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Attractors in coherent systems of differential equations

David Angeli, Dip. di Sistemi e Informatica, University of FirenzeMorris W. Hirsch, Dep. of Mathematics, University of California Berkeley

Eduardo D. Sontag, Dep. of Mathematics, Rutgers University

Abstract

Attractors of cooperative dynamical systems are particularly simple; for example,a nontrivial periodic orbit cannot be an attractor. This paper provides characterizationsof attractors for the wider class ofcoherentsystems, defined by the property that nodirected feedback loops are negative. Several new results for cooperative systems areobtained in the process.

Contents

Introduction 2Statement of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Structure of proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Local semiflows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Attractors and attracting sets . . . . . . . . . . . . . . . . . . . . . . . .. . . . 6Ordered spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Cascades 7

Graphs 8Graphs and systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Spin assignments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Monotone dynamics 12

Proofs of the main theorems 16

Appendix: Notes on the development of the concept “attractor” 17

1

Introduction

We consider differential equations

dxdt= F(x), x ∈ X, t ≥ 0, (1)

whereX ⊂ Rn is convex, its interior is dense inX, and the vector fieldF : X→ Rn extendsto aC1 vector field on an open set. The maximally defined solutionst 7→ Φt(a), t ≥ 0, a ∈ Xgenerate the local semiflowΦ := Φtt∈R+ . We refer toF (or (F,X,Rn), or (F,X,Rn,Φ)) asasystem. Dynamical notions are applied interchangeably toF andΦ.

Many biological situations are modeled bycooperativesystems:∂F j

∂xi≥ 0 if j , i. The

biological interpretation is that an increase of speciesi tends to increase the populationgrowth rate of every other speciesj. In this caseΦ is monotone, meaning it preserves thevector ordering. This causes the crude dynamics of a cooperative system to be compara-tively simple; for example, there are no attracting cycles and every orbit is nowhere dense(Hadeler & Glas [13], Hirsch [16]).

Here we show that some of the dynamical advantages of cooperative systems extendto systems having a significantly weaker property:F is coherent(another name ispositivefeedback system) if wheneveri0, . . . , iν, ν ∈ 1, . . . , n are such that

iν = i0, ik−1 , ik and∂Fik−1

∂xik

. 0 (1≤ k ≤ ν)

then,∂Fik−1

∂xik

(x) does not change sign (1≤ k ≤ ν)

and∂Fi0

∂xi1

(x) · · ·∂Fiν−1

∂xiν

(x) ≥ 0, (∀ x ∈ X). (2)

Our chief combinatorial result, Theorem 10, shows that by permuting the variablesxi

and changing the signs of some of them, any coherent system can be transformed into adynamically equivalent system (F,X,Rn,Φ) with the following properties:

• F is not merely coherent, it has the stronger property of beingquasicooperative: forany (i1, . . . , im) as above, each factor in the left hand side of (2) is≥ 0

• if F is not cooperative, there exists a cooperative system (F1,X1,Rn1,Φ1), 1≤ n1 < n,such that the the natural projection

Π : Rn → Rn1, (x1, . . . , xn) 7→ (x1, . . . , xn1)

mapsX ontoX1 and semiconjugatesF to F1 andΦ toΦ1:

Π F(x) = F1 Π(x), Π Φt(x) = Φ1t Π(x) if Φt(x) is defined

2

Mild geometrical conditions onX guarantee that for each equilibriump of F1, the restric-tion of F to Xp := X ∩ Π−1(p) is equivalent to a quasicooperative system (Fp, Xp,R

n−n1).This is the basis for inductive proofs of our main results.

We turn to our main topic, attractors. Anattractor for F is a nonempty invariant con-tinuumA ⊂ X that uniformly attracts all points in some neighborhood ofA. If the attractionis not necessarily uniform we talk instead of anattracting set. Three types of attractorsAhave received special attention:

Point attractors: A is single point, necessarily an equilibrium.

Periodic attractors: A is a cycle, i.e., a periodic orbit that is not an equilibrium.

Strange attractors, often called “chaotic”. This somewhat vague term signifiesthatA isneither an equilibrium nor a cycle, and usually thatA is topologically transitive and exhibits“sensitive dependence on initial conditions”. Some authors also require that periodic orbitsbe dense inA.

This paper is motivated by the question: What kind of nonequilibrium attractorsA canexist in coherent systems? Theorem 1 shows thatA cannot be topologically transitive;Theorems 2 and 3 give further dynamical information. Other results apply to more generalmonotone local semiflows.

Statement of results

A set isfinitely transitivefor a system (or a local semiflow) if it is the union of the omegalimit sets of finitely many of its points.

Theorem 1 A finitely transitive attracting set A for a system(F,X,Rn) reduces to an equi-librium in the following cases:

(i) F is coherent, and X is open inRn or relatively open in a coordinate half-space

(ii) F is quasicooperative, and every point of A is strongly accessible in X from above, orevery point of A is strongly accessible in X from below

(iii) F is cooperative, and each point of A is strongly accessible in X from above or below

A stronger conclusion, Theorem 16, holds for cooperative systems.The following result requires no additional geometrical conditions onX:

Theorem 2 If (F,X,Rn), n ≥ 2 is a coherent system, every orbit is nowhere dense.

