We systematically explore a simple class of global attractors, called Sturm due to nodal properties, for the semilinear scalar parabolic partial differential equation (PDE) on the unit interval , 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 . 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 , 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 between equilibrium vertices of adjacent Morse index. The global attractor turns out to be a ball of dimension , given as the closure of the unstable manifold of the unique equilibrium with maximal Morse index . Surprisingly, for parabolic PDEs based on irreversible diffusion, the connection graph indicates time reversibility on the ( )-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.
It is, therefore, not our intention, in the present paper, to contribute just another analysis or simulation, for this or that particular nonlinearity , arising in one or the other highly specialized applied context. For general -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 . See the following Sec. II. The permutations themselves, as introduced by Fusco and Rocha,56 are based on the discrepancies between the orderings of the equilibria at the boundaries and , respectively; see (2.4) and (2.5). Although each of the permutations will be represented by an open class of nonlinearities , in principle, we will provide specific nonlinearities only in exceptional cases; but see (4.1) and Sec. IV for cubic . 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 .
In the present paper, we describe the global dynamics of the full PDE (1.2), for a certain subclass of permutations . 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 , standard theory of strongly continuous semigroups provides local solutions of (1.2) in suitable Sobolev spaces , for and given initial data at time . See Refs. 64, 89, and 99 for a general PDE background.
We assume the solution semigroup generated by the nonlinearity to be dissipative: any solution exists globally in forward time and eventually enters a fixed large ball in . Explicit sufficient, but by no means necessary, conditions on which guarantee dissipativeness are sign conditions , for large , together with subquadratic growth in . For large times , any large ball in then limits onto the same maximal compact and invariant subset of which is called the global attractor. In general, the global attractor consists of all solutions which exist globally, for all positive and negative times , and remain bounded in . Of course, , 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 is quite general. However, it provides practically no information concerning the specific dynamics on . 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.
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 of the PDE linearization of (1.2) at any equilibrium are algebraically simple and real. In fact , for the eigenfunction of . We assume all equilibria are hyperbolic, i.e., all eigenvalues are nonzero. The Morse index of then counts the number of unstable eigenvalues . In other words, the Morse index is the dimension of the unstable manifold of . Let denote the set of equilibria. Our generic hyperbolicity assumption and dissipativeness of imply that := is odd; see also (5.5).
Using a shooting approach to the ODE boundary value problem (1.1), the Sturm permutations have been characterized, purely combinatorially, as dissipative Morse meanders in Ref. 35. Here, the dissipativeness property, abstractly, requires fixed and . In fact, the shooting meander emanates upwards, toward , from the leftmost (or lowest) equilibrium at , and terminates from below, , at x=1. The meander property requires the formal path of alternating upper and lower half-circle arcs defined by the permutation , as in Fig. 1(c), to be Jordan, i.e. non-selfintersecting. For dissipative meanders, the recursion in (2.6), and , define all Morse numbers . Note how and are always of opposite parity, . In particular, is odd, and =0 follows automatically. The Morse property, finally, requires non-negative Morse indices in the formal recursion (2.6), for all . 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” , briefly, in Proposition 3.1 and Sec. VI. See also our sequel.46
More geometrically, global Sturm attractors and of dissipative nonlinearities with the same Sturm permutation are orbit-equivalent.36 Only locally, i.e., for -close nonlinearities and , 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 and with different Sturm permutations .
Clearly, any heteroclinic orbit implies adjacency: by (2.2), any blocking equilibrium would force to drop strictly at the Neumann boundary , for some . This contradicts the equal values of at the limiting equilibria of , for .
We encode the above heteroclinic structure in the directed connection graph . See Fig. 1(d) for an example. The connection graph is graded by the Morse index of its equilibrium vertices. Directed edges are the heteroclinic orbits running downwards between equilibria of adjacent Morse index. Uniqueness of such heteroclinic orbits, given , 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 on is transitive, by Morse–Smale transversality and the -Lemma.87 Therefore, any directed path from to also defines a direct heteroclinic orbit . Given , conversely, the cascading principle first described in Ref. 13 asserts an interpolating sequence of heteroclinic orbits between equilibria of adjacent Morse indices, from to .
