, concerning isotropy $ f \gamma =f$ of nonlinearities under some trivial equivalence $\gamma \u2208\u27e8\kappa ,\rho \u27e9$. Such $f$-isotropy implies permutation-isotropy $ \sigma \gamma =\sigma $, of course. However, we never proved the converse. Although some nonlinearities $f$ will always realize isotropic permutations $\sigma = \sigma \gamma $, by Ref. 35, we never proved realization by an $f$ with isotropy $\gamma $, i.e., such that $f= f \gamma $.

We define the *suspension* $ M ~$ as an augmentation of $ M$ by two overarching arcs [black in Fig. 3(d)]: an upper arc from the first new vertex $ E ~ 0$ to the last old vertex $ E ~ N= E N$, and a lower arc from the first old vertex $ E ~ 1= E 1$ to the last new vertex $ E ~ N + 1$. This extends the previous definition of $ h ~ 1$ to $ h ~ 1(j):= E ~ j \u2212 1$ for $j=1,\u2026,N+2$.

By construction, the number of meander-noses is invariant under suspension, for $N\u22653$. In the Sturm case, i.e., if our dissipative meanders $ M ~$ are also Morse, our definition also extends to define the suspensions $ A ~$ and $ C ~$ of their attractors $ A$ and connection graphs $ C$.

More abstractly, however, our definition of suspension generalizes to dissipative meanders $ M$, which are not necessarily Sturm. Indeed, they may violate the Morse property $ i j\u22650$ and hence may also violate $ z j k\u22650$. Abstractly, however, dissipative meanders $ M$ still determine their permutations $\sigma $, Morse numbers $ i j$, and zero numbers $ z j k$ via (2.6) and (2.7)—even when those numbers lack any ODE or PDE interpretation. Sturm “attractors” $ A$ with actual “equilibria” and actual “heteroclinic orbits” $ v 1\u21dd v 2$ cannot exist, of course, once negative “Morse indices” $ i j$ are involved. By $z$-adjacency (2.8), and blocking (2.9), however, we can still define connection graphs $ C$. Quite radically, indeed, we abuse the notation $i( E j),z( E j\u2212 E k)$ here, and even $ E j\u21dd E k$, to denote the recursively defined quantities $ i j, z j k$, and the relation $\u21dd$ defined abstractly via (2.8) and (2.9). In particular, our definition of meander suspensions readily extends to define the suspensions $ M ~, \sigma ~$, and $ C ~$, even in non-Morse cases. Of course, these remarks also extend the notions of trivial equivalences to merely dissipative non-Morse meanders, algebraically, by (3.4), (3.6), and (3.8) instead of the explicit maps (3.1) and (3.2).

The following proposition justifies the name “suspension.” Indeed, we may view the suspension $ A ~$ of a global Sturm attractor $ A$ as the double cone suspension of $ A$ itself, with respect to the two added polar cone vertices $ E ~ 0$ and $ E ~ N + 1$. See Fig. 3.

For dissipative, but not necessarily Morse, meanders $ M$, the suspension defined above has the following properties, for all $1\u2264j,k\u2264N,j\u2260k:$

$ \sigma ~(1)=1$ and $ \sigma ~(N+2)=N+2,$

$ \sigma ~(j+1)=N+2\u2212\sigma (j)=\kappa \sigma (j)+1,$

$i( E ~ 0)=i( E ~ N + 1)=0$,

$i( E ~ j)=i( E j)+1,$

$z( E ~ j\u2212 E ~ 0)=z( E ~ j\u2212 E ~ N + 1)=0,$

$z( E ~ j\u2212 E ~ k)=z( E j\u2212 E k)+1,$

$ E ~ j\u21dd E ~ k\u27fa E j\u21dd E k,$

$ E ~ j\u21dd E ~ 0, E ~ N + 1$, in case all $ i j\u22650$.

Consider suspensions $ h ~ \iota :{1,\u2026,N+2}\u2192 E ~$ and $ \sigma ~\u2208 S N + 2$ of abstract “boundary orders” $ h \iota :{1,\u2026,N}\u2192 E$ which fix $1$ as well as $N$. Define the dissipative meander permutation $ \sigma ~= h ~ 0 \u2212 1 h ~ 1\u2208 S N$, as in (2.5).

Claim (i) then holds by construction. To prove claim (ii), first note that $ h ~ 1(j+1)= E ~ j= E j= h 1(j)$. Since the orders $ h ~ 0$ and $ h 0$ follow the shared part of the meanders $ M ~$ and $ M$, in opposite directions, we also have $k+ k ~=N+2$ for $ h 0(k):= E j$ and $ h ~ 0( k ~):= E ~ j$. Together, this proves (ii), if we substitute the flip $\kappa $ from (3.4). Properties (iii)–(vi) can be derived from the explicit recursions (2.6) and (2.7). In particular, (iv) enters in (vi) via the term $ z k k$ which gets raised by 1 after suspension.

Property (vii) follows from Wolfrum blocking (2.8) and (2.9). Indeed, (vi) implies that blocking (2.9) between lifted old vertices $ v 1, v 2\u2208 E$ by any new vertex $ w ~\u2208{ E ~ 0, E ~ N + 1}$ cannot occur, because the $ h ~ \iota $-position of those new vertices is extremal and never between $ v 1, v 2$. By (vi), in contrast, any old blocking remains in effect. This proves claim (vii).

In claim (viii), we assume $ M$ to be Morse, and hence Sturm. In particular, this implies $ z j k\u22650$, for all zero numbers. Therefore, (v),(vi) prevent blocking (2.9), and (viii) follows from (iii),(iv) with (2.8).

For Sturm meanders $ M$, the following holds true.

The suspension $ \sigma ~\u2208 S N + 2$ of any Sturm permutation $\sigma \u2208 S N$ is Sturm.

All $i=1$ equilibria connect heteroclinically, in $ C ~$, toward the two polar $i=0$ sinks $ E ~ 0, E ~ N + 1$ in the bottom row.

The connection graph $ C ~$ of the suspension contains the connection graph $ C$, lifted to the rows $i\u22651$.

Claim (i) follows from Proposition 3.1 (iii), (iv). With (viii), this also proves claim (ii). Claim (iii) then follows from (vii).

