Global Attractors: Topology and Finite-Dimensional Dynamics

Transcription

1 This paper has appeared in Journal of Dynamics and Differential Equations 11 (1999) Global Attractors: Topology and Finite-Dimensional Dynamics James C. Robinson 1,2 Proposed running head: Attractors: topology and dynamics Many a dissipative evolution equation possesses a global attractor A with finite Hausdorff dimension d. In this paper it is shown there is an embedding X of A into IR N, with N = [2d + 2], such that X is the global attractor of some finite-dimensional system on IR N with trivial dynamics on X. This allows the construction of a discrete dynamical system on IR N which reproduces the dynamics of the time T map on A, and has an attractor within an arbitrarily small neighbourhood of X. If the Hausdorff dimension is replaced by the fractal dimension, a similar construction can be shown to hold good even if one restricts to orthogonal projections rather than arbitrary embeddings. KEY WORDS: Global attractors, inertial manifolds, exponential attractors, connectedness. 1 Institute for Scientific Computing & Applied Mathematics, Indiana University, Bloomington, IN 47405, U. S. A. 2 Current address: Mathematics Institute, University of Warwick, Coventry, CV4 7AL. U.K. j.c.robinson@damtp.cam.ac.uk 1

2 1. INTRODUCTION AND SUMMARY The application of methods from dynamical systems to the study of partial differential equations has raised many interesting questions and prompted a variety of new approaches. By considering the flow generated by the solutions of an equation on a suitable infinitedimensional phase space, one can begin to apply the geometric arguments and ideas more familiar in the classical study of ordinary differential equations and finite-dimensional systems. As in the study of ordinary differential equations, it has proved extremely useful to concentrate on the asymptotic behaviour of solutions, and properties of sets which are globally attracting in the original phase space. Many dissipative partial differential equations have recently been shown to possess globally attracting sets which are finite-dimensional, even though the phase space is infinite dimensional. Such equations include reaction-diffusion equations, the 2D Navier-Stokes equations, the Kuramoto-Sivashinsky equation, the Cahn-Hilliard equation, the Ginzburg- Landau equation, the Sine-Gordon equation, and a host of other examples (see Hale (1988) or Temam (1988)). In section 2 some very general standing assumptions are given which cover all these examples, and in section 3 various measures of the dimension of compact sets are recalled. The existence of such an attractor naturally leads one to consider the question of whether there is a finite-dimensional system that will adequately capture the asymptotic nature of the original flow (cf. Temam (1988) (Introduction), Eden et al. (1994)). This is formulated more precisely in section 2. One approach to this problem is to show the existence of an inertial manifold (Foias et al. (1988)), a finite-dimensional, positively invariant Lipschitz manifold which attracts 2

3 all orbits exponentially. This technique is briefly reviewed in section 4, and two problems highlighted. Section 5 recalls an elementary result of Langa & Robinson (1999) which guarantees that the asymptotic behaviour is in some sense determined by the behaviour on the attractor alone. This justifies the attempt in the following sections to reproduce the dynamics of the PDE by concentrating on the attractor dynamics. Since the attractor is initially a subset of an infinite-dimensional space, one has to be sure that it can be embedded into a finite-dimensional Euclidean space of sufficiently high dimension, and this forms the subject of section 6. A classical theorem (due to Hurewicz & Wallman (1941)) guarantees that a set A with finite topological dimensional dim A = d can be embedded in IR 2d+1. However, it will prove more convenient to deal with a distinguished set of embeddings, namely orthogonal projections. A theorem of Mañé (1981) ensures that if the fractal dimension of A is finite, d F (A) d, then a dense set of orthogonal projections of rank [2d + 1] are injective on A. This has recently been strengthened by Foias & Olson (1996) who guarantee that these projections have a Hölder continuous inverse on P A. Two partial solutions to the problem are given in section 7. Using this projection theorem, Eden et al. (1994) constructed a set of ordinary differential equations which reproduce the dynamics on A and have P A as a global attractor, and this is recalled in section 7.1. However, their equations do not have unique solutions, and uniqueness is an eminently desirable property. Dropping the requirement of exactly reproducing the solutions on A, in section 7.2 it is shown (theorem 7.1) that any trajectory on the attractor can be approximated for a large (but finite) length of time by the solutions of a Lipschitz continuous differential equation on IR 3. 3