The basin of attraction of an sink vertex in , for example, consists of itself, and all heteroclinic orbits . The basin boundary consists of just those other equilibria , and all heteroclinic orbits among them. The connection graph 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).
We call the dimension of , or of the complex . Then, at least one equilibrium has maximal Morse index , i.e., for all other Morse indices. If is the closure of a single -cell, then the Sturm complex turns out to be a closed -ball.41 We call this case a Sturm -ball.
A three-dimensional Sturm complex , for example, is the regular Thom–Smale complex of a three-dimensional Sturm global attractor . See Fig. 1(c) for the Sturm complex of the Sturm three-ball associated with the meander in Fig. 1(a).
In the Sturm-ball trilogy,42–44 we have characterized all Sturm 3-balls . Earlier, the trilogy37–39 had characterized all planar Sturm complexes , i.e., the case . The case , i.e., with odd , is a trivial line with alternating sinks and saddles. Global asymptotic stability of a unique sink equilibrium is the case of .
Conversely, we have described in Refs. 45 and 94 how the boundary orders of (2.4), and therefore the Sturm permutation of (2.5), are determined uniquely by the signed hemisphere decomposition. This is a slight refinement of the Sturm complex , which we do not pursue in further detail here. In Fig. 1, for example, the signed hemisphere complex (c) determines how the boundary orders [red in (a)] and (blue) traverse the equilibrium vertices, from the North pole to the South pole . The predecessors and successors, on , of the repelling sphere barycenter are marked by small annotated red and blue circles, everywhere in Fig. 1.
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 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 with two nose arcs above the horizontal axis, each as the innermost of and nested upper arcs, respectively. Below the horizontal axis, the only remaining nose is centered as the innermost of the complementing 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, and 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 is Sturm if, and only if, , for some . Let denote the associated Sturm permutations. The resulting global attractors are all distinct—except for the not immediately obvious “trivial” linear flow equivalence upon interchange of and ; see Corollary 5.3. In Theorem 5.4, the Sturm complex turns out to be a Sturm ball of dimension . The three-ball attractor of Fig. 1, for example, is trivially equivalent to the simple case , in the sense of Sec. III.
We prove Theorem 5.1 in Sec. VI. To circumvent tiresome mathematical pedantry, we only provide proofs for the simplest interesting case of our remaining results, in Sec. VII. This includes the explicit connection graphs for ; see Theorem 7.6.
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 , meanders , permutations , and connection graphs , 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 of the entourage . See also previous accounts in Refs. 36, 67, and 94.
A small subtlety arises, concerning isotropy of nonlinearities under some trivial equivalence . Such -isotropy implies permutation-isotropy , of course. However, we never proved the converse. Although some nonlinearities will always realize isotropic permutations , by Ref. 35, we never proved realization by an with isotropy , i.e., such that .
We define the suspension as an augmentation of by two overarching arcs [black in Fig. 3(d)]: an upper arc from the first new vertex to the last old vertex , and a lower arc from the first old vertex to the last new vertex . This extends the previous definition of to for .
By construction, the number of meander-noses is invariant under suspension, for . In the Sturm case, i.e., if our dissipative meanders are also Morse, our definition also extends to define the suspensions and of their attractors and connection graphs .
More abstractly, however, our definition of suspension generalizes to dissipative meanders , which are not necessarily Sturm. Indeed, they may violate the Morse property and hence may also violate . Abstractly, however, dissipative meanders still determine their permutations , Morse numbers , and zero numbers via (2.6) and (2.7)—even when those numbers lack any ODE or PDE interpretation. Sturm “attractors” with actual “equilibria” and actual “heteroclinic orbits” cannot exist, of course, once negative “Morse indices” are involved. By -adjacency (2.8), and blocking (2.9), however, we can still define connection graphs . Quite radically, indeed, we abuse the notation here, and even , to denote the recursively defined quantities , and the relation defined abstractly via (2.8) and (2.9). In particular, our definition of meander suspensions readily extends to define the suspensions , and , 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 of a global Sturm attractor as the double cone suspension of itself, with respect to the two added polar cone vertices and . See Fig. 3.
For dissipative, but not necessarily Morse, meanders , the suspension defined above has the following properties, for all
, in case all .
Consider suspensions and of abstract “boundary orders” which fix as well as . Define the dissipative meander permutation , as in (2.5).