In the Sturm case, the realization of suspensions by nonlinearities $f(x,u,p)$ may be of applied interest in design. For example, we may append a region $x\u2208[1,2]$ to the $x$-domain of (1.1) and (1.2). Then, suspension can be effected, in terms of $x$-profiles of equilibria like Fig. 1(b), if $f$ reverses the order of equilibria at the right boundary, as $x$ increases from $x=1$ to $x=2$. This agrees well with Proposition 3.1(vi). Dissipativeness, of course, will require two new equilibria, e.g., homogeneous throughout $0\u2264x\u22642$: one at the top and one at the bottom.

## IV. TWO NOSES: THE CHAFEE–INFANTE PARADIGM

^{15}Via the abstract two-nose Sturm meander $ M d$, we will see how our abstract global attractor $ A d$ is actually orbit equivalent to the Chafee–Infante attractor of that explicit original example.

To pursue this program, let us start from just $d$ upper arcs, separately and without meanders in mind as yet. Equivalently, the arcs define a balanced structure of $d$ pairs of opening and closing parentheses, “ $($” and “ $)$,” also know as *Dyck words* of length $2d$, as counted by the Catalan numbers. For a historical reference, see the habilitation thesis by Dyck on the word problem in combinatorial group theory.^{21} Upper noses correspond to innermost pairs “ $()$.” Any nonempty Dyck word has to contain at least one nose. If the Dyck word only contains a single nose, then all parenthesis pairs, alias arcs, must be nested. In Sec. II, we already called such a total nesting a *rainbow*. Proceeding for lower arcs, analogously, we obtain another rainbow of $d$ nested lower arcs. Dissipativeness requires the lower rainbow to be shifted one entry to the right, with respect to the upper rainbow. See Fig. 4(a). Joining the two rainbows defines a unique double spiral which, automatically, turns out to be a dissipative meander $ M d$, for any $d\u22651$. By construction, $ M d$ possesses $d$ upper and $d$ lower arcs, each, over its $N=2d+1$ intersections with the horizontal axis. Alas, we do not know yet whether $ M d$ is Morse, and therefore Sturm.

*Chafee–Infante meander of dimension*$d$.

We have already mentioned that the 1974 Chafee–Infante version $ A d$ of (1.2) had been studied for the cubic nonlinearity $f(u)= \lambda 2u(1\u2212 u 2)$, originally, albeit under Dirichlet boundary conditions; see (4.1) and the original paper.^{15} Their method was local bifurcation analysis of the trivial equilibrium $ O:v\u22610$. Note $i( O)=d$, for $(d\u22121)\pi <\lambda <d\pi $, under Neumann boundary conditions, by elementary linearization. The second order ODE (1.1) is Hamiltonian integrable, for nonlinearities $f=f(u)$. For the hard spring cubic nonlinearities $f$, the minimal periods $p(a)$ of $v(x)$ at $\lambda =1$ grow monotonically with their amplitude $a=v(0)\u2208(0,1)$ at $ v x(0)=0$. Note the limit $p(0)=2\pi $. Rescaling $x$ as in $v(\lambda x)$, we see that $v$ reappears as a rescaled solution at $\lambda =np(a)/2$, for any nonzero integer $n$. See Fig. 4(b) for such rescaled equilibrium profiles in case $d=3$. In particular, this produces a (stylized) shooting meander which, by monotonicity of the periods, coincides with the Sturm meander $ M d$, and hence determines the Sturm permutation $ \sigma d$ of (4.9).

For an early geometric description of the Chafee–Infante attractor $ A d$, for low dimensions $d=1,2,3$, see Section 5.3 of Ref. 64. In 1985, Henry achieved the first description of $ A d$ for general $d$.^{65} His description was based on a nodal property akin to (2.2), and on a careful geometric analysis of unstable and center manifolds at the sequence of pitchfork bifurcations from $ O:v\u22610$, at $\lambda =n\pi $. See Fig. 4(c).

In Sec. 5 of Ref. 45, we have discussed the Sturm complex of the Chafee–Infante attractors in the more refined setting of signed hemisphere decompositions, which also leads to Fig. 4(c). This also provides extremal characterizations of the Chafee–Infante attractor $ A d$, among all Sturm attractors:

*Among all Sturm attractors with $N=2d+1\u22653$ equilibria, $ A d$ is the unique Sturm attractor with the maximal possible dimension, $d$.**Among all Sturm attractors of dimension $d\u22651$, $ A d$ is the unique Sturm attractor with the smallest possible number of equilibria, $N=2d+1$.*

The two claims follow, e.g., from the connection graph. In fact, each unstable hyperbolic equilibrium $v$ must connect, heteroclinically, to at least two other equilibria $ v \xb1$, such that $z( v \xb1\u2212v)=i(v)\u22121$ and $ v \u2212<v< v +$ at $x=0$. See also Ref. 23.

Topological Conley index and the connection matrix have been employed by Mischaikow,^{83} to establish heteroclinic orbits in larger classes of gradient-like PDEs with equilibrium configurations of Chafee–Infante type. This technique establishes the existence of some (possibly non-unique) heteroclinic orbit $u(t,x)$ between the sets ${ A j, B j}$ and ${ A j \u2212 1, B j \u2212 1}$. Acting on $u(t,x)$ with the Klein four-group of symmetries $\u27e8\kappa ,\rho \u27e9$, generated by (3.1) and (3.2) in the Sturm setting, we obtain the four required heteroclinic orbits (4.6). Indeed $\kappa $, alias $\u2212 id$, interchanges each $ A j$ with $ B j$; see (3.1) and Fig. 4(b). Inversion $\rho $, in contrast, performs the same interchange for odd $j$, only; see (3.2). Since the Morse levels $j$ and $j\u22121$ are of opposite even/odd parity, this generates the four required heteroclinic orbits. The argument for the heteroclinic orbits emanating from the equilibrium $ O$, of top Morse index, is analogous.

From an applied point of view,^{83} greatly extends the Chafee–Infante paradigm beyond the requirement of Sturm zero numbers—as long as a variational structure remains intact, with the same (minimal) configuration of equilibria, symmetries, and Morse indices. This includes damped wave equations and other applications. See also Ref. 63.