4 Any improvement on the result of Eden et al. will rely on the topological properties of the global attractor, and these are discussed in detail in section 8, collecting together various scattered results from the literature - in particular, results of Geoghegan & Summerhill (1973), Garay (1991), and Günther (1995) are combined to show that there is a differential equation on a finite-dimensional Euclidean space that admits P A as an attractor, although the dynamics on P A are trivial. As a corollary (corollary 8.7) it is shown that all topological properties of finite-dimensional global attractors can be found by considering only finite-dimensional systems. It seems likely that all the topological properties of the global attractor (and its projections) that might be required to produce the finite-dimensional system of ordinary differential equations discussed in section 2, are contained in the results summarised in section 8. In section 9 a discrete dynamical system is constructed which reproduces the time T map on the global attractor and itself has an attractor lying within an arbitrarily small neighbourhood of the original one. Note that this is in fact a closer analogue of the methods used to compute the solutions of a PDE on a computer than a continuous system would be, and shows that the behaviour of some computable finite system will indeed reproduce the asymptotic behaviour of a dissipative partial differential equation, indicating that one can expect to see all the complexities of such systems within appropriate computer experiments. 2. STANDING ASSUMPTIONS In order to keep the results as general as possible, consider a general dissipative PDE written as an ordinary differential equation on a Hilbert space H, du/dt = F(u), (2.1) 4

5 which generates a strongly continuous semigroup S(t) (defined only for t 0) S(0) = id S(s)S(t) = S(s + t) S(t)u 0 continuous in t and u 0, (2.2) such that there exists a unique solution u(t; u 0 ) through any initial condition u 0 H, given by u(t; u 0 ) = S(t)u 0. (It may be that the semigroup is only well-defined on a subspace V of H. For example, for the equation du/dt = Au + f(u), where A is a sectorial operator and f a locally Lipschitz nonlinearity from D(A α ) into H, it is natural to define the semigroup on V = D(A α ) rather than H, see Henry (1981).) The main object of study is the global attractor A, the maximal compact invariant set which attracts the orbits of all bounded sets at a uniform rate: S(t)A = A dist(s(t)b, A) 0 as t, where B is any bounded set in H. It will be assumed that the flow is injective on the attractor A. This implies that the flow is defined on A for all t IR (see Hale (1988) Theorem 3.4.6, and also Temam (1988) for a discussion of the injectivity properties of the semiflow), and that on the attractor the time T map S(T ) is a homeomorphism for every T IR. An ideal solution of the problem would be to find an embedding ϕ of A into some IR N, and a system of ordinary differential equations ẋ = F (x) x IR N (2.3) 5

6 with unique solutions and a global attractor X = ϕ(a) on which the dynamics are conjugate to those on A via the homeomorphism ϕ, i.e. if T (t) is the solution operator for (2.3), then T (t) = ϕ S(t) ϕ 1. (2.4) It is shown in section 4 that inertial manifolds provide just such a system. General embeddings are discussed in sections 6 and 8, and in section 9 a discrete dynamical system is produced which comes close to providing this ideal solution. 3. MEASURES OF DIMENSION As generalisations of the fundamental topological dimension, which will be denoted by dim X (see Hurewicz & Wallman, 1941), there are several fractional measures of dimension applicable to sets which do not have the regular structure of, for example, a manifold. The two most frequently used are the Hausdorff and fractal (or box-counting) dimension. The Hausdorff dimension is based on approximating the d-dimensional volume of a set X by a covering of a finite number of balls with radius smaller than ɛ, { µ(x, d, ɛ) = inf ri d : r i ɛ and X i i } B(x i, r i ), where B(y, s) is a ball centred at y and with radius s. The d-dimensional Hausdorff measure is then defined as m(x, d) = lim ɛ 0 µ(x, d, ɛ), and the Hausdorff dimension of X, d H (X), is (essentially) the value of d for which m(x, d) is finite, d H (X) = inf {d : m(x, d) = 0}. d>0 One has (Hurewicz & Wallman, 1941) dim X d H (X), (3.1) 6

7 and there are examples of sets in IR N which have dim X = 0 but d H (X) = N. In the definition of the fractal dimension, a stronger measure still, all the balls in the covering are required to have the same radius. Given ɛ > 0, let N(ɛ, X) denote the number of balls of radius ɛ necessary to cover X. Then the fractal dimension d F (X) is defined as d F (X) = lim inf ɛ 0 log N(ɛ, X). log(1/ɛ) It is straightforward to show that d H (X) d F (X), and in general this inequality is strict: there are sets with d H (X) = 0 but d F (X) = (Eden et al., 1994). Mallet-Paret (1976) and Mañé (1981) showed that a large class of compact invariant sets will have finite fractal dimension, and reliable methods of estimating the dimensions of the globally attracting sets for given equations have been developed by Douady & Oesterlé (1980), Constantin et al. (1985), Doering & Gibbon (1996) (a nice exposition) and Hunt (1996), the last three expressed elegantly in terms of the Lyapunov exponents of the flow. These results have been applied to yield finite-dimensional attractors for all the above examples (see Temam, 1988) 4. INERTIAL MANIFOLDS For some of the equations mentioned, most interestingly the Kuramoto-Sivashinsky (Temam, 1988; Foias et al., 1988a; Robinson, 1994; Temam & Wang, 1994), Ginzburg- Landau (Temam, 1988), and reaction-diffusion equations in space dimension 1 (Temam, 1988) (and some special domains in dimension 2 and 3, Mallet-Paret & Sell (1988)), one can show that there exists a finite-dimensional system that reproduces the long-term dynamics, by proving the existence of an inertial manifold. These have been generally studied for systems of the form du/dt + Au = f(u), (4.1) 7