Claim (i) then holds by construction. To prove claim (ii), first note that . Since the orders and follow the shared part of the meanders and , in opposite directions, we also have for and . Together, this proves (ii), if we substitute the flip 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 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 by any new vertex cannot occur, because the -position of those new vertices is extremal and never between . By (vi), in contrast, any old blocking remains in effect. This proves claim (vii).
In claim (viii), we assume to be Morse, and hence Sturm. In particular, this implies , for all zero numbers. Therefore, (v),(vi) prevent blocking (2.9), and (viii) follows from (iii),(iv) with (2.8).
For Sturm meanders , the following holds true.
The suspension of any Sturm permutation is Sturm.
All equilibria connect heteroclinically, in , toward the two polar sinks in the bottom row.
The connection graph of the suspension contains the connection graph , lifted to the rows .
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 may be of applied interest in design. For example, we may append a region to the -domain of (1.1) and (1.2). Then, suspension can be effected, in terms of -profiles of equilibria like Fig. 1(b), if reverses the order of equilibria at the right boundary, as increases from to . This agrees well with Proposition 3.1(vi). Dissipativeness, of course, will require two new equilibria, e.g., homogeneous throughout : one at the top and one at the bottom.
IV. TWO NOSES: THE CHAFEE–INFANTE PARADIGM
To pursue this program, let us start from just upper arcs, separately and without meanders in mind as yet. Equivalently, the arcs define a balanced structure of pairs of opening and closing parentheses, “ ” and “ ,” also know as Dyck words of length , 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 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 , for any . By construction, possesses upper and lower arcs, each, over its intersections with the horizontal axis. Alas, we do not know yet whether is Morse, and therefore Sturm.
We have already mentioned that the 1974 Chafee–Infante version of (1.2) had been studied for the cubic nonlinearity , originally, albeit under Dirichlet boundary conditions; see (4.1) and the original paper.15 Their method was local bifurcation analysis of the trivial equilibrium . Note , for , under Neumann boundary conditions, by elementary linearization. The second order ODE (1.1) is Hamiltonian integrable, for nonlinearities . For the hard spring cubic nonlinearities , the minimal periods of at grow monotonically with their amplitude at . Note the limit . Rescaling as in , we see that reappears as a rescaled solution at , for any nonzero integer . See Fig. 4(b) for such rescaled equilibrium profiles in case . In particular, this produces a (stylized) shooting meander which, by monotonicity of the periods, coincides with the Sturm meander , and hence determines the Sturm permutation of (4.9).
For an early geometric description of the Chafee–Infante attractor , for low dimensions , see Section 5.3 of Ref. 64. In 1985, Henry achieved the first description of for general .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 , at . 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 , among all Sturm attractors:
Among all Sturm attractors with equilibria, is the unique Sturm attractor with the maximal possible dimension, .
Among all Sturm attractors of dimension , is the unique Sturm attractor with the smallest possible number of equilibria, .
The two claims follow, e.g., from the connection graph. In fact, each unstable hyperbolic equilibrium must connect, heteroclinically, to at least two other equilibria , such that and at . 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 between the sets and . Acting on with the Klein four-group of symmetries , generated by (3.1) and (3.2) in the Sturm setting, we obtain the four required heteroclinic orbits (4.6). Indeed , alias , interchanges each with ; see (3.1) and Fig. 4(b). Inversion , in contrast, performs the same interchange for odd , only; see (3.2). Since the Morse levels and are of opposite even/odd parity, this generates the four required heteroclinic orbits. The argument for the heteroclinic orbits emanating from the equilibrium , 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 ; 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 reduces to the sequences ; see Theorems 5.1 and 5.2. As usual, these come with their entourage of Sturm permutations , associated Sturm attractors , and connection graphs (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 and coincide; see Corollary 5.2. In fact, the Sturm attractors turn out to be trivially equivalent to , by Theorem 5.3 and Corollary 5.3. Geometrically, these are Sturm balls of dimension (Theorem 5.4). Finally, Theorem 5.5 asserts that the connection graph remains time reversible on the invariant boundary sphere .
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 , i.e., and , in Sec. VII. For , see our sequel.46
With the above notation, the following holds true for .
is a dissipative meander if, and only if, are co-prime and .
For , any dissipative meander fails to be Morse.