Later work in the Sturm context addressed general autonomous nonlinearities $f(u)$; see for example, Refs. 12, 13, and 48. The paradigm of pitchfork bifurcations has been beautifully extended by Karnauhova, with many pictures, in Ref. 67. With the pitchforkable class essentially well-understood since,^{65} however, the simplest non-pitchforkable example had been discovered in Ref. 93. Since none of our three-nose meanders of dimension three or higher will fall into the pitchforkable class, either, we have to take another approach instead. We will progress further along the more promising abstract path which, as a warm-up, we have just sketched for the Chafee–Infante problem.

## V. THREE NOSES: MAIN RESULTS

In this section, we present our main results on meanders with three noses. The Chafee–Infante case of only two noses, discussed in the previous Sec. IV, will serve as a paradigm not to be skipped. The general case of Sturm meanders $ M p q$ reduces to the sequences $p=r(q+1),r,q\u22651$; see Theorems 5.1 and 5.2. As usual, these come with their entourage of Sturm permutations $ \sigma r q$, associated Sturm attractors $ A r q$, and connection graphs $ C r q$ (Definition 5.1). In Theorem 5.2, we determine the Morse polynomial, i.e., we count the number of equilibria for each Morse index. The Morse polynomials of $ A r q$ and $ A q r$ coincide; see Corollary 5.2. In fact, the Sturm attractors $ A r q$ turn out to be trivially equivalent to $ A q r$, by Theorem 5.3 and Corollary 5.3. Geometrically, these are Sturm balls $ A r q= clos W u( O)$ of dimension $i( O)=r+q$ (Theorem 5.4). Finally, Theorem 5.5 asserts that the connection graph remains time reversible on the invariant boundary sphere $\u2202 A r q= \Sigma r + q \u2212 1$.

To not clutter our conceptual approach by baroque notation, we will refrain from proving our results in full generality. Instead, we only address the simplest interesting case $r=1$, i.e., $ A 1 q$ and $ A q 1$, in Sec. VII. For $r>1$, see our sequel.^{46}

With the above notation, the following holds true for $r,p,q\u22651$.

$ M p q$ is a dissipative meander if, and only if, $(p\u22121,q+1)$ are co-prime and $p\u22652$.

For $p\u2260r(q+1)$, any dissipative meander $ M p q$ fails to be Morse.

We will prove Theorem 5.1 in Sec. VI. Note that the non-Morse three-nose cases (i), with $p\u2260r(q+1)$ are not a lost cause, from the Sturm PDE point of view (1.2). Indeed, suspension Proposition 3.1(iv) always provides a minimal number $s=\u2212mini( E j)\u22650$ of suspensions after which $ M p q$ becomes Morse, and hence Sturm. See Fig. 5 for the non-Morse three-nose example $p=q=2\u2260r(q+1)$. We will pursue those cases further in our sequel.^{46}

Let us now focus on the three-nose cases $p=r(q+1)$, which are complementary to Theorem 5.1(ii). Then, $p\u22121,q+1$ are automatically co-prime, because $r(q+1)\u2212(p\u22121)=1$. The following theorem shows that all cases $p=r(q+1)$ do lead to Morse meanders $ M p q$ and, therefore, to Sturm attractors. The rotation $\sigma :=\kappa \sigma 11\kappa $ of the simplest case $r=q=1$ has already served in Figs. 3(a)–3(c), to illustrate suspension. We, therefore, assume $r,q\u22651,rq>1$, for the rest of this paper. Proofs of the next four theorems will be given in Sec. VII, for the simplest interesting case $r=1$, only. See Ref. 46 for general $r\u22651$.

The Morse count functions $i\u21a6 m r q(i)$ have the following symmetry properties.

Up to ordering, the subscript set ${r,q}$ is determined by $ m r q$.

Conversely, the subscript set determines $ m r q= m q r$.

For all $0\u2264i<r+q$, we have $ m r q(i)= m r q(r+q\u22121\u2212i)$.

To prove (i), just note ${r,q}={min{r,q},max{r,q}}$. Claim (ii) follows from $r+q=min{r,q}+max{r,q}$. To prove (iii), insert (5.2).

For any $r,q\u22651$, we call $ M r ( q + 1 ) , q$ a *primitive* three*-nose meander.* For the Sturm entourage of $ M r ( q + 1 ) , q$, we denote the associated *primitive* Sturm permutation as $ \sigma r q$, the *primitive* Sturm attractor as $ A r q$, and the *primitive* connection graph as $ C r q$.

Suppose $ A r q$ and $ A r \u2032 q \u2032$ are orbit equivalent. Then, their Morse counts coincide, and the first claim follows from Corollary 5.2.

Conversely, suppose their subscript sets coincide, but with reversed order. Then, the trivial equivalence of the attractors follows from Theorem 5.3 and Sec. III.

In the “symmetric” case $r=q$, Theorem 5.3 reveals the only nontrivial isotropy $\kappa \rho $, in the Klein four-group of trivial linear equivalences. In particular, the rainbow argument above shows that $f$ still cannot be $\kappa $-isotropic. Admittedly, (5.3) suggests to study $f$ which commute with $\kappa \rho $, i.e., $f(x,u,p)=\u2212f(1\u2212x,\u2212u,p)$. However, $f=f(u)$ is still excluded, because $f(x,u,p)$ and $f(1\u2212x,u,\u2212p)$ must remain distinct.

Note the Morse count $ m r q(r+q)=1$ at maximal $i=r+q$; see (5.2). Let $ O$ denote that unique equilibrium in $ A r q$ of maximal Morse index $i( O)=r+q=dim\u2061 A r q$.

The primitive Sturm attractor $ A r q$ is the closure of the unstable manifold of the single equilibrium $ O\u2208 A r q$. i.e., $ A r q$ is a Sturm ball of dimension $r+q$.

^{59}For odd $d$, this useful test of (5.2) is less trivial to check. Taken $ mod2$, of course, it again implies that the total number $N$ of equilibria must be odd.

With dimension replaced by $r+q+s,s>0$, Theorem 5.4 remains valid for any $s$-fold suspension of $ A r q$.

By Corollary 3.1, suspensions of Sturm balls are Sturm balls.