8 where A is a positive, linear, self-adjoint operator with compact inverse, and f is a Lipschitz function from D(A α ) (the domain of A α ) into H, for some 0 α < 1. Since A has a compact inverse, there is a set of orthonormal eigenfunctions {w n } of A with corresponding eigenvectors λ n, which one can order such that Aw n = λ n w n λ n+1 λ n λ n, see Renardy & Rogers (1992), for example. One can define the finite-dimensional projection operators P n and their orthogonal complements Q n by P n u = n (u, w j )w j Q n u = (u, w j )w j, 1 n+1 where (, ) is the scalar product in H. An inertial manifold M is a finite-dimensional, positively invariant Lipschitz smooth manifold which exponentially attracts all trajectories (Foias et al., 1988b), so that S(t)M M dist(s(t)u 0, M) C(u 0 )e kt. All current existence proofs give the manifold as a Lipschitz (or smoother) graph over one of the finite-dimensional subspaces P n H, i.e. M = {p + φ(p) : p P n H}. (4.2) Restricting the flow from (4.1) to the manifold given by (4.2) immediately yields the set of ordinary differential equations for p = P u, ṗ + Ap = P f(p + φ(p)). (4.3) Since p P n H IR n and φ is Lipschitz, it follows that (4.3) has unique solutions (see Hartman (1964), chapter II, theorem 1.1, for example). Clearly the solutions of (4.3) on P n A are precisely those projected down from A, i.e. p(t) = P n S(t)[p(0) + φ(p(0))], 8

9 and since M is an invariant manifold in H, P n A is the global attractor for (4.3). This is exactly a system of the kind specified in (2.3), since Pn 1 x is given by x + φ(x). However, there are two problems with the inertial manifold approach. Outstandingly, the conditions known to be sufficient to prove the existence of such an object are restrictive, and although satisfied for the examples above, there are many situations in which one can prove the existence of a finite-dimensional global attractor but not (at present) of an inertial manifold - of greatest interest, perhaps, are the 2D Navier Stokes equations. Secondly, the dimension of the inertial manifold, and hence of the differential system (4.3) can be much greater than that of the attractor. For example, for the Kuramoto-Sivashinsky equation u t + u xxxx + u xx + uu x = 0 u(x + L, t) = u(x, t) the best estimate of the dimension of the attractor is d F (X) L (Temam, 1988; Collet et al., 1993), whereas the best estimate of the dimension of the inertial manifold is dim M L 1.64 (ln L) 0.2 (Temam & Wang, 1994). It is thus of interest, even for systems which possess an inertial manifold, to try to produce a system of ODEs of dimension comparable to that of the global attractor. The method adopted in this paper is to work directly with the global attractor, and to embed the dynamics on this set into a discrete finite-dimensional system. (Note that since Eden et al. (1994) show that it is often possible to find an exponential attractor with a dimension comparable to that of the attractor, their programme is similar to that adopted here - for a comparison with their approach and that of inertial manifolds, see chapter 9 of their monograph). 5. ASYMPTOTIC DYNAMICS AND THE GLOBAL ATTRACTOR If the aim is to embed the dynamics on the attractor only (unlike the inertial manifold which includes some of the transient behaviour too), it is necessary to know that the 9

10 asymptotic behaviour of solutions of the equation (2.1) is to some extent determined by the dynamics on the global attractor. This is the case in that the solutions follow trajectories on A more and more closely for longer and longer times, as is stated more formally in the following proposition. Proposition 5.1. (Langa & Robinson, 1999). Given a solution u(t) of (2.1), there exists a sequence of errors {ɛ n }, ɛ n > 0 ɛ n 0 as n, a sequence of times {t n }, t n+1 > t n 0 t n+1 t n as n, and a sequence of points {v n }, v n A, such that u(t) S(t t n )v n ɛ n for all t n t t n+1. Furthermore the jumps decrease to zero as n. v n+1 S(t n+1 t n )v n The simple proof relies soley on the attracting properties of A and the continuity of solutions of the equation with respect to initial conditions. This result guarantees that an understanding of the dynamics on the attractor provides an understanding of the dynamics of the full equation. (Note, however, that it does not guarantee that a trajectory of the PDE will be asymptotic to some particular trajectory on the attractor - indeed, for this some stricter conditions are necessary (see Foias et al. (1989) or Robinson (1996) for inertial manifolds, and Langa & Robinson (1999) for more general attracting sets).) 10