Conjecture In a coherent system with n> 1, every orbit closure has measure zero.Evenfor cooperative systems this is known only forn = 2.

An attractor isglobal if it attracts all points ofX. An equilibrium isglobally asymptot-ically stableif it is the global attractor. The following theorem needsX to be open:

3

Theorem 3 Let (F,X,Rn) be a coherent system with X open inRn. Assume there existsa global attractor A. Then there exists an equilibrium, and if it is unique it is globallyasymptotically stable.

Proposition 14 extends a basic result previously known onlyfor strongly order-preservinglocal semiflows. The development of the concept “attractor”is discussed in the Appendix.

Motivations

A coherent system is one whose interaction graph (defined below) has no directed negativeloops. A more restrictive condition, for graphs that are notnecessarily strongly connected,is the requirement that the graph has noundirected negative loops: in that case, one mayalways perform an elementary change of variables (defined below) that transforms sucha system into a cooperative one. In a classical and often-quoted 1981 paper, R. Thomasconjectured that coherent systems do not have any periodic attractors: “the presence of atleast one negative loop in the logical structure appears as anecessary (but not sufficient)condition for a permanent periodic behavior” [48]. It has often been claimed (see e.g. [30])that Thomas’ conjecture was settled in [41, 12]. However, these references only dealt withthe more restricted monotone case. Theorem 1 in this paper settles the question. We referthe reader to [42] for further comments on the relevance of these concepts to molecularsystems biology, and to [43] for numerical simulations which suggest that systems thatare “close” to having the coherence property might have, in some statistical sense, simplerattractors.

Structure of proofs

The proofs of Theorems 1, 2 and 3 have a common pattern which wenow discuss. LetTstand for one of these theorems. It is proved first for a cooperative system, which includesthe casen = 1. The proof proceeds by induction onn. A coherent system which is notcooperative is transformed, by permuting and changing signs of variables, to a system(F,X,Rn) having the following properties:

• F is quasicooperative

• there is a system (F1,X1,Rn1) with n1 < n, such that the natural projectionΠ : Rn →

Rn1 satisfies

Π(X) = X1, Π F(x) = F Π(x), (x ∈ X) (3)

• F1 is cooperative

It follows thatΠ semiconjugates the local semiflowΦ of F to the local semiflowΦ1 of F1:

Π Φt(x) = Φ1t Π(x) if Φt(x) is defined (4)

4

We summarize this by saying thatΠ : (F,X,Rn) ։ (F1,X1,Rn1) (or Π : F ։ F1) is acascade. We also allow thetrivial cascade, for whichF = F1.

For each equilibriump of F1 the affine subspaceEp := Π−1(p) is a coset of the kernelof Π. Thecanonical chart

Tp : Ep ≈ Rn−n1, (x1, . . . , xn) 7→ (xn1+1, . . . , xn) (5)

is an affine automorphism.The vector fieldF, being tangent toEp alongXp, restricts to a vector fieldFp in Xp :=

Xp := X∩Ep, andΦ restricts to a local semiflowΦp in Xp. The hypothesis ofT will ensurethat the relative interior ofXp in Ep is dense inXp. The canonical chart converts thefibresystem(Fp,Xp,Ep) into a system (Fp, Xp,R

n−n1).We identify each fibre system(Fp,Xp,Ep) with (F, Xp,R

n−n1) by means of the canonicalchart. ThusFp has an interaction graphΓ(Fp) := Γ(Fp). We ascribe toFp the property ofbeing cooperative, quasicooperative or coherent wheneverthat property holds forFp.

TheoremT holds for the cooperative systemF1, and it holds for all fibre systems bythe inductive assumption. The induction is completed by showing that this impliesT alsoholds for (F,X,Rn).

There is a delicate point regarding the domains of these systems. The proofs for coop-erative systems use special properties ofX, such as every point being strongly accessiblefrom above. These properties are postulated in the hypotheses of the main theorems. Tomake the induction work, the same properties must be verifiedfor the systems obtained byelementary coordinate changes, and also for fibre systems. This means that the class ofdomainsX referred to in the theorems must be preserved by permuting and changing signsof variables, and by intersectingX with the affine subspacesEp. For this reasonX is usuallyrequired to be an open set inRn or a relatively open subset of a coordinate halfspace.

Local semiflows

A local semiflowΦ in a metrizable spaceZ is a collectionΦ = Φtt∈R+ of continuous mapsΦt : Dt → Rt between nonempty subsets ofZ, with Dt open. The notationΦtx indicatesx ∈ Dt, absent contraindications.Φ is required to have the following properties:

• The setΩ := (t, x) ∈ R+ × Z : x ∈ Dt is an open neighborhood of0 × Z in R+ × Z,and the mapΩ→ Z, (t, x) 7→ Φtx is continuous.

• x ∈ (Φs)−1Dt =⇒ Φs Φt(x) = Φs+t(x)

• Φ0 is the identity map ofZ.

We also say that (Φ,Z) is a local semiflow. WhenΦ is obtained by solving Equation (1)each mapΦt is a homeomorphism, but this is not assumed for general localsemiflows.

5

Theorbit andomega limit setof x are respectively

γ(x) := Φtx: x ∈ Dt, ω(x) :=⋂

t≥0

γ(Φtx)

p is anequilibriumif Φt p = p for all t. The set of equilibria is denoted byE(Φ), and byE(F) whenΦ is generated by the vector fieldF.