More surprisingly than in the Chafee–Infante case, we still observe time reversibility on the sphere boundary of the primitive three-nose Sturm global attractors $ A r q$—in spite of the parabolic, diffusion-dominated nature of the underlying original PDE (1.2).

The connection graph $ C r q$ is reversible on the flow-invariant boundary sphere $ \Sigma r + q \u2212 1=\u2202 A r q=\u2202 W u( O)$ of the primitive Sturm ball $ A r q= clos W u( O)$.

The reversibility on the boundary sphere $ \Sigma r + q \u2212 1$, of course, is a much deeper reason for the symmetry of the Morse count function $i\u21a6 m r q(i)$, for $0\u2264i<r+q$, which we have already noticed in Corollary 5.2(iii). Indeed, the reversor $ R$ on $ C r q\u2216 O$ swaps equilibria of Morse indices $i$ and $r+q\u22121\u2212i$.

## VI. NON-MORSE MEANDERS WITH THREE NOSES

In this section, we prove Theorem 5.1.

Claim (i) states that the dissipative arc configuration of $p$ nested upper arcs followed by $q$ nested upper arcs, and a right shifted lower $(p+q)$-rainbow, is a meander if, and only if, $p\u22652$ and $(p\u22121,q+1)$ are co-prime.

The case $p=1$ is trivially discarded: all upper arcs of the nonempty $q$-nest close up to become circles, with the corresponding inner arcs of the lower rainbow. This contradicts the meander property.

For $p\u22652$, let us remove the outermost arc of the upper $p$-nest and, instead, stack it onto the upper $q$-nest. The resulting closed arc configuration now features upper nests of $p\u22121$ and $q+1$ arcs over the same lower rainbow. This closing construction has been described and studied in Ref. 28, in terms of certain Cartesian billiards. See also Refs. 19, 20, and 67, and the many references there. The closing provides a closed Jordan curve if, and only if, the original dissipative arc configuration is a meander. In other words, we obtain *closed meanders* from dissipative meanders, and vice versa.

Let us now return to the dissipative arc configuration of $ M p q$ with $p\u22652$. By (6.1) of Ref. 28, the greatest common divisor of $p\u22121$ and $q+1$ counts the connected components of the resulting closed arc configuration. The proof was recursive, via the Euclidean algorithm for $(p\u22121,q+1)$. This proves claim (i).

It remains to show, (ii), that the dissipative meander $ M p q$ fails to be Morse, if $2\u2264p\u2260r(q+1)$ for any integer $r\u22651$.

We first consider the case $2\u2264p<q+1$. We label equilibria such that $ h 1= id$. Then, $A:=2p+1$ and $B:=2p+2q$ are the left and right end points of the uppermost arc in the upper $q$-nest. By (2.6), Morse numbers of $ h \iota $-adjacent vertices are adjacent. Obviously, $B$ is $ h 1$ adjacent to $N=2p+2q+1$. By dissipativeness, $i(N)=0$. Adjacency implies $i(B)=\xb11$. In case $i(B)=\u22121$, we are done.

In case $i(B)=+1$, we obtain $i(A)=0$ because the meander arc $AB$ turns left from $B$ to $A$; see (2.6). Now consider the preceding lower rainbow arc from $ B \u2032$ to $ A \u2032:=A=2p+1$. Since $a+b=2p+2q+3$ for the two end points $a,b$ of any lower rainbow arc, our assumption $p<q+1$ implies $ B \u2032> A \u2032$: the lower arc $ A \u2032 B \u2032$ turns left, from $ A \u2032$ to $ B \u2032$. But we already know $i( A \u2032)=i(A)=0$. Therefore, (2.6) implies a negative Morse index $i( B \u2032)=\u22121$, and we are done again.

## VII. THE SIMPLEST INTERESTING CASE

In this section, we address the remaining four Theorems 5.2, 5.3, 5.4, and 5.5, of Sec. V, on the primitive three-nose Sturm attractors $ A r q$, their dissipative Morse meanders $ M r ( q + 1 ) , q$, and their entourage of Sturm permutations $ \sigma r q$ and connection graphs $ C r q$. For brevity and simplicity, we restrict our proofs to the simplest interesting case $r=1$. We skip the trivial case $r=q=1$, already treated in Figs. 3(a)–3(c). In Sec. VII A, we use conspicuous nose locations to identify the action of trivial equivalences among these objects. In particular, we prove the trivial equivalence of $ A q 1$ and $ A 1 q$ claimed in Theorem 5.3, for $r=1$. Theorem 7.6 in Sec. VII B identifies the connection graphs. This will easily prove the remaining three theorems, in Secs. VII C–VII E. As an afterthought, we conclude with explicit expressions for the Sturm permutations $ \sigma 1 q$ and their trivially equivalent relatives, in Proposition 7.2 of Sec. VII F.

### A. Proof of Theorem 5.3

To locate noses $AB$ of equilibria, we use the matrix notation $( a 1, b 1 | a 0, b 0)$ for locations $ a \iota := h \iota \u2212 1(A), b \iota := h \iota \u2212 1(B)$ and $\iota =0,1$. Note how noses are characterized by adjacency $ | a \iota \u2212 b \iota |=1$ under both boundary orders $ h \iota $.

The following are corresponding nose locations of the indicated Sturm permutations, for any fixed $q\u22652:$

the upper right nose $(4q+1,4q+2 |2q+1,2q+2)$ of $ \sigma q 1,$

the lower left nose $(2,3 |2q+2,2q+3)$ of $ \sigma q 1 \kappa ,$

the nose $(2q+1,2q+2 |4q+1,4q+2)$ of the upper rainbow of $ \sigma 1 q \kappa ,$

the nose $(2q+2,2q+3 |2,3)$ of the lower rainbow of $ \sigma 1 q$.

*Figs. 2(a)–2(d)*for the special case $q=2$. In particular, the four permutations are trivially equivalent and (5.3) holds, for $r=1$.

The lower rainbow nose (d) of $ \sigma 1 q$, i.e., for $p=q+1$, is obviously located at $(2p,2p+1 |2,3)=(2q+2,2q+3 |2,3)$, by arc counting. Similarly, the upper rainbow nose (c) for the rotated meander associated with $ \sigma 1 q \kappa $ is just as obviously located at the rotated position $(2p\u22121,2p |2p+2q\u22121,2p+2q)=(2q+1,2q+2 |4q+1,4q+2)$.