11 6. EMBEDDING FINITE-DIMENSIONAL SETS IN IR D Although the global attractor is finite dimensional, it is naturally embedded in the infinite dimensional Hilbert space H. The first step towards obtaining a finite dimensional system is to find a well-defined way in which to embed the attractor into a finite-dimensional space, IR D for some sufficiently large D. It is a classical result (see Hurewicz & Wallman (1941), page 57) that a set with dim X = d can be embedded in IR 2d+1 (and in fact the set of such projections form a dense G δ in the set of all maps from X IR 2d+1 with the supremum norm). However, the form of this embedding and the properties of its inverse are not known. Because of inequality (3.1), any set with finite Hausdorff dimension can thus be embedded in IR [2dH(X)+1], by some map ϕ (here, [k], denotes the smallest integer greater than or equal to k). In some cases it may be more convenient to work with a distinguished set of embeddings, namely orthogonal projections, and the topological results of section 8 will also be proved for this more restrictive case. Although the Hausdorff dimension has nice mathematical properties (See Falconer (1990) for more details), one can construct subsets of an infinite-dimensional Hilbert space with finite Hausdorff dimension that cannot be embedded by an orthogonal projection into IR N for any N (see Eden et al. (1994) and Kan in Sauer et al. (1991)). Thus the stronger notion of fractal dimension will be more useful in what follows. In 1981 Mañé showed that most projections of a set with a finite fractal dimension d onto subspaces of dimension [2d+1] are injective. This result has recently been improved by Foias & Olson (1996) to guarantee that the inverse of the projection is a Hölder continuous function (a result previously proved for H = IR M by Ben-Artzi et al. (1993) and Eden et al. (1994)). 11

12 Theorem 6.1. (Mañé; Foias & Olson). Let H be a Hilbert space, A a compact subset of H with d F (A) d, and P 0 a projection of rank [2d + 1]. Then, for every δ > 0 there exists an orthogonal projection P = P (δ) (also of rank [2d + 1]) and θ = θ(δ) such that P is injective on A, P P 0 δ, and P 1 x P 1 y C x y θ x, y P A. Using this result and a standard extension theorem (e.g. the corollary of theorem VI.3 in Stein (1970), or theorem 11.3 in Wells & Williams (1975)) it is immediate that the attractor A lies within a finite-dimensional Hölder manifold in H. One can go further and use the density of the projections to construct approximate inertial manifolds of arbitrary accuracy provided the dimension is large enough (see Robinson (1997) for details). Ideally, one would now use this theorem to construct a dynamical system on IR [2d+1] which has P A as an attractor, and on which the dynamics coincide with those projected down from A. In what follows, P A will be denoted by X. To give an indication of the problems with this approach, first make the additional assumption that the nonlinear term F(u) from (2.1) is Hölder continuous from H into H on the attractor, so that F(u) F(v) D u v α u, v A, (6.1) for some D > 0, 0 < α 1. This certainly follows, for example (cf. Eden et al., 1994), when F(u) = Au + f(u), 12

13 as in (4.1), provided that the attractor is bounded in D(A 1+r ). In this case F(u) F(v) c Au Av c u v r/(1+r) A 1+r (u v) 1/(1+r) K u v r/(1+r), for u, v A, using a standard interpolation inequality (e.g. Temam, 1988). In general the projected vector field on X is given by ẋ = f(x) = P F(P 1 x) x X, (6.2) and when (6.1) holds f is Hölder continuous: f(x) f(y) F(P 1 x) F(P 1 y) D P 1 x P 1 y α (6.3) CD x y αθ. Since standard uniqueness results for ordinary differential equations require the vector field to be at least locally Lipschitz (see Hartman, 1964), uniqueness for the solutions of (6.2) is not guaranteed, even on the projection of the attractor. If one were not to be restricted to projections, it would be reasonable to hope that by considering all possible C 1 maps h from H into IR D one could find at least one map which was injective on A and had a Lipschitz continuous inverse. Although Sauer et al. (1991) show that most such C 1 maps are injective on A, there is an example due to Movahedi-Lankarani (1992) of a compact set with finite fractal dimension which cannot be embedded in any finite-dimensional space using a bi-lipschitz function (where h and h 1 are Lipschitz). Even if one could find such a function, the continuity of F on A cannot be guaranteed to be better than Hölder. Thus one is forced in general to work with a flow on X whose vector field is not Lipschitz continuous. The bad properties of f on X are a constant problem if one tries to extend f to a vector field on IR D in such a way that the solutions are unique. 13