Attractors and attracting sets

We callA positively invariantfor if Φt(a) is defined and belongs toA for all t ≥ 0, a ∈ A,and invariant if in addition A is nonempty andΦt(A) = A for all t ≥ 0. We say thatAattracts xif γ(x) is compact andω(x) ⊂ A. The set of such pointsy is thebasinof A.

A is topologically transitiveif it is the omega limit set of one of its points, andfinitelytransitive if it is the union of the omega limit sets of finitely many of itspoints.

We callA attractingif it is invariant, connected and compact, and its basin is a neigh-borhood ofA. If in additionA has arbitrarily small positively invariant neighborhoods, A isanattractor.1

Ordered spaces

By anordered spacewe mean a topological spaceZ together an order relationR ⊂ Z × Zthat is topologically closed. Ifx, y ∈ Z we write:

x y andy x if ( x, y) ∈ R, x ≻ y andy ≺ x if x y, x , y (6)

Thevector orderin any subspace ofRn is defined by

u v ⇐⇒ u− v ∈ Rn+

whereRn+

denotes the the positive orthant [0,∞)n ⊂ Rn.A subset of an ordered space isunorderedif none of its points are related by≻.Every subspaceX ⊂ Z inherits an order relation fromZ. If M ⊂ Z thenx ≻ M means

x ≻ y for all y ∈ M, and similarly for the other relations in (6). Forx, y ∈ X we write

x⊲X y if x ≻ N, y ∈ IntX(N),

x⊳X y if x ≺ N, y ∈ IntX(N)

for some openN ⊂ X. Note the notational anomaly thatx⊲X y andy⊳X x are not equivalentstatements for general ordered spaces. They are equivalent, however, ifX ⊂ Rn is open andhas the vector ordering. For example, inX = R2

+we have (0, 0)⊳X (0, 1) but (0, 1) 6 ⊲X(0, 0).

1There are many definitions of “attractor” in current use, notmutually consistent. The one adopted hereis equivalent to that of Conley [10], and (for compact invariant sets) those of Hale [14] and Sell & You [38].It is analogous to the definitions for discrete-time systemsin Smale [39] and Akin [1].

6

Let X be a subset of an ordered spaceZ. We callq ∈ X strongly accessible in X fromabove (respectively, from below)if every neighborhood ofq in X contains a pointx ⊲X q(respectively,x⊳X q).2

All our results are valid whenX is an open set inRn, and some are valid for special kindsof nonopen sets, especially open subsets of acoordinate halfspaceof Rn, which means aset

x ∈ Rn : αxl ≥ cl

for some choice ofl ∈ 1, . . . , n, α ∈ ±1, (c1, . . . , cn) ∈ Rn. We rely on the following fact,whose proof is left to the reader:

Lemma 4 Assume X⊂ Rn has the vector ordering. If X is an open subset ofRn, or arelatively open subset of a coordinate halfspace, every point of X is strongly accessiblefrom above and below in X.

Note also that ifX is an open subset ofRn+, all points ofX are strongly accessible inX from

above.

Cascades

Let (F,X,Rn) and (F1,X1,Rn1) be systems with 1≤ n1 < n and assumeΠ : F ։ F1 is acascade (see (3)). This implies

∂Fi

∂xj= 0 if i ≤ n1 < j, (i, j ∈ 1, . . . , ν) (7)

and the Jacobian matrices ofF have lower triangular block decompositions of the form

F′(x) =

[

M11(x) OM21(x) M22(x)

]

(8)

whereM11(x) = (F1)′(Πx) ∈ Rn1×n1, andO stands for a matrix of zeroes. The followingdiagrams commute for eacht ≥ 0:

D(Φ1t ) R(Φ1

t )

D(Φt) R(Φt)

-Φ

1t

-Φt

?

Π?

Π

For p ∈ E(F1) let Tp : Ep ≈ Rn−n1 be the canonical chart. SetXp = Tp(Xp) and define

Fp : Xp→ Rn−n1 to be the unique vector field transformed by (Tp)−1 to Fp, that is,

2Slightly stronger properties with the same names are used inHirsch & Smith [20].

7

F(Tpx) = TpF(x), (x ∈ Xp) (9)

The local semiflows ofFp andFp are conjugate underTp. For (Fp, Xp,Rn−n1) to be a system

it is necessary and sufficient that the relative interior ofXp in Ep be dense inEp. When thisholds we call (Fp,Xp,Ep) thefibre systemoverp and identify it with (Fp, Xp,R

n−n1) by Tp,The interaction graphΓ(Fp) := Γ(Fp) is determined by the signs of the entries in the

block M22(x) in (8). The next lemma gives convenient conditions ensuring this.Consider the following conditions:

• C1(X,Rn): X is open inRn

• C2(X,Rn): X is open in a coordinate halfspace ofRn

• C3(X,Rn): X is open inRn+

• C4(X,Rn): X is a rectangle

Lemma 5 AssumeΠ : F ։ F1 a cascade as above and p∈ E(F1). Suppose p∈ E(F1),and Cd(X,Rn), is satisfied for some d∈ 1, 2, 3, 4. Then(Fp, Xp,R

n−n1) is a system, andCd(Xp,R

n−n1) holds.