We will prove Theorem 5.1 in Sec. VI. Note that the non-Morse three-nose cases (i), with 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 of suspensions after which becomes Morse, and hence Sturm. See Fig. 5 for the non-Morse three-nose example . We will pursue those cases further in our sequel.46
Let us now focus on the three-nose cases , which are complementary to Theorem 5.1(ii). Then, are automatically co-prime, because . The following theorem shows that all cases do lead to Morse meanders and, therefore, to Sturm attractors. The rotation of the simplest case has already served in Figs. 3(a)–3(c), to illustrate suspension. We, therefore, assume , for the rest of this paper. Proofs of the next four theorems will be given in Sec. VII, for the simplest interesting case , only. See Ref. 46 for general .
The Morse count functions have the following symmetry properties.
Up to ordering, the subscript set is determined by .
Conversely, the subscript set determines .
For all , we have .
To prove (i), just note . Claim (ii) follows from . To prove (iii), insert (5.2).
For any , we call a primitive three-nose meander. For the Sturm entourage of , we denote the associated primitive Sturm permutation as , the primitive Sturm attractor as , and the primitive connection graph as .
Suppose and 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 , Theorem 5.3 reveals the only nontrivial isotropy , in the Klein four-group of trivial linear equivalences. In particular, the rainbow argument above shows that still cannot be -isotropic. Admittedly, (5.3) suggests to study which commute with , i.e., . However, is still excluded, because and must remain distinct.
Note the Morse count at maximal ; see (5.2). Let denote that unique equilibrium in of maximal Morse index .
The primitive Sturm attractor is the closure of the unstable manifold of the single equilibrium . i.e., is a Sturm ball of dimension .
With dimension replaced by , Theorem 5.4 remains valid for any -fold suspension of .
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 —in spite of the parabolic, diffusion-dominated nature of the underlying original PDE (1.2).
The connection graph is reversible on the flow-invariant boundary sphere of the primitive Sturm ball .
The reversibility on the boundary sphere , of course, is a much deeper reason for the symmetry of the Morse count function , for , which we have already noticed in Corollary 5.2(iii). Indeed, the reversor on swaps equilibria of Morse indices and .
VI. NON-MORSE MEANDERS WITH THREE NOSES
In this section, we prove Theorem 5.1.
Claim (i) states that the dissipative arc configuration of nested upper arcs followed by nested upper arcs, and a right shifted lower -rainbow, is a meander if, and only if, and are co-prime.
The case is trivially discarded: all upper arcs of the nonempty -nest close up to become circles, with the corresponding inner arcs of the lower rainbow. This contradicts the meander property.
For , let us remove the outermost arc of the upper -nest and, instead, stack it onto the upper -nest. The resulting closed arc configuration now features upper nests of and 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 with . By (6.1) of Ref. 28, the greatest common divisor of and counts the connected components of the resulting closed arc configuration. The proof was recursive, via the Euclidean algorithm for . This proves claim (i).
It remains to show, (ii), that the dissipative meander fails to be Morse, if for any integer .
We first consider the case . We label equilibria such that . Then, and are the left and right end points of the uppermost arc in the upper -nest. By (2.6), Morse numbers of -adjacent vertices are adjacent. Obviously, is adjacent to . By dissipativeness, . Adjacency implies . In case , we are done.
In case , we obtain because the meander arc turns left from to ; see (2.6). Now consider the preceding lower rainbow arc from to . Since for the two end points of any lower rainbow arc, our assumption implies : the lower arc turns left, from to . But we already know . Therefore, (2.6) implies a negative Morse index , 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 , their dissipative Morse meanders , and their entourage of Sturm permutations and connection graphs . For brevity and simplicity, we restrict our proofs to the simplest interesting case . We skip the trivial case , 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 and claimed in Theorem 5.3, for . 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 and their trivially equivalent relatives, in Proposition 7.2 of Sec. VII F.
A. Proof of Theorem 5.3
To locate noses of equilibria, we use the matrix notation for locations and . Note how noses are characterized by adjacency under both boundary orders .
The following are corresponding nose locations of the indicated Sturm permutations, for any fixed
the upper right nose of
the lower left nose of
the nose of the upper rainbow of
the nose of the lower rainbow of .