14 7. TWO PARTIAL RESULTS This section contains two results that go some way toward obtaining a system of ordinary differential equations as discussed in section 2. Section 7.1 reviews the approach of Eden et al. (1994), which obtains a system of ODEs with the required properties, but whose vector field is not continuous. Thus the solutions are not unique, and one does not have continuous dependence on initial conditions. In section 7.2 it is shown that the solutions on A can be approximately recovered from a smooth system of ODEs on IR A non-smooth system system In their monograph on exponential attractors, Eden et al. (1994) obtain a differential ẋ = F (x) = α(x ν(x)) + f(ν(x)) (7.1) for which X is an exponential attractor (attracting at rate α), and on which the dynamics agree with those projected from A. (In fact Eden et al. consider the more general case of exponential attractors and not just attractors. It will be shown in section 8 that the nice topological properties of global attractors provide some hope that a more regular construction might be possible.) The function f is defined for y X by f(y) = P F(P 1 y), and ν(x) maps any point x IR D to one of the points y X such that x y = dist(x, X). Thus the first term of (7.1) forces X to be attracting, and the second term reproduces the dynamics on A. In general such a function ν is only continuous at points where the closest point ν(x) is unique. Thus solutions of (7.1) have to be defined as solutions of the corresponding 14

15 integral equation x(t) = x 0 + t 0 F (x(s)) ds, and these cannot be guaranteed to be unique, even on X. Since equations without uniqueness can exhibit all manner of pathologies (see Rubel (1981) for a particularly all-encompassing example), this is certainly not an ideal solution. 7.2 An approximate solution in IR 3 If one drops the requirement of following the trajectories on A exactly and for all time, it is possible to reproduce the dynamics approximately in a system of ODEs on IR 3. Note that the assumption in the theorem that F(u) is bounded on A follows if A is bounded in D(A) (cf. the argument near (6.1)). Theorem 7.1. Assume that F(u) is bounded and continuous on A. Then given T > 0 and ɛ > 0, there exist functions g : IR 3 IR 3 and Φ : IR 3 H, Lipschitz and Hölder continuous respectively, such that given any solution u(t) A, there exists a solution x(t) of ẋ = g(x) with Φ(x(t)) u(t) ɛ for all 0 t T. (7.2) Proof. By continuous dependence on initial conditions (2.2) and the compactness of A, given ɛ and T as in the statement, there exists an η > 0 such that u(0) v(0) η u(t) v(t) ɛ/2 for all 0 t T. 15

16 Now take a cover of A by balls of radius η. Since A is compact there exists a finite sub-cover. Denote the centres of the balls in this finite sub-cover by {u k }, where 0 k κ <. Then given u A, there is a u k with u u k η. (7.3) Now consider the trajectory through u k between t = 0 and t = T, τ k = S(t)u k. 0 t T Note that d F (τ k ) = 1, since the boundedness of F(u) on A implies that S(t)u k is Lipschitz in t. Take the union of these sets over k, T = k τ k, and note that if, for some k j, τ k τ j, then τ k τ j = S(t)v, 0 t T where T T 2T, and v is either u k or u j, since solutions on A are unique in both forwards and backwards time. Thus T consists of disjoint portions of trajectories τ j = S(t)ũ j, 0 t T j with ũ j = u k for some k and T T j κt : T = j τ j. 16

17 Since T is the finite union of sets with fractal dimension 1, d F (T ) = 1. Thus theorem 6.1 guarantees the existence of an orthogonal projection P : H IR 3, injective on T and with Hölder continuous inverse on P T. Using, for example, the extension theorem in Stein (1970, corollary of theorem VI.3), extend the Hölder function P 1 : P T H to a Hölder function Φ defined on the whole of IR 3, and choose ζ such that x y ζ Φ(x) Φ(y) ɛ/2. (7.4) Now it is necessary to construct a Lipschitz function g, such that the equation ẋ = g(x) has trajectories close to each projected solution x j (t) = P S(t)ũ j. Since each P τ j is compact and P τ j P τ k = for j k, there exists a σ such that x y 3σ x P τ j, y P τ k j k. The problem thus reduces to defining g on a σ-neighbourhood of each P τ j, N j = {x + y : x P τ j, y σ}. Clearly x y σ x N j, y N k j k. (7.5) Such a Lipschitz function can easily be constructed, by making small alterations to the trajectory x j (t) to produce a smooth trajectory x a j (t) with x a j (t) x j (t) ζ, 17

18 and defining g(x a j (t)) = ẋa j (t). That one can make adjustments such that g is Lipschitz on the trajectory x a j follows from the argument in Robinson (1999). Since g is Lipschitz on each trajectory x a j, there is a constant K 1 such that if x and y lie in the same x a j (for any j) g(x) g(y) K 1 x y x, y x a j. If g = max j max g(x), x x a j then if x x a j and y xa k with j k, g(x) g(y) 2g 2g σ x y K 2 x y, using (7.5). Thus g is Lipschitz on the closed set x a j, j with Lipschitz constant max(k 1, K 2 ); hence g can be extended to a function on the whole of IR 3 that is globally Lipschitz, using Stein s theorem as above. Note that ẋ = g(x) therefore has unique solutions. Finally, to prove the theorem, take a solution u(t) A. Then by (7.3), there is a u k with u k u(0) η, and so a solution v(t) = S(t)u k in T which satisfies v(t) u(t) ɛ/2 for all 0 t T. (7.6) Since there is a solution of ẋ = g(x) 18