Inversion $\rho $ of $\sigma $ interchanges the roles of $ h 0$ and $ h 1$. This swaps the entries of the nose matrix before and after the separator “ $ |$.” Therefore, the noses corresponding to the rainbow noses in (c) and (d) become $(4q+1,4q+2 |2q+1,2q+2)$ and $(2,3 |2q+2,2q+3)$ in (a) and (b), respectively. The first $ h 1$-entries locate these noses at the extreme right and left of the horizontal $ h 1$ axis, respectively.

It remains to show that the permutation $ \sigma q 1$ in (a) is indeed the inverse of the Sturm permutation $ \sigma 1 q \kappa $ in (c). [The other pair (b), (d) is treated analogously.] From Sec. III, we already know that inversion $\rho $ preserves the number of noses and, up to $\kappa $, commutes with suspension; see (3.11). Therefore, the inverse $ \sigma 1 q \rho \kappa $ of $ \sigma 1 q \kappa $ must also be a primitive three-nose Sturm permutation $ \sigma r \u2032 q \u2032$. The upper nose in (a) is located rightmost, at $ h 1\u2208{4q+1,4q+2}$, and hence cannot sit inside any larger nest. Therefore, $ \sigma 1 q \kappa \rho = \sigma r \u2032 1$ for some $ r \u2032$. This implies $ r \u2032=q$, since the total number $2(p+q)+1=4q+3=4 r \u2032+3$ of vertices is also preserved under inversion $\rho $. This proves the lemma, (5.3), and Theorem 5.3.

### B. The connection graphs $ C 1 q$ and $ C q 1$

With the case $q=2$ already settled, we proceed by induction on $2\u2264q\u22121\u21a6q$. We may, therefore, assume that the theorem already holds true for the ( $q$-1) meander $ M \u2032$ and its connection graph $ C \u2032$, as illustrated in Figs. 7 and 8(a). Starting from $q\u22121$, our first step is by suspension to $ M \u2032 ~, C \u2032 ~$ as in Figs. 7 and 8(b). Our second step, leading to the $q$-meander $ M$ and its connection graph $ C$, is by nose insertion; see Figs. 7 and 8(c).

Suspension, our first step, invokes Proposition 3.1. The equilibria $ E ~ j \u2032$ of the suspension have been labeled $ L ~ j \u2032$, to correspond to our notation $ L j \u2032$ for $ E j \u2032$. Suspension raises Morse indices by 1, due to Proposition 3.1(iv). Only for the cone vertices $ E ~ 0 \u2032$ and $ E ~ N + 1 \u2032$ of the suspension, at the lowest Morse level $i=0$, we have substituted the new labels $ A 0, D 0$ in Figs. 7 and 8(b). The connection graph $ C \u2032 ~$ of (b) then follows from the suspension Corollary 3.1.

Our second step is the nose insertion of Figs. 7 and 8(c). First note our substitution $ L ~ j \u2032\u21a6 L j + 1$, for equilibria inherited by (c) from (b). This ensures $i( L j)=j$, for $j\u22651$, as claimed in (7.10). The cone vertices $ A 0, D 0$ have not been relabeled. However, we now have to address three possible effects of the newly inserted nose arc $ B 0 C 1$ on heteroclinic edges (purple) in Fig. 8,

previous edges of (b) blocked by nose equilibria $ B 0, C 1$,

new edges in (c) emanating from the nose $ B 0, C 1$,

new edges in (c) terminating at the nose $ B 0, C 1$.

We start with blocking of type (i). By (2.8) and (2.9), new blockings of $ v 1\u21dd v 2$, i.e., purple edges in (b), only arise through nose equilibria $w\u2208{ B 0, C 1}$ which are located between other $ v 1$ and $ v 2$ along the meander order $ h 0$ of $ M$, and which satisfy (2.9). Since $ B 0 C 1$ is a nose arc, blocking by $ C 1= h 0(4q+2)$ is equivalent to blocking by $ B 0= h 0(4q+1)$. Except for the last equilibrium $ D 0= h 0(4q+3)$, all equilibria $v= h 0(j)$ inherited by (c) from (b) have $ h 0$-position $j$ less than the second to last $ h 0$-position $4q+2$ of $ C 1$. Therefore, $ C 1$ (or $ B 0$, equivalently) cannot block any of the heteroclinic edges inherited from (b), by (c), except possibly for edges from ${ A 1, B 1, D 1}$ to $ D 0$. The edge between $ D ~ 0 \u2032= D 1$ and the sink $ A 0$, for example, cannot be blocked, because they are $ h 0$-neighbors on the suspension arc $ A 0 D 1$. Similarly, the edge between the $ h 1$-neighbors $ D ~ 0 \u2032= D 1$ and the sink $ D 0$ remains non-blocked. However, $z( A 1\u2212 B 0)=z( B 0\u2212 D 0)=z( A 1\u2212 D 0)=0$ implies that $w= B 0$ blocks $ A ~ 0 \u2032= A 1=: v 1\u21dd v 2:= D 0$. Here and below, we refer to (2.7) along the orders of $ h 0$ or $ h 1$, equivalently, for the calculation of zero numbers. Similarly, $ B ~ 0 \u2032= B 1=: v 1\u21dd v 2:= D 0$ is blocked by $ B 0$ at $z=0$. This settles the effects of blocking, (i).

Next, we address new heteroclinic edges (ii) emanating from the nose. Obviously, edges cannot emanate from the sink $i( B 0)=0$. Just as obviously, $ C 1$ connects heteroclinically to its nose neighbor $ B 0$, and to its $ h 0$-neighbor $ D 0$. However, $z( C 1\u2212 B 0)=z( B 0\u2212 A 0)=z( C 1\u2212 A 0)=0$ implies that $w= B 0$ blocks $ C 1=: v 1\u21dd v 2:= A 0$. This identifies all edges emanating from the nose, (ii).