Proof The verification that Cd(X,Rn) implies Cd(Xp,Rn−n1), and also that relative interior

of Xp in Ep is dense inXp, is straightforward.

Graphs

By a directed graphΓ := (VΓ,EΓ) we mean a nonempty finite setV := VΓ (the set ofvertices) together with a binary relationE := EΓ ⊂ V×V (the set ofdirected edges, usuallyreferred to simply as “edges”). We always assumeE is totally nonreflexive i.e., (i, i) < E.

An isomorphismbetween a pair of directed graphs is a bijectionf between their vertexsets such thatf × f restricts to a bijectionf∗ between their edge sets.

Our chief tool for analyzing the crude dynamics of systems (F,X,Rn) is theinteractiongraph Γ := Γ(F). This is the labeled directed graph with vertex set isV = V(Γ) :=1, . . . , n, whose set of (directed) edges is

E = E(Γ) := ( j, i) ∈ V × V : j , i and∂Fi

∂xjis not identically 0 inX

Edge (j, i) is assigned the labelh( j, i) ∈ +1,−1, θ according to the rule:

h( j, i) =

1 if ∂Fi

∂xj (x) ≥ 0 for all x ∈ X,

−1 if ∂Fi

∂xj (x) ≤ 0 for all x ∈ X,

θ otherwise

(10)

8

and is respectively called positive, negative or ambiguous. A loop ispositiveif each of itsedges is labeled+1 or−1 and the product of these labels is+1.

We define three types of graphs in increasing order of generality:

Γ is positiveif every edge is positive,

Γ is quasipositiveif every loop has only positive edges,

Γ has thepositive loop propertyif every loop is positive.

Paraphrasing some of the earlier definitions, we define corresponding types of systemsFin terms ofΓ(F):

F is cooperativeif Γ(F) is positive

F is quasicooperativeif Γ(F) is quasipositive,

F is coherentif Γ has the positive loop property

Evidentlycooperative=⇒ quasicooperative=⇒ coherent.

The term “graph” is shorthand for “finite directed graph having edges labeled in1,−1, θ.”Graphs are denoted by Greek capitalsΓ,Λ, perhaps with indices. The sets of vertices andedges ofΓ are denoted byV(Γ) andE(Γ), respectively, and the labeling function is denotedby hΓ : V(Γ)→ 1,−1, θ. Two graphsΓ,Λ areisomorphicif there there is an isomorphismf : V(Γ)→ V(Λ) between the underlying directed graphs such thathΛ f∗ = hΓ.Λ is asubgraphof Γ provided

V(Λ) ⊂ V(Γ), E(Λ) ⊂ E(Γ), hΛ = hΓ|E(Λ),

We abuse notation and denote this byΛ ⊂ Γ, saying thatΛ contained inΓ.If Λ,Λ′ are subgraphs theirgraph unionis the subgraph with vertex setV(Λ) ∪ V(Λ′)

and edge setE(Λ) ∪ E(Λ′).A pathof lengthk ∈ N+ is a sequence (u0, . . . , uk) of vertices such that (vj−1, vj) is an

edge forj = 1, . . . , k. Theconcatenationof an ordered pair (λ, µ) of paths,

λ = (u0, . . . , uk), µ = (uk, . . . , uk+l),

is the pathλ · µ := (u0, . . . , uk, uk+1, . . . , uk+l)

obtained by transversing firstλ and thenµ.A loopof lengthµ ∈ N+ is a sequence ofµ ≥ 2 edges having the form

(i0, i1), (i1, i2), . . . , (iµ−1, iµ), iµ = i0

As our graphs are totally nonreflexive, there are no self-loops: i j , i j−1, j = 1, . . . , µ.A loop ispositive(respectively,negative) if each of its edges is labeled 1 or−1 and the

product of these labels is+1 (respectively,−1). All other loops areambiguous.In the next three definitions the labeling plays no role. A graph is called:

connectedif for each pair of distinct verticesj, k there is a sequence of verticesj = i0, . . . , im = k, m ∈ N+ such that (i l−1, i l) or (i l, i l−1) is an edge ofΛ, (l = 1, . . . ,m)

9

strongly connectedif for any ordered pair (a, b) of distinct vertices there is a pathin Λ from a to b,

primary if every edge belongs to a loop,

These definitions imply:

• A graph with no edges is primary, but a graph with only one edgeis not primary.

• The graph union of primary subgraphs is primary.

• A strongly connected subgraph is primary, and a primary connected subgraph havingmore than one vertex is strongly connected. IfΓ is quasipositive, every primarysubgraph is positive.

A subgraphΛ ⊂ Γ is called:

• full provided it contains all edges inΓ joining vertices ofΛ,

• initial if no edge ofΓ is directed from a vertex outsideΛ to a vertex ofΛ,

• terminal if if no directed edge ofΓ joins a vertex ofΛ to a vertex not inΛ,

• fundamentalif is connected, primary and initial, and no other subgraph containingΛ has these properties.