19 which satisfies x(t) P v(t) ζ, for 0 t T, then for this solution Φ(x(t)) v(t) ɛ/2 by (7.4), and so, using (7.6), (7.2) holds. The implications of this result for both ordinary and partial differential equations are discussed in detail in Robinson (1998). 8. TOPOLOGICAL PROPERTIES OF THE GLOBAL ATTRACTOR In the construction of section 7.1 due to Eden et al. (1994) the system of ordinary differential equations obtained is not continuous, and indeed there may be topological obstructions to constructing a smooth system of equations which have an arbitrary set as an attractor. For example, a circle (or any image of an N-sphere) cannot be the attractor of a smooth flow, since this would involve a tear in the flow. Although it appears at first to be likely, it is unfortunately not true that any global attractor must be a retract of the phase space. Recall that A is a retract of H if one can find a continuous map r : H A such that r = id on A. Such a map could replace the closest point map ν(x) used in the construction above, and would imply that global attractors were simply connected (and, more generally, connected in dimension N, see Dold (1980) or Kuratowski (1968)). However, Günther & Segal (1993) comment that the pseudoarc (see Hocking & Young (1961), page 142) is the global attractor for a flow, and this set is not even arcwise connected, so it cannot be a retract. 19

20 All that is known in general (Temam, 1988; Hale, 1988) is that global attractors are connected. (For hyperbolicity conditions on the flow guaranteeing that the attractor is a retract, see Langa & Robinson (1999).) What one can show in finite-dimensional systems of ordinary differential equations, is that the global attractor X must satisfy IR N \ X IR N \ {0}, (8.1) or in other words, the set X must be pointlike (precisely (8.1)). This is a classical result, due to Bhatia & Szegö (1967, theorem ). Theorem 8.1. If X is the global attractor of a dynamical system on IR N then X is pointlike and connected. Garay (1991, theorem 2.7), Günther & Segal (1993), and Sanjurjo (1994) show that any connected pointlike set can be made the attractor for some dynamical system (but not necessarily a set of ODEs) on IR N. With an additional topological condition, that X is embedded in IR N in standard position (see definition 3.1 of Chapman (1972b)), Günther (1996) shows that one can construct a smooth (C r for any 0 r < ) differential equation ẋ = φ(x) x IR N, (8.2) where φ = 0 φ = 0 on X and which has X as a global attractor. φ > 0 φ 0 on IR N \ X, Thus to guarantee that an embedding ϕ(a) of A in IR N is to be the attractor of a finite-dimensional system of ordinary differential equations, it is sufficient that X = ϕ(a) 20

21 satisfy (8.1), be connected, and be in standard position. This is not a priori obvious, and it is necessary to proceed via several topological notions, the most important of which is shape theory (Borsuk, 1970b; Dydak & Segal, 1978). This theory arose (Borsuk, 1970a) in an attempt to understand and classify the global homotopy properties of arbitrary compacta, extending results valid for nice compact sets (the absolute neighbourhood retracts of Borsuk (1967)). The potential of this theory in the study of dynamical systems was first noticed by Hastings (1978 & 1979), and has been most usefully developed (in the context of this article) by Garay (1991), who has shown that, in infinite dimensions, a globally attracting set must have the shape of a point, Sh(X) = Sh({0}) (see also Bogatyj & Gutsu (1989), Sanjurjo (1994 & 1995)). Garay also shows, conversely, that any set with the shape of a point can be made the attractor for some infinite-dimensional dynamical system. Theorem 8.2. (Garay). (i) A global attractor has the shape of a point. (ii) Any set with the shape of a point can be the global attractor of some infinitedimensional dynamical system. Proof. (Summary of (i)). Since the domain of attraction of X is all of H, then X is contractible in its every neighbourhood in H (Garay, 1991, theorem 2.1). That X is contractible in its every neighbourhood in H implies that X is a fundamental absolute retract (FAR) (Borsuk, 1969, corollary 9.5). A compact set is an FAR iff it has the shape of a point (Borsuk, 1970a, theorem 7.1). So X has the shape of a point. In essence, as was shown by Chapman (1972a), the notion of shape is equivalent to studying the complement of the set. In the following theorem, Q denotes the Hilbert cube [0, 1], and ω its pseudo-interior (0, 1). 21