The lower rainbow nose (d) of , i.e., for , is obviously located at , by arc counting. Similarly, the upper rainbow nose (c) for the rotated meander associated with is just as obviously located at the rotated position .
Inversion of interchanges the roles of and . 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 and in (a) and (b), respectively. The first -entries locate these noses at the extreme right and left of the horizontal axis, respectively.
It remains to show that the permutation in (a) is indeed the inverse of the Sturm permutation in (c). [The other pair (b), (d) is treated analogously.] From Sec. III, we already know that inversion preserves the number of noses and, up to , commutes with suspension; see (3.11). Therefore, the inverse of must also be a primitive three-nose Sturm permutation . The upper nose in (a) is located rightmost, at , and hence cannot sit inside any larger nest. Therefore, for some . This implies , since the total number of vertices is also preserved under inversion . This proves the lemma, (5.3), and Theorem 5.3.
B. The connection graphs and
With the case already settled, we proceed by induction on . We may, therefore, assume that the theorem already holds true for the ( -1) meander and its connection graph , as illustrated in Figs. 7 and 8(a). Starting from , our first step is by suspension to as in Figs. 7 and 8(b). Our second step, leading to the -meander and its connection graph , is by nose insertion; see Figs. 7 and 8(c).
Suspension, our first step, invokes Proposition 3.1. The equilibria of the suspension have been labeled , to correspond to our notation for . Suspension raises Morse indices by 1, due to Proposition 3.1(iv). Only for the cone vertices and of the suspension, at the lowest Morse level , we have substituted the new labels in Figs. 7 and 8(b). The connection graph 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 , for equilibria inherited by (c) from (b). This ensures , for , as claimed in (7.10). The cone vertices have not been relabeled. However, we now have to address three possible effects of the newly inserted nose arc on heteroclinic edges (purple) in Fig. 8,
previous edges of (b) blocked by nose equilibria ,
new edges in (c) emanating from the nose ,
new edges in (c) terminating at the nose .
We start with blocking of type (i). By (2.8) and (2.9), new blockings of , i.e., purple edges in (b), only arise through nose equilibria which are located between other and along the meander order of , and which satisfy (2.9). Since is a nose arc, blocking by is equivalent to blocking by . Except for the last equilibrium , all equilibria inherited by (c) from (b) have -position less than the second to last -position of . Therefore, (or , equivalently) cannot block any of the heteroclinic edges inherited from (b), by (c), except possibly for edges from to . The edge between and the sink , for example, cannot be blocked, because they are -neighbors on the suspension arc . Similarly, the edge between the -neighbors and the sink remains non-blocked. However, implies that blocks . Here and below, we refer to (2.7) along the orders of or , equivalently, for the calculation of zero numbers. Similarly, is blocked by at . 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 . Just as obviously, connects heteroclinically to its nose neighbor , and to its -neighbor . However, implies that blocks . 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 , first. Obviously, there are heteroclinic edges toward the sink from its -neighbors , all at Morse level . The hypothetical edge is blocked by , at . This settles the three edges toward target .
Finally, consider the target of (iii). We proceed by checking the potential sources , , in alphabetical order. The hypothetical edge is blocked by , at . Indeed, implies . Similarly, blocks the hypothetical edge , at . Obviously, there is a heteroclinic edge toward the saddle from its -neighbor at Morse level . To show , just note that the only equilibria -between and are and . However, the latter pair precedes the former, along , and therefore cannot be blocking.
We can now prove the remaining three main Theorems 5.2, 5.4, and 5.5, for . We repeat that Lemma 7.1, which already established Theorem 5.3, allows us to base our proofs on the trivially equivalent connection graphs , instead of 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 , , and , in Proposition 7.2.
C. Proof of Theorem 5.2
D. Proof of Theorem 5.4
E. Proof of Theorem 5.5
F. Explicit sturm permutations
We derive the explicit primitive three-nose Sturm permutations .
Claim (5.3) of Theorem 5.3 holds true, for and all , due to the following explicit expressions of the relevant permutations.
- With arguments , as appropriate, the permutation satisfies
- The inverse permutation is given explicitly by
- The permutation is given explicitly by
- With arguments , as appropriate, the permutation satisfies
Obviously, the 16 expressions (7.14)–(7.19) define permutations in . Just for the moment, let us denote by and the expressions in (i) and (ii). Then, is obvious, by definition. Therefore, (ii) actually defines the inverse permutation of (i).