It only remains to address new heteroclinic edges (iii) terminating at the nose. Consider the target $ B 0$, first. Obviously, there are heteroclinic edges toward the sink $i( B 0)=0$ from its $ h \iota $-neighbors $ A 1, B 1, C 1$, all at Morse level $i=1$. The hypothetical edge $ D 1\u21dd B 0$ is blocked by $ C 2$, at $z=1$. This settles the three edges toward target $ B 0$.

Finally, consider the target $ C 1$ of (iii). We proceed by checking the potential sources $ A 2, B 2, C 2$, $ D 2$, in alphabetical order. The hypothetical edge $ A 2\u21dd C 1$ is blocked by $ B 0$, at $z=0$. Indeed, $z( A 2\u2212 D 0)=0$ implies $z( A 2\u2212 B 0)=z( B 0\u2212 C 1)=z( A 2\u2212 C 1)=0$. Similarly, $ B 0$ blocks the hypothetical edge $ B 2\u21dd C 1$, at $z=0$. Obviously, there is a heteroclinic edge toward the saddle $i( C 1)=1$ from its $ h 1$-neighbor $ C 2$ at Morse level $i=2$. To show $ h 0 ( 4 q \u2212 1 ) = D 2 \u21dd C 1 = h 0 ( 4 q + 2 )$, just note that the only equilibria $ h 0$-between $ D 2$ and $ C 1$ are $ h 0(4q)= A 1$ and $ h 0(4q+1)= B 0$. However, the latter pair precedes the former, along $ h 1$, and therefore cannot be blocking.

This establishes the connection graph $ C$ of $ M$, as illustrated in Figs. 7 and 8(c). By induction on $q$, the theorem is now proved.

We can now prove the remaining three main Theorems 5.2, 5.4, and 5.5, for $r=1$. We repeat that Lemma 7.1, which already established Theorem 5.3, allows us to base our proofs on the trivially equivalent connection graphs $ C= C 1 q \kappa $, instead of $ C 1 q$ itself. All three theorems will become easy corollaries of Theorem 7.6; see also Fig. 8. We conclude with an explicit proof of equivalence Theorem 5.3 which is independent of our more abstract approach via Lemma 7.1. Instead, it will be based on a direct, explicit, and elementary computation of the Sturm permutations $ \sigma q 1, \sigma 1 q$, $ \sigma q 1 \u2212 1$, and $ \sigma 1 q \u2212 1$, in Proposition 7.2.

### C. Proof of Theorem 5.2

### D. Proof of Theorem 5.4

^{41}it is sufficient to prove that the single equilibrium $ O= D q + 1$ of the top Morse index $i( O)=q+1=dim\u2061 A 1 q$ connects heteroclinically to all other equilibria $E$. In symbols, $ D q + 1\u21ddE$. By transitivity of the directed edge relation $\u21dd$, this amounts to showing that there exists a di-path from $ D q + 1$ to any $E$, in the connection di-graph $ C$. This is obvious from (7.4) to (7.9), which coarsen to

### E. Proof of Theorem 5.5

### F. Explicit sturm permutations

We derive the explicit primitive three-nose Sturm permutations $ \sigma q 1, \sigma q 1 \u2212 1, \sigma 1 q, \sigma 1 q \u2212 1\u2208 S 4 q + 3$.

Claim (5.3) of Theorem *5.3* holds true, for $r=1$ and all $q\u22651$, due to the following explicit expressions of the relevant permutations.

- With arguments $0\u2264j\u2264q$, as appropriate, the permutation $ \sigma q 1$ satisfies$ \sigma q 1 ( 4 j ) = 2 q \u2212 2 j + 2 , \sigma q 1 ( 4 j + 1 ) = 2 j + 1 , \sigma q 1 ( 4 j + 2 ) = 4 q \u2212 2 j + 2 , \sigma q 1 ( 4 j + 3 ) = 2 q + 2 j + 3. $
- The inverse permutation $ \sigma q 1 \u2212 1$ is given explicitly by$ \sigma q 1 \u2212 1 ( 2 j \u2032 + 1 ) = { 4 j \u2032 + 1 , for 0 \u2264 j \u2032 \u2264 q , 4 ( j \u2032 \u2212 q \u2212 1 ) + 3 , i for q j \u2032 \u2264 2 q + 1 ,$$ \sigma q 1 \u2212 1 ( 2 j \u2032 ) = { 4 ( q + 1 \u2212 j \u2032 ) , for 1 \u2264 j \u2032 \u2264 q , 4 ( 2 q + 1 \u2212 j \u2032 ) + 2 , for q j \u2032 \u2264 2 q + 1.$
- The permutation $ \sigma 1 q$ is given explicitly by$ \sigma 1 q ( 2 j \u2032 + 1 ) = { 4 j \u2032 + 1 , for 0 \u2264 j \u2032 \u2264 q , 4 ( j \u2032 \u2212 q \u2212 1 ) + 3 , for q j \u2032 \u2264 2 q + 1 ,$$ \sigma 1 q ( 2 j \u2032 ) = { 4 ( q + 1 \u2212 j \u2032 ) + 2 , for 1 \u2264 j \u2032 \u2264 q + 1 , 4 ( 2 q + 2 \u2212 j \u2032 ) , for q + 1 j \u2032 \u2264 2 q + 1.$
- With arguments $0\u2264j\u2264q$, as appropriate, the permutation $ \sigma 1 q \u2212 1$ satisfies$ \sigma 1 q \u2212 1 ( 4 j ) = 4 q \u2212 2 j + 4 , \sigma 1 q \u2212 1 ( 4 j + 1 ) = 2 j + 1 , \sigma 1 q \u2212 1 ( 4 j + 2 ) = 2 q \u2212 2 j + 2 , \sigma 1 q \u2212 1 ( 4 j + 3 ) = 2 q + 2 j + 3. $

Obviously, the 16 expressions (7.14)–(7.19) define permutations in $ S 4 q + 3$. Just for the moment, let us denote by $ \sigma q 1$ and $ \sigma q 1 \u2212 1$ the expressions in (i) and (ii). Then, $ \sigma q 1\xb0 \sigma q 1 \u2212 1= id$ is obvious, by definition. Therefore, (ii) actually defines the inverse permutation of (i).

Finally, we obtain (iv) via $ \sigma 1 q \u2212 1=\kappa \sigma q 1\kappa $. Alternatively, we may check the inversion (iv) of (iii) formally, as we did for the pair (i),(ii).