22 Theorem 8.3. (Chapman). Let X, Y be two compact sets, and ˆX and Ŷ homeomorphic images of X and Y in ω. Then Sh(X) = Sh(Y ) iff Q \ ˆX Q \ Ŷ. The condition in the theorem is stronger than just an embedding of X and Y into a Hilbert space H with H \ ˆX H \ Ŷ ; indeed, in infinite dimensions, any compact set is pointlike (Klee (1956), see below), i.e. satisfies H \ A H \ {0}. This condition alone does not imply that the set A is connected, but (as remarked above) this follows from the standard theory of global attractors (and indeed, McCoy (1973) shows that any set with the shape of a point is connected). (In fact Klee (1956) shows that in infinite dimensional spaces, H \{0} is homeomorphic to H, and thus pointlike sets are those which satisfy H \ A H. Such sets are termed negligible, and it is shown in the same paper that any compact set is negligible, and thus pointlike.) When A is finite-dimensional, it is the properties of embeddings of A into finite dimensional Euclidean spaces that are the main concern here. The first result in this direction was due, again, to Chapman (1972b). The dimension used in the theorem is the topological dimension. Theorem 8.4. (Chapman). Let X, Y be two finite-dimensional compact sets, with Sh(X) = Sh(Y ). Then, if N max (dim X, dim Y ) there exist embeddings ˆX, Ŷ of X and Y in IR N such that ˆX and Ŷ are in standard position, and IR N \ ˆX IR N \ Ŷ (8.3) where the homeomorphism is piecewise linear. There is a problem with this result, in that it only asserts the existence of some 22

23 embedding that ensures (8.3). If one wishes to be able to apply such a result to projections, some further details are needed. These are provided by Geoghegan & Summerhill (1973), who give explicit conditions for the embedding of X and Y into IR N to be sufficiently nice for the homeomorphism (8.3) to hold (but in may not be piecewise linear). Essentially one needs IR N \ X and IR N \ Y to be locally simply connected (Z is locally simply connected if, given a δ > 0 there is an ɛ > 0 such that any map of S 1 into a subset of Z with diameter ɛ can be extended to a map of D 1 (the disc) into a subset of Z with diameter δ). This property holds for any embedding which is pseudo-polyhedral (their definition 3.0), and this includes piecewise linear embeddings. Since orthogonal projections are linear, they are covered by this result. Furthermore, it is easy to show that any set which is embedded in IR N by a pseudo-polyhedral embedding is in standard position in IR N, and so one can obtain a piecewise linear homeomorphism as in theorem 8.4. In fact Geoghegan & Summerhill (1973) prove that (their theorem 3.3) Theorem 8.5. (Geoghegan & Summerhill). The set of all pseudo-polyhedral embeddings is a dense G δ in the set of all embeddings of A into IR N (with the supremum norm) provided N 2 +2 dim A. In particular piecewise linear embeddings (which include the orthogonal projections) are pseudo-polyhedral. Thus any pseudo-polyhedral embedding ϕ (including orthogonal projections) of a finite- dimensional global attractor into IR N, with N [2d + 2], will satisfy the conditions required by Günther (1995) to produce the set of ordinary differential equations (8.2) with X = ϕ(a) as an attractor. Denote the solution operator for this system, ẋ = φ, (8.4) by S φ (t), so that x(t; x 0 ) = S φ (t)x 0. 23

24 This will be useful in what follows. (Note that if one wishes to try to construct a differential system, one would hope to be able to use the nicer properties of orthogonal projections to help. See, for example, the construction of Eden et al. (1994), where they ensure that P is injective on a larger set that avoids introducing spurious steady solutions on P A.) As a corollary, it is worth formulating in a formal way the statement that the set of finite-dimensional attractors of dissipative PDEs is the same as the set of attractors of finite sets of ODEs. Here, of course, the attractors are only being viewed as sets (independent of the dynamics), and the same means homeomorphic. Corollary 8.6. If A is the global attractor of a dynamical system on a Hilbert space H, with d H (A) <, then there exists a finite-dimensional ordinary differential equation with a global attractor X A. Conversely, if X is the global attractor for a finite system of ODEs, then there exists an infinite-dimensional dynamical system on H with a global attractor A X, and with d H (A) <. Proof. If A has d H (A) d then there is a pseudo-polyhedral embedding ϕ into IR [2d+1] such that X = ϕ(a) is the global attractor in IR [2d+2] for the ODE system (8.4). Conversely, if X is a global attractor in IR N then X has the shape of a point, and any image ψ(x) in H, A, has the shape of a point (see McCoy, 1973). The result of Garay (theorem 8.2 (ii), above) then guarantees an infinite-dimensional dynamical system on H with ˆX as the global attractor, and taking ψ to be Lipschitz ensures that d H (A) N. 9. A FINITE-DIMENSIONAL DISCRETE DYNAMICAL SYSTEM Although one would ideally construct a set of ordinary differential equations on IR N with X as an attractor, the fact that f is only Hölder continuous on X has proved an 24