Lemma 6 The following hold for all subgraphs:

(a) fundamental subgraphs are full

(b) if fundamental subgraphsΛ1,Λ2 share a vertex, they coincide

(c) every connected, primary, initial subgraph is contained ina unique fundamental sub-graph

Proof (a) and (b) follow directly from definitions. (c) is proved byshowing that the graphunion of a maximal nested family of connected, primary, initial subgraphs is fundamen-tal.

Graphs and systems

Let (F,X,Rn) be a system.

Proposition 7 If Π : F ։ F1 is a cascade having a fibre system Fp, then:

(a) Γ(F1) is a full subgraph ofΓ(F).

(b) Γ(Fp) is isomorphic to a subgraph ofΓ(F).

10

(c) when F is cooperative, quasicooperative or coherent, F1 and Fp have the same prop-erty.

Proof (a) and (b), which imply (c), are proved by inspecting the block decomposition (8)of the matrix of functionsF′(x).

Proposition 8 Let Γ1 ⊂ Γ(F) be an initial full subgraph such that V(Γ1) = 1, . . . , n1.Then:

(i) there is a cascadeΠ : (F,X,Rn)։ (F1,X1,Rn1) such thatΓ(F1) = Γ1.

(ii) When F is quasicooperative, F1 and all fibre systems are quasicooperative, and ifΓ1

is primary then F1 is cooperative

Proof Initiality and fullness ofΓ1 means that (7) holds. Therefore (3) defines a cascadesatisfying (i). The first assertion in (ii) follows from Proposition 7(c). The second assertionholds becauseΓ1 is quasipositive, and if it is primary all its edges are in loops and henceare positive.

Spin assignments

A spin assignmentfor a graphΓ is any functionσ : V(Γ) → ±1. It is consistentifh(u, v) = σ(u)σ(v) for every edge (u, v) belonging to a loop. (This terminology is not thesame as in [42], where it was required that every edge be consistent. With that strongerrequirement, the theorem given below would become a characterization of monotonicitywith respect to an orthant order, a more restrictive property than coherence.)

Theorem 9 Γ has the positive loop property if and only if it has a consistent spin assign-ment.

Proof AssumeΓ has the positive loop property. LetΓ′ be obtained fromΓ by keeping thesame vertices but deleting the edges not contained in loops.ClearlyΓ′ has the positive loopproperty, and ifσ is a consistent spin assignment onΓ′ it is also consistent onΓ. Thereforewe can assume every edgeebelongs to a loop and is thus positive.

Claim: If λ1, λ2 are paths froma to b thenh(λ1) = h(λ2) ∈ ±1. To see this, choosea pathµ from b to a, which can be done because each edge belongs to a loop. Since everyloop is positive by hypothesis, forj = 1, 2 we have

1 = h(λ j · µ) = h(λ j)h(µ)

Thereforeh(λ1) = h(µ) = h(λ2).Now fix a vertexp of Γ and for each vertexv choose a pathλv from p to v. Define

σ(p) = 1 andσ(v) = h(λv), which by the claim is independent of the choice ofλv. For anyedgee= (u, v) we can fixλu and defineλv := λu · e. Then have:

σ(u) = h(λu), σ(v) = h(λu · e) = h(λu)h(e),

which impliesh(e) = σ(u)σ(v). The converse implication is left to the reader.

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Remark The foregoing proof can be expressed homologically. LetΛ denote the 1-dimensional cell complex corresponding to a prime subgraphΛ ⊂ Γ having the verticesof Λ for 0-cells and the directed edges ofΛ for 1-cells. In the cellular chain groups ofΛ with coefficients inZ2 (identified with the multiplicative group±1), a labelingh is a1-cochain, spin assignments are 0-cocycles, and a spin assignmentσ is consistent forh ifits coboundary isδσ = h. As the evaluation of cochains on chains induces a dual pairingH1(Λ;Z2) × H1(Λ;Z2) → Z2, the positive loop property makes the cohomology class ofhtrivial. Thush = δσ, proving thatσ is consistent.

A change of variablesx 7→ y is calledelementaryif there is a permutationi 7→ i′ of1, . . . , n and ann-tupleρ ∈ ±1n such thatyi = ρi xi′ .

Theorem 10 If a system is coherent, there is an elementary change of variables trans-forming it to a quasicooperative system admitting a cascadeover a cooperative system forwhich all fibre systems are quasicooperative.

Proof Assume (F,X,Rn) is a coherent system, which by Theorem 9 has a consistent spinassignmentσ. The elementary change of variablesL : Rn→ Rn,

y = Lx, yi := σ(i)xi

transforms (F,X,Rn) into a system

(G, L(X),Rn), L G ∗ ∗∗

such thatΓ(G) andΓ(F) have the same undirected edges. For every directed edge (j, i) ofΓ(G):

hΓ(G)( j, i) = sign(∂G j

∂yi) = σ jσi sign(∂F j

∂xi) σ jσihΓ(F)( j, i)

If ( j, i) belongs to a loop thenhΓ(F)( j, i) = σ jσi by the consistency condition. Therefore

sign(∂G j

∂yi)(σ jσi)2

= (±1)2 = 1,

showing thatG is quasicooperative. After reindexing variables we assumethere is a funda-mental subgraphΓ1 ⊂ Γ(F) with vertex set1, . . . , n1, 1 ≤ n1 ≤ n. Now apply Proposition8.