Finally, we obtain (iv) via . Alternatively, we may check the inversion (iv) of (iii) formally, as we did for the pair (i),(ii).
This proves the proposition.
We discuss some broader settings for our results. See Sec. VIII A for the cases 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 of global attractors, rather than for just their connection graphs .
A. The cases r > 1
The proof of Theorem 5.1 in Sec. VI gives an indication on how to proceed inductively for ; see Fig. 6. Of course, we may perform successive nose insertions as in Fig. 6(b) for as well, coming from . In case , this inserts just one nose of two equilibria, reminiscent of—but, technically, slightly different from—our insertion of the nose in Figs. 7 and 8(c). That insertion occurred at Morse levels . In case , more ambitiously, we are inserting a -nest of equilibria, at the lowest Morse levels . This makes it more demanding, technically and notationally, to perform the requisite induction step for the connection graphs . As our starting point , for any , however, we may use the connection graphs 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 is a very intriguing subject, even in the semilinear case. For general blow-up in finite time, see the monograph91 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 , 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
D. Time reversal and reversibility
One elementary formal operation on a Sturm meander is a vertical flip, to some meander , by reflection at the horizontal -axis. Let denote the associated meander permutations, respectively. The flipped meander emanates below the horizontal axis, from vertex , but remains otherwise dissipative, formally. Inspection of Morse numbers (2.6), however, now replaces any by . Indeed, right turns on become left turns, on . Induced by 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 reverses all arrows in the associated formal connection graph of . However, what does such algebraic trickery mean, in terms of actual equilibria of (1.2) and (1.1), which cannot possibly possess negative Morse indices ?
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 denote the dimension of the original Sturm attractor associated with . Then, has - as its minimal Morse number. Therefore, becomes Sturm, first, after suspensions. In fact, the suspended connection di-graph of will contain the time reversed, i.e., inversely oriented, original connection di-graph , as a subgraph; see Proposition 3.1(vii).
Alas, such time reversal does not provide time reversibility, i.e., an involutive time reversor within one and the same connection graph, as we have encountered on the boundary spheres of within the Sturm balls . 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 .
Time reversibility of the connection graph on the boundary sphere , however, is not all that exceptional either, for Sturm balls of dimension . Before the three-nose examples , with , we had already encountered the Chafee–Infante balls . Other examples of such are all planar -gons,38 and the solid tetrahedron.40 The methods of Ref. 44 provide any self-dual graph on , and any solid -simplex.
All these examples exhibit a weak reversor , as in (2.12), just on the vertices of in the connection graph . However, such a weak reversor need not extend to a strong reversor, automatically, on all of . Indeed, only establishes that certain heteroclinic orbits possess twins, someplace else, which run in a reverse direction.
To prove strong time reversibility, the strong reversor 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 itself. In general, duality on is fraught with the additional difficulty that may contain pieces of different local dimension. The above cases of a sphere boundary have been more benign.
There remain several steps toward elusive strong reversibility on . 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 of Fig. 8(c), as the global attractor does, on the boundary sphere . An intriguing feature of on are the two full Chafee–Infante sub-graphs : one with tags , on the left, and the other—upside down, i.e., time-reversed—on the right with tags . 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 sinks . We only describe the basins within the sphere , stereographically projected to . The basin boundary of is the invariant ( -1)-sphere . The invariant hyperplane , inside the -ball of radius , is the shared basin boundary of the other two sinks , at angles . The same hyperplane splits the sphere into two closed hemispheres, which are the shared basin boundaries of with , respectively. The equilibria in their intersection, i.e., in the equatorial sphere of dimension , 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 connect to . Similarly, all non-sink equilibria of any tags, except , connect to . For the analogous exception is .
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 from the targets of their reversor-related sources 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 are targets of . Similarly, all non-source equilibria of any tags, except , are targets of . For , the analogous exception is .
Whether or not the same detailed geometry describes strong time reversibility in the sphere boundary of the Sturm attractor, , remains open at present. The explicit reversor 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.”
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.
Conflict of Interest
The authors have no conflicts to disclose.
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).
The data that support the findings of this study are available within the article.