25 obstruction to all attempts to extend f to a vector field on the whole of IR N with unique solutions. Instead, theorem 9.2 guarantees the existence of a discrete dynamical system on IR N which reproduces the time T -map on A, and has a global attractor arbitrarily close to X. Once again topology is important, and what enables the construction of a homeomorphism on IR N rather than just a continuous (but not necessarily invertible map) is a result used in the argument of Geoghegan & Summerhill (1973), concerning the extension of homeomorphisms. In the statement of the result, N(X, ɛ) denotes the ɛ-neighbourhood of X, N(X, ɛ) = {x + y : x X, y ɛ}. Theorem 9.1. (Geoghegan & Summerhill). Let A be a compact set with dim A <, and ϕ a pseudo-polyhedral embedding of A into IR N (N dim A). If X = ϕ(a) and g : X X is a homeomorphism such that g(x) x ɛ then there exists a homeomorphism h : IR N IR N such that h X = g, h(x) x ɛ, and h = id on IR N \ N(X, ɛ). So h extends f in such a way that h moves points no more than f and leaves points outside a neighbourhood of X unchanged (actually, Geoghegan & Summerhill show further that there is an isotopy i : IR N [0, 1] such that i 0 (x) = id and i 1 (x) = h(x), which is tantalisingly close to a flow agreeing with that projected on the attractor). Note that one 25

26 can always extend a homeomorphism f to a continuous function h (via Stein s theorem in VI, section 2.2 (1970), for example), but to obtain a homeomorphism requires either topological properties of X or the guarantee of more smoothness of f X. By combining this theorem with the flow S φ (t) from (8.4), one obtains a homeomorphism which reproduces the time T map on A, and has an attractor within an ɛ neighbourhood of X. Theorem 9.2. Let A have finite Hausdorff dimension, d H (A) d. Then given T > 0 and ɛ > 0, for any N [2d + 2] there exists an embedding ϕ : A IR N and homeomorphism f from IR N into itself, such that X = ϕ(a) is an invariant set for f and f X = ϕ S(T ) ϕ 1. Furthermore, the discrete dynamical system generated by f, {f n }, has a global attractor X f which satisfies X X f N(X, ɛ). Proof. First choose a pseudo-polyhedral embedding ϕ of A into IR N by theorem 8.5, and set X = ϕ(a). Now given ɛ, use the uniform continuity of ϕ on A to find an η > 0 such that if u v η u, v A then ϕ(u) ϕ(v) ɛ. Now choose an integer m such that S(T/m)u u η, u A and consider the homeomorphism g : X X given by g = ϕ S(T/m) ϕ 1. 26

27 Then since m was chosen such that S(T/m) [ ϕ 1 (x) ] ϕ 1 (x) η x X, the choice of η gives ϕ S(T/m) ϕ 1 (x) x ɛ x X, which is g(x) x ɛ for all x X. Thus g satisfies the conditions of theorem 9.1, and so can be extended to a homeomorphism h which is equal to g on X and is the identity outside N(X, ɛ). Clearly the map G = h m satisfies G = ϕ S(T ) ϕ 1 on X, and is still the identity outside N(X, ɛ). Now define f as the composition of G with with S φ (T ) (in fact, any argument would do as well as T ), which is another homeomorphism from IR N to itself: f = S φ (T ) G. Then as S φ (T ) is the identity on X, f X = ϕ S(T ) ϕ 1, and since G is the identity on IR N \ N(X, ɛ), f IR N \N(X,ɛ) = S φ (T ), which ensures that N(X, ɛ) is an absorbing set for the dynamical system {f n }. Thus the global attractor X f for this dynamical system lies within N(X, ɛ) (Hale, 1988). That X is invariant and that X X f are obvious. Note that if one wishes to use an orthogonal projection P in the theorem, replacing d H (A) by d F (A) enables one to use theorem 6.1 and theorem 8.5 to ensure that P is 27

28 a pseudo-polyhedral embedding. The argument is then the same, producing a discrete dynamical system on IR N with N [2d F (A) + 2]. CONCLUSION By using results which characterise the topology of global attractors, and topological embedding theorems, it has been shown that all topological information about global attractors can be obtained from finite-dimensional sets of ordinary differential equations (corollary 8.7). Furthermore, the dynamics on the global attractor can be embedded into a discrete dynamical system which has a global attractor arbitrarily close to the true attractor (theorem 9.2). This has important implications for the validity of computational studies of partial differential equations. Finally, a conjecture. Reformulated using discrete instead of continuous time, the problem appears to be much simplified. It seems reasonable to expect, therefore, that a more refined construction could produce a discrete dynamical system as in theorem 9.2, but whose global attractor is precisely X. ACKNOWLEDGMENTS Many thanks to José Langa, Eric Olson, and Ciprian Foias, with whom I have had very many interesting and stimulating conversations while working on these problems, which would have been much longer coming to fruition without their help. Thanks also to Trinity College, Cambridge, for all their financial support, and to Roger Temam and the Institute of Scientific Computing & Applied Mathematics for their hospitality. 28

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