Monotone dynamics

A local semiflowΦ is monotoneif x y =⇒ Φtx Φty. Throughout this section weassume:

• Φ := Φtt≥0 is a monotone local semiflow in an ordered space X

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To simplify notation we may writex(t) := Φtx wheneverΦtx is defined. It is well knownfor the data in Equation (1) that ifF is cooperative andX is convex, the correspondinglocal semiflowΦ is monotone. This is a corollary of the Muller-Kamke theorem [28, 21]on differential inequalities (Hirsch [16]).

Proposition 11 The following are true for all x∈ X:

(a) No points ofω(x) are related by⊲X or ⊳X

(b) ω(x) is a singleton in the following cases:

(i) γ(x) is compact and there exist t∗ ≥ 0, ε > 0 such that

t∗ < t < t∗ + ε =⇒ Φtx ≺ x orΦtx ≻ x

(ii) γ(x) is compact and there exist t> 0 such that

Φtx⊲X x orΦtx⊳X x

Proof (a) and (b)(i) are sharpenings of Hirsch & Smith [20, Theorems 1.8, 1.4], respec-tively. Assertion (b)(ii) follows from (b)(i).

Proposition 12 Assume A⊂ X is attracting.

(a) If each point of A is strongly accessible in X from either above or below, then A containsan equilibrium.

(b) If each point of A is strongly accessible in X from both above and below and A∩E = pthen A= p.

Proof This is a slight generalization of Hirsch [17, Theorems III.3.1 and III.3.3], and thesame proofs work here.

Proposition 13 Assume A⊂ ω(x). Let q ∈ A be a minimal (respectively, maximal) pointof A having a neighborhood N⊂ X such that there is a point y≺ N (respectively, y≻ N) isattracted to A. Then q= inf A (respectively, q= supA).

Proof To fix ideas we assumeq is a minimal point ofA andy ≺ N. Notation is simplifiedby settingΦtw = w(t) wheneveverw ∈ X, t ≥ 0.

Some point onγ(x) lies inN its omega limit set containsA. Replacingx by such a pointwe assumex ∈ N. Thereforey ≺ x and

y(t) ≺ x(t), (t ≥ 0) (11)

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There is a sequencetn→∞ such thatx(tn) ∈ N and

x(tn)→ q (12)

Becauseω(y) meetsA we can choose this sequence so that also

y(tn)→ a ∈ A (13)

It follows from (11), (12), (13) and closedness of the order relation thata q, so minimalityof q impliesa = q. Thus

y(tn)→ q ∈ E (14)

Choosen0 so thaty(tn0) ∈ N. If I ⊂ R+ is a sufficiently small open interval abouttn0

then s ∈ I =⇒ y(s) ∈ N, hencey(s) ≻ y. The dual of Proposition 11(b)(i) now showsthatω(y) is an equilibrium, henceω(y) = q. It follows from (11) thatω(x) ≻ q, henceA q.

In the rest of this section we assume:

• X ⊂ Rn with the vector ordering.

Proposition 14 Assume x∈ ω(A), ω(x) = A. If inf A = p or supA = p then A= p.

This result also holds whenX is ordered by a solid polyhedral cone, but it is has not beenproved for more general ordered spaces. For strongly order-preserving local semiflowsa stronger conclusion holds: Every omega limit set is unordered (Hirsch & Smith [20,Corollary 1.9]).

Proof For anyΣ ⊂ 1, . . . , n the correspondingfaceof Rn+

is

J := J(Σ) = z ∈ Rn+

: zi > 0 =⇒ i ∈ Σ

WhenΣ , ∅ the correspondingopen faceis

Jo := Jo(Σ) = z ∈ Rn+

: zi > 0 ⇐⇒ i ∈ Σ

It can be seen thatJo = J andJo is relatively open in its linear span. Moreover

(∀z ∈ Jo) (∃δ > 0) z≻ J ∩ Nδ(0) (15)

Fix x, p ∈ X such that infω(x) = p or supω(x) = p; we have to proveω(x) = p. Tofix ideas we assumep = 0 = inf ω(x). Claim: Φt(x) is defined for allt ≥ 0. It is wellknown that this is the case if the orbit closure ofx is compact. If it is not compact, the orbitintersects the boundary of some open ball centered at 0 in an infinite set. Consequentlyω(x) contains a point, p, which implies the claim.

For anyI ⊂ [0,∞) setΦ(I , x) := Φtx: t ∈ I . By the Baire category theorem there is adense open subsetS ⊂ [0,∞) such that for each componentI of S there is a unique openfaceJo

I ⊃ Φ(I , x).

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There is a sequenceIk of these components and pointstk ∈ Ik such that ask→ ∞ wehave

tk→ ∞, x(tk) ≻ 0, x(tk)→ 0

After passing to a subsequence we can assume there is an open faceKo such thatJoIk= Ko

for all k. Choose such aKo having the largest possible dimension. Thenx(t) ∈ Ko forsufficiently larget. For if x(t0) ∈ Ko andε > 0 is such thatx(t) < Ko for t ∈ (t0, t0 + ε],thenx(t′) for somet′ ∈ (t0, t0+ ε] belongs to an open face of larger dimension, and this canonly happen finitely many times. Set dimKo

= m ∈ 1, . . . , n and relabel variables so thatKo= Ko(1, . . . ,m).By (15) there existst∗ > 0 such that

t > t∗ =⇒ x(t∗) ≻ x(t) ≻ 0

By Proposition 11(b)(i) the trajectory ofx(t∗) converges, necessarily to 0. Thereforeω(x) =ω(x(t∗)) = 0.

Corollary 15 Assume0 ∈ X ⊂ Rn+. If 0 ∈ ω(x), then0 = ω(x).

Proof Follows from Theorem 14 because 0= inf ω(x).

Remark We digress to interpret this result biologically. Letxi ≥ 0 stand for the “size” ofspeciesi (population, biomass, density, . . . ) and call

∑ni=1 xi the “total size”. Assume that

from each initial statex(0) ∈ Rn+

the species develop along a curvex(t) = (x1(t), . . . , xn(t)) ∈R

n+, t ≥ 0 governed by a cooperative system (suggesting symbiosis orcommensalism) in

Rn+. Then:

• If the total population does not die out, the total size is bounded above0.

This follows from the contrapositive of the Corollary.

The next result will be used to start the inductive proof of Theorem 1. It applies onlyto cooperative systems, but the assumptions onΦ, X andA are weaker than in Theorem 1.Recall that every nonempty compact set in an ordered space contains a maximum point anda minimum point (Ward [51]),

Theorem 16 Assune X⊂ Rn has the vector ordering andΦ is a monotone local semiflowin X. Let A⊂ X be attracting and finitely transitive forΦ. If every point of A is stronglyaccessible in X from above or below, then A∈ E.

More precisely: If q∈ A is maximal and strongly accessible in X from above thenA = q. Likewise if q∈ A is minimal and strongly accessible in X from below.

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Proof It suffices to assumeq ∈ A is maximal and strongly accessible inX from above.Under the current assumptions there existx ∈ A, y ∈ X and neighborhoodU ⊂ X of q suchthat thatq ∈ ω(x), y ≻ q andy is attracted toA. Evidentlyq is maximal inω(x), henceq = supω(x) by Proposition 13, and thereforeq = ω(x) by Theorem 14.

Supposez ∈ X andω(z)∩U , ∅. There existsl ∈ N+ with z(tl) ∈ U, hencey(t)⊲X z(t+l), (t ≥ 0), and monotonicity proves

ω(z) ∩ U , ∅ =⇒ ω(z) q (16)

Now we prove for allv ∈ X:q ∈ ω(v) =⇒ q = ω(v) (17)

For there existsz ∈ γ(v)∩U and Equation (16) impliesq = supω(z), henceq = ω(z) = ω(v)by Theorem 14.

Let ak be any sequence inU ∩ A converging toq. By hypothesis there is a finite setS ⊂ X such that eachak is an omega limit point of some member ofS. By finiteness ofSthere is a subsequencebk of ak andv ∈ S such thatbk ⊂ ω(v). Evidentlyq ∈ ω(v),whenceq = ω(v) by (17). This can only happen ifbk = q for all k ∈ N+. It follows thatq isisolated in the connected setA, entailingA = q.

Proofs of the main theorems

Proof of Theorem 1 Let the system (F,X,Rn) be as in Theorem 1, with a finitely transi-tive attracting setA ⊂ X.

Step (i) Consider first the case thatF is cooperative. ThenΦ is monotone becauseX isconvex, and each of the assumptions (i), (ii) implies each point of A is strongly accessiblein X from above or below. The conclusion for this case follows from Proposition 16.

Step (ii) We proceed by induction onn, the casen = 1 following from the cooperativecase. Assume inductively thatn > 1 and that the conclusion holds for smaller values ofn. By Step (i) we can assumeF is not cooperative, whence by Theorem 10 there is acooperative system (F1,X1,Rn1) and a cascadeΠ : F ։ F1 with 1 ≤ n1 < n, whose fibresystems are quasicooperative. Lemma 5 shows that (Fp,Xp,Ep) is a fibre system for eachp ∈ E(F1).

The setΠ(A) ⊂ X1 is finitely transitive for the cooperative systemF1, henceΠ(A) =p ∈ E(F1) by Step (i). ThusA lies in the invariant setXp = X ∩ Q−1(A), andA is attractingand finitely transitive forΦp := Φ|Xp. The inductive hypothesis applied to (Fp,Xp,Ep)shows thatA is an equilibrium, completing the induction.

Proof of Theorem 2 Consider first the case thatF is cooperative. Assumeper contrathat the orbit closure ofx ∈ X contains a nonempty open subsetU ⊂ X. As some opensubset ofRn is dense inX we can assumeU is open inRn. The orbitγ(x), being a smooth

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curve, is nowhere dense inU becausen ≥ 2. ThereforeU ⊂ ω(x), henceω(x) containspointsa, b such thata⊲X b. But this contradicts Proposition 11(a).

Now assumeF is not cooperative. By Theorem 10 there is a cascadeΠ : F ։ F1 withF1 cooperative. IfW ⊂ X is open andγ is an orbit ofF, thenΠ(W) is open inX1 andΠ(γ)is an orbit ofF1. The cooperative case shows thatΠ(γ) ∩ Π(W) is not dense inΠ(W) andthereforeγ ∩W is not dense inW.

