Two dimensional heteroclinic attractor in the generalized Lotka-Volterra system
aa r X i v : . [ m a t h . D S ] S e p Two dimensional heteroclinic attractor in the generalizedLotka-Volterra system.
Valentin S. Afraimovich ∗ , Gregory Moses † , Todd Young ‡ August 21, 2018
Abstract
We study a simple dynamical model exhibiting sequential dynamics. We show thatin this model there exist sets of parameter values for which a cyclic chain of saddleequilibria, O k , k = 1 , . . . , p , have two dimensional unstable manifolds that containorbits connecting each O k to the next two equilibrium points O k +1 and O k +2 in the chain( O p +1 = O ). We show that the union of these equilibria and their unstable manifoldsform a 2-dimensional surface with boundary that is homeomorphic to a cylinder if p iseven and a M¨obius strip if p is odd. If, further, each equilibrium in the chain satisfiesa condition called “dissipativity,” then this surface is asymptotically stable. In the last decade it became clear that typical processes in many neural and cognitivenetworks are realized in the form of sequential dynamics (see [23], [14], [24], [26], [25],[2] and references therein). The dynamics are exhibited as sequential switching amongmetastable states, each of which represents a collection of simultaneously activated nodesin the network, so that at most instants of time a single state is activated. Such dynamicsare consistent with the winner-less competition principle [26, 2]. In the phase space ofa mathematical model of such a system each state corresponds to an invariant saddleset and the switchings are determined by trajectories joining these invariant sets. In thesimplest case these invariant sets may be saddle equilibrium points coupled by heteroclinictrajectories, and they form a heteroclinic sequence (HS). This sequence can be stable if allsaddle equilibria have one-dimensional unstable manifolds [2], in the sense that there is anopen set of initial points such that trajectories going through them follow the heteroclinicones in the HS, or unstable if some of the unstable manifolds are two-or-more dimensional.In the latter case properties of trajectories in a neighborhood of the HS were studied in[2] and [3]. General results on the stability of heteroclinic sequences have been obtained ∗ Universidad Autonoma de San Luis Potosi, IICO † Corresponding author, Department of Mathematics, Ohio University, [email protected] ‡ Ohio University, Department of Mathematics
1y Krupa and Melbourne [17, 18] in the form of necessary conditions that may also besufficient in the presence of certain algebraic conditions.An instability of a HS may be caused if some initial conditions follow trajectories on theunstable manifold different from the heteroclinic ones. M. Rabinovich suggested [22] con-sidering the case when all trajetories on the unstable manifold of any saddle in a HS areheteroclinic to saddles in the HS, which implies an assumption that the HS is, in fact, aheteroclinic cycle. In this case one can expect some kind of stability, not of the HS, ofcourse, but of the object in the phase space formed by all heteroclinic trajectories of thesaddles in the HS. We study in our paper the Rabinovich problem. We deal here withthe generalized Lotka-Volterra model [2] that is a basic model of sequential dynamics forwhich unstable sets are realized as saddle equilibrium points. All variables and parametersmay take only nonnegative values, so we work in the positive orthant of the phase space R n . We impose some restrictions on parameters under which all unstable manifolds of thesaddle point are two-dimensional and all trajectories on them (in the positive orthant) areheteroclinic in some specific way (see below). We prove that they form a piece-wise smoothmanifold homeomorphic to the cylinder if the number of the saddle points in the HS is evenor to the M¨obius band if it is odd. We prove also that under the additional assumptionthat each equilibrium is dissipative (see below), then this manifold is the maximal attractorfor some absorbing region (in the positive orthant). Trajectories in this region may followdifferent heteroclinic trajectories and may manifest some kind of weak complex behavior.Although our motivations are neurological, we observe that heteroclinic networks (andthus, potentially, high-dimensional counterparts of the same) are ubiquitous, appearing