This proves the proposition.

## VIII. DISCUSSION

We discuss some broader settings for our results. See Sec. VIII A for the cases $r>1$ of our main results in Sec. V, which Sec. VII did not provide proofs for. In Sec. VIII B, we briefly mention some pertinent literature on fully nonlinear equations, grow-up, and blow-up. ODE variants of the PDE (1.2), like cyclic monotone feedback systems and Jacobi systems, arise by finite difference discretization. See Sec. VIII C. In Sec. VIII D, we conclude with some more topological aspects of our results, and the open question of time reversal for full boundary spheres $\Sigma =\u2202 A$ of global attractors, rather than for just their connection graphs $ C\u2216 O$.

### A. The cases *r* **>** 1

The proof of Theorem 5.1 in Sec. VI gives an indication on how to proceed inductively for $r>1$; see Fig. 6. Of course, we may perform $q$ successive nose insertions as in Fig. 6(b) for $p=(r+1)(q+1)$ as well, coming from $ p \u2032=r(q+1)$. In case $q=1$, this inserts just one nose of two equilibria, reminiscent of—but, technically, slightly different from—our insertion of the nose $ B 0 C 1$ in Figs. 7 and 8(c). That insertion occurred at Morse levels $i=0,1$. In case $q>1$, more ambitiously, we are inserting a $q$-nest of $2q$ equilibria, at the lowest Morse levels $i=0,\u2026,q$. This makes it more demanding, technically and notationally, to perform the requisite induction step $r\u21a6r+1$ for the connection graphs $ C r q$. As our starting point $r=1$, for any $q>1$, however, we may use the connection graphs $ C= C 1 q \kappa $ already established in Theorem 7.6 and Fig. 8(c). We postpone the details to our sequel.^{46}

### B. Nonlinear PDEs, grow-up, and blow-up

Technical groundwork for generalizations to fully nonlinear equations, including nonlinear boundary conditions, has been laid by Lappicy and co-workers.^{70–74} An interesting class of geometric applications is curve-shortening flows in the plane.^{4}

The qualitative behavior of parabolic global “attractors” of non-dissipative nonlinearities $f$ is a very intriguing subject, even in the semilinear case. For general *blow-up* in finite time, see the monograph^{91} and, in the Sturm setting, also Ref. 57. For an attempt to describe the development of sign-changing blow-up profiles by zero numbers, in one space dimension, see Ref. 33. Alternatively, solutions may exhibit *grow-up* to infinity, taking infinite time. The set of bounded global solutions $u(t,.),t\u2208 R$, of (1.2) will still consist of only equilibria and heteroclinic orbits. The question how global solutions may connect to infinity, “heteroclinically,” is attracting increasing attention; see for example, Refs. 8, 90, 75, and 14 and the references there.

### C. Jacobi systems

*Jacobi systems*are a spatially discrete analog, including a zero number dropping property (2.2); see Ref. 55. Motivated by, but much more general than, a semi-discretized finite-difference version of the PDE (1.2), they take the ODE form

*Cyclic monotone feedback systems*are a limiting case of spatially periodic subscripts $j$ mod $n$, with $ f j$ independent of $ u j + 1$. See for example, Refs. 97, 78, and 26 with applications to gene feedback cycles and oscillations, and Ref. 77 for an extension which includes an additional time-delay. For “Neumann” (or other separated) boundary conditions like $ u 0:= u 1, u n + 1:= u n$, but not necessarily for periodic boundary conditions, the system is still gradient-like.

^{29}

^{36}The role of the horizontal $ h 1$-axis is then taken over by the diagonal $ u n + 1= u n$, in the $( u n, u n + 1)$-plane.

*pair*, i.e., $ u j\u21a6\u2212 u j$, for each $j\u22612,3mod4$. That recovers a properly negative partial derivative of $g$ with respect to $ u j \u2212 1$. For odd length $n$, we can define meanders and permutations with respect to the “anti-Neumann” off-diagonal $ u n + 1=\u2212 u n$, in the $( u n, u n + 1)$-plane, rather than the diagonal. The specific meander permutations which arise in such standard examples, by shooting, have never been addressed in any systematic way, to our knowledge. For some related remarks in the context of Anosov maps, see Ref. 25.

### D. Time reversal and reversibility

One elementary formal operation on a Sturm meander $ M$ is a vertical flip, to some meander $ W$, by reflection at the horizontal $ h 1$-axis. Let $\sigma ,\u03f1$ denote the associated meander permutations, respectively. The flipped meander $ W$ emanates below the horizontal axis, from vertex $ h 0(1)= h 1(1)$, but remains otherwise dissipative, formally. Inspection of Morse numbers (2.6), however, now replaces any $ i j$ by $\u2212 i j$. Indeed, right turns on $ M$ become left turns, on $ W$. Induced by $ z k k= i k$ in (2.7), the zero numbers also reverse sign. Adjacency and blocking (2.9), however, remain unaffected. In terms of formal connectivity (2.8), the sign reversal of the Morse numbers $i$ reverses all arrows in the associated formal connection graph of $ W$. However, what does such algebraic trickery mean, in terms of actual equilibria $v$ of (1.2) and (1.1), which cannot possibly possess negative Morse indices $i(v)$?

We have already observed in Sec. III, how repeated suspensions raise Morse numbers and zero numbers, but preserve formal connectivity; see Proposition 3.1 and Corollary 3.1. Let $d=dim\u2061 A= max j i j$ denote the dimension of the original Sturm attractor associated with $ M$. Then, $ W$ has - $d$ as its minimal Morse number. Therefore, $ W$ becomes Sturm, first, after $s=d$ suspensions. In fact, the suspended connection di-graph of $ W$ will contain the time reversed, i.e., inversely oriented, original connection di-graph $ C$, as a subgraph; see Proposition 3.1(vii).

Alas, such *time reversal* does not provide *time reversibility*, i.e., an involutive time reversor $ R$ *within* one and the same connection graph, as we have encountered on the boundary spheres of $ C r q$ within the Sturm balls $ A r q$. Surprising as time reversal may be, it only shows how

any time-reversed Sturm connection graph appears within some larger Sturm connection graph, of the same dimension $d$.