Proof of Theorem 3 If F is cooperative, as whenn = 1, the conclusion follows fromProposition 12. We proceed by induction onn, assuming thatn > 1 and the theorem holdsfor smaller values.

We can assumeF is not cooperative. By Theorem 10 there is a cascadeΠ : F ։ F1

cooperative system (F1,X1,Rn1) with F1 cooperative and 1≤ n1 < n, such that ifp ∈ E(F1)then (Fp,Xp,Ep) is a quasicooperative system. Applying the inductive hypothesis twice,we conclude that there existsp ∈ E(F1) andq ∈ E(Fp) ⊂ E(F).

AssumeE(F) = q and setΠ(q) = p′ ∈ E(F1). Then p′ = p. For we showed abovethat every fibre system contains an equilibrium ofF, which must beq. ThusΠ−1(p′) andΠ−1(p) are not disjoint, hence they coincide andΠ maps both of them top.

By the inductive hypothesisp is the global attractor forF1, thereforeXp attracts allpoints ofX by Equation (4). This impliesA is the global attractor forΦ|Xp, and the induc-tive hypothesis applied to (Fp,Xp,Ep) shows thatA = q.

Appendix: Notes on the development of the concept “at-tractor”

In spite of the fact that everyone who is interested in dynamics has a more or less vagueintuition of what an attractor of a map f: M → M should be, there is no generallyaccepted mathematical definition for this concept even if M is a smooth manifold andf is also smooth. —H. Bothe [4]

The first mathematical use of the word “attractor” may be in Coddington & Levinson’s1955 book [9], where it refers to an asymptotically stable equilibrium. The term was subse-quently extended to include an attracting cycles. Today there are many definitions, usuallymeaning an invariant set (of some kind) that is approached uniformly (in some sense) bythe forward orbits of all (or most) points in some neighborhood of the set.

Attractors do not occur explicitly in the work of Poincare or Birkhoff. These authorswere primarily interested in Hamiltonian systems, which have no attractors because theypreserve volume.

An early proof of existence of a unique attracting periodic orbit for a general class ofsystems is in the 1942 paper of N. Levinson and O. Smith [23].3

3Thanks to George Sell for this reference.

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Early computer simulations revealed what appear to be attractors. As far back as 1952,Turing [49] published pictures of numerical simulations ofa nonlinear dynamical modelof cell development, exhibiting striking pattern formation. Simulations by Stein & Ulam[44, 45] and Lorenz [24] gave persuasive pictorial evidenceof complicated structure inattractors, but attracted little attention when they were published. Hamming’s review [15]of [45] was unenthusiastic:

Many photographs of cathode ray tube displays are given, a fondness for citing largenumbers of iterations and machine time used is revealed, anda crude classification ofthe limited results is offered, but there appears to be no firm new results of generalmathematical interest. . .

One can only wonder what will happen to mathematics if we allow the undigestedoutputs of computers to fill our literature. The present paper shows only slight tracesof any digestion of the computer output.

Much of the early theoretical work on attractors on global analysis was concerned withcharacterizing them in terms of Liapunov functions and topological dynamics (e.g., Ura[50], Auslanderet al. [2], Mendelson [25], Bhatia [3]). Little was known of their internaldynamics beyond the existence of fixed points in global attractors for flows in Euclideanspace (Bhatia & Szego [5]).

In the 1960s a number of articles on attractors and related forms of stability were in-spired by Sell [37]. In his seminal 1967 work on global analysis, Smale gave detailedconstructions and analyses of hyperbolic attractors and other invariant sets, which wouldlater be called “chaotic” and “fractal”, and proved them structurally stable. He called at-tention to the vast mixture of periodic, almost periodic, homoclinic and other phenomenafound in structurally stable attractors, even in rather simply given systems.

“Strange attractors” were proposed in 1971 as a model of turbulence by Ruelle andTakens [35, 36, 32], Newhouseet al. [29]). The physical significance of this route to chaosis still debated.

In his controversial 1972 book on morphogenesis ([46, 47]) the late Rene Thom issueda bold manifesto proclaiming the fundamental scientific role of attractors:

1. Every object, or every physical form, can be represented by an attractor C of adynamical system in a spaceM of internal variables.

2. Such an object possesses no stability, and for this reasoncannot be perceived, unlessthe corresponding attractor isstructurally stable.

3. Every creation or destruction of forms, every morphogenesis, can be described bythe disappearance of the attractors representing the initial forms and their replacementthrough capture by the attractors representing the final forms. This process, calledcatastrophe, can be described in a space ofexternal variables. . . .

In recent years much work has been devoted to analysis of attractors in specific classesof chaotic systems, such as those named after Duffing, Lorenz, Henon and Chua, and to at-tractors having particular topological properties, such as R. Williams’ expanding attractors

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(Williams [52], Plykin & Zhirov [31]). A novel measure-theoretic type of attractor due toMilnor [27] has stimulated several papers.

Many authors have investigated attractors in infinite-dimensional systems, especiallyfor partial differential equations, a prime desideratum being finite dimensional gllobal at-tractors. The large literature includes books by Constantin et al.[11], Hale [14], Ladyzhen-skaya [22], Ruelle [33], Sell & You [38], and others.

Attractors, being objects defined by topological limiting processes, resist classificationand even description. A general theory appears quite distant.

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