inapplications that range from celestial dynamics [16] to evolutionary game theory [11]. We begin our consideration of two-dimensional heteroclinic channels with the study of aseries of Lotka-Volterra equations. Lotka-Volterra models are widely used in the context ofheteroclinic sequences where all the unstable manifolds are one dimensional, so that eachequilibrium is connected to exactly one subsequent equilibrium (e.g. [13, 28, 10, 9]). It hasbeen recently shown that they ask provide a general method for embedding directed graphsinto a system of ordinary differential equations [6]. We remark that although [6] allowsgraphs of high valency to be modeled by heteroclinic networks, and some work has beendone on the stability of such systems (e.g. [15], which considers the competing dynamicsof the “overlapping” channels λ → λ → λ → λ and λ → λ → λ → λ ), studyof heteroclinic networks has usually only considered one-dimensional unstable manifolds.In keeping with the discussion of the introduction, we consider the dynamics of a systemsuch that initial conditions arbitrarily close to any of p ≤ n saddle nodes may be mappedinto neighborhoods of either one of two other saddle nodes. In particular, a simple generalmodel for heteroclinic sequential dynamics was given in [4] by˙ x i = F i ( x ) = x i ( σ i − n X j =1 ρ ij x j ) for i = 1 , ..., n, (1)2igure 1: A rough representation of the dynamics of the system defined by Equation 1,with p = 5.where all of the parameters are assumed to be positive. The constants ρ ij have biologicalmeaning, representing inhibition of mode i by mode j . For the sake of simplicity, we assumethat ρ ii = 1 for all i . Further, since the variable x is assumed to encode biological informa-tion that is necessarily non-negative, e.g. activation levels or chemical concentrations, werestrict the system to the first closed orthant, R n + = { x ∈ R n : x i ≥ , ≤ i ≤ n } . Thesystem is so constructed that it contains saddle points lying on the axes, with σ i being the i -th coordinate of the i -th saddle along the i -th axis ( σ i > n equilibrium points of the form O k = (0 , ..., , σ k , , ... ), for k = 1 , .., n . There may be otherequilibria as well, but they are not relevant for our purposes; we only study transitionsbetween the n equilibria just defined.In the present work we suppose that the first p equilibria points are sequentially connectedby a set of 2-dimensional unstable manifolds. For each k , 1 ≤ k ≤ p , there will be aheteroclinic orbit connecting O k to O k +1 and a heteroclinic orbit connecting O k to O k +2 .Furthermore, the system is closed in the sense that O p + i = O i , i.e. p is the modulus ofthe subscript. In the following, we will consider the restrictions necessary to enforce suchdynamics.To ensure that there are heteroclinic trajectories between O k and both O k +1 and O k +2 ,we apply eigenvalue conditions to the system. Consider first O . The linearization of thevector field F ( x ) (1) at O is given by the upper triangular matrix: DF ( O ) = − σ − σ ρ − σ ρ − σ ρ · · · − σ ρ n σ − ρ σ · · ·
00 0 σ − ρ σ · · ·
00 0 0 σ − ρ σ · · · ...... ... ... 0 . . . ...0 0 0 0 · · · σ n − ρ n σ , (2)and so the eigenvalues appear on the diagonal. The matrices DF ( O k ) have a similar simplestructure, zeros everywhere except on the diagonal and on the k -th row. It is then easy tosee that the eigenvalues of DF at O k are λ kk = − σ k and λ kj = σ j − ρ jk σ k , for j = k. In particular, the eigenvalues are all real. One can also see from the structure of (2) that DF ( O k ) has a full set of eigenvectors, even if some eigenvalues are repeated.3ote that because of the particular form of the equations, all coordinate axes, planes andhyperplanes are invariant. Thus, in order that trajectories can travel from O k to O k +1 or O k +2 , it is sufficient to put the restriction 0 < λ kk +2 , λ kk +1 , and λ kj < k , 1 ≤ k ≤ p , we requirethat 0 < min i =1 , { σ k + i − ρ k + i,k σ k } (indices mod p ) , (3) λ j = σ j − ρ jk σ k < , for j = k, k + 1 , k + 2 mod p. (4)Note also that − σ k <
0. These inequalities guarantee that each equilibrium is a hyperbolicsaddle with 2 unstable directions and n − f W uk = W u ( O k ) ∩ R n + bethe unstable manifold of O k restricted to the positive orthant. We show below thatΓ ≡ p [ k =1 ( f W uk ∪ O k )forms a piecewise smooth surface that we will classify topologically as follows. Theorem