Time reversibility of the connection graph on the boundary sphere $ \Sigma d \u2212 1=\u2202 A$, however, is not all that exceptional either, for Sturm balls $ A= clos W u( O)$ of dimension $d$. Before the three-nose examples $ A r q$, with $d=r+q$, we had already encountered the Chafee–Infante balls $ A d$. Other examples of such $ A$ are all planar $n$-gons,^{38} and the solid tetrahedron.^{40} The methods of Ref. 44 provide any self-dual graph on $ \Sigma 2$, and any solid $d$-simplex.

All these examples exhibit a weak reversor $ R$, as in (2.12), just on the vertices of $ \Sigma d \u2212 1=\u2202 A$ in the connection graph $ C$. However, such a weak reversor $ R$ need not extend to a strong reversor, automatically, on all of $ \Sigma d \u2212 1$. Indeed, $ R$ only establishes that certain heteroclinic orbits possess twins, someplace else, which run in a reverse direction.

To prove strong time reversibility, the strong reversor $ R$ needs to define an orbit equivalence, mapping PDE orbits to PDE orbits, but reversing their time direction. Poincaré (self-)duality of the Thom–Smale complex (2.10) may provide a first step in that direction. Reversing time in fact interchanges the roles of stable and unstable manifolds, in the Thom–Smale complex. Although this seems problematic in our infinite-dimensional PDE setting, finite-difference semi-discretization allows us to consider Jacobi systems (8.1), instead, where duality becomes finite-dimensional. Alternatively, we could work inside the global attractor $ A$ itself. In general, duality on $ A$ is fraught with the additional difficulty that $ A$ may contain pieces of different local dimension. The above cases of a sphere boundary $\u2202 A r q= \Sigma r + q \u2212 1$ have been more benign.

There remain several steps toward elusive strong reversibility on $ \Sigma r + q \u2212 1$. We have to show that the dual Thom–Smale complex is equivalent to its original, at least combinatorially. We then have to establish topological equivalence of the complexes. And, finally, we have to carefully adapt the duality construction such that the topological equivalence actually maps orbits to orbits, in the underlying Jacobi system.

Only in the special Chafee–Infante case of Sec. IV it seems fairly clear how to achieve that. The general task certainly lies beyond the scope of the present paper. Instead, we present a simple ODE model which, at least, features the same reversible connection graph $ C= C 1 q \kappa $ of Fig. 8(c), as the global attractor $ A= A 1 q \kappa $ does, on the boundary sphere $ \Sigma q$. An intriguing feature of $ C\u2216 O$ on $ \Sigma q$ are the two full Chafee–Infante sub-graphs $ C q$: one with tags $ L j,L\u2208{A,B}$, on the left, and the other—upside down, i.e., time-reversed—on the right with tags $L\u2208{C,D}$. Compare (4.5), (4.6), and Fig. 4, with (7.4)–(7.9) and Fig. 8(c).

The basins of attraction are easily described, for the three $i=0$ sinks $ A 0, B 0, D 0$. We only describe the basins within the sphere $ \Sigma q$, stereographically projected to $ R d$. The basin boundary of $ D 0=\u221e$ is the invariant ( $q$-1)-sphere $ r=2$. The invariant hyperplane $\u27e8 e 1,\u2026, e q \u2212 1\u27e9$, inside the $q$-ball of radius $ r=2$, is the shared basin boundary of the other two sinks $ A 0, B 0$, at angles $\Phi =\xb1 e 0$. The same hyperplane splits the $ r=2$ sphere into two closed hemispheres, which are the shared basin boundaries of $ A 0, B 0$ with $ D 0$, respectively. The equilibria in their intersection, i.e., in the equatorial $ r=2$ sphere of dimension $q\u22122$, and only those, possess heteroclinic orbits to all three sinks.

It is a useful exercise to locate all those equatorial and hemisphere equilibria, in our geometric description, and to verify their heteroclinic orbits to the respective sinks in the connection graph of Fig. 8(c). Indeed, all other equilibria with tags $C,D$ connect to $ D 0$. Similarly, all non-sink equilibria of any tags, except $ C 1$, connect to $ A 0$. For $ B 0$ the analogous exception is $ D 1$.

Upon time reversal, equilibria in basin boundaries of sinks become heteroclinic targets of sources, instead. Therefore, we can read off the basin boundary equilibria of $ D 0, A 0, B 0$ from the targets of their reversor-related sources $ A q, D q, C q$ in Fig. 8(c), respectively. See (8.7) and (8.8) for the precise tags and subscripts involved. Again we see how, analogously, all other equilibria with tags $A,B$ are targets of $ A q$. Similarly, all non-source equilibria of any tags, except $ B q \u2212 1$, are targets of $ D q$. For $ C q$, the analogous exception is $ A q \u2212 1$.

Whether or not the same detailed geometry describes strong time reversibility in the sphere boundary of the Sturm attractor, $\u2202 A 1 q \kappa = \Sigma q$, remains open at present. The explicit reversor $ R$ of (8.7) in our model (8.6) certainly cautions us that this is not a trivial task. But this is just one of the many curiosities, intricacies, and mysteries surrounding time reversal for global attractors of even the simplest of parabolic PDEs—which diffusion, supposedly, governs ever so “irreversibly.”

## ACKNOWLEDGMENTS

This work has been supported, most generously, by the Deutsche Forschungsgemeinschaft, Collaborative Research Center 910 *“Control of self-organizing nonlinear systems: Theoretical methods and concepts of application”* under project A4: *“Spatio-temporal patterns: control, delays, and design.”* We are grateful for the numerous inspirations, lively discussions, and excitingly active working atmosphere to which our speakers Eckehard Schöll and Sabine Klapp contributed so much. We are also much indebted for enlightening discussions on meanders with very patient Piotr Zograf and for the warm hospitality at the Mathematical Institute of Sankt Petersburg University. Support by FCT/Portugal through Project Nos. UID/MAT/04459/2019 and UIDB/04459/2020 is also gratefully acknowledged.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Bernold Fiedler:** Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Project administration (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). **Carlos Rocha:** Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Project administration (equal); Visualization (supporting); Writing – original draft (supporting); Writing – review & editing (supporting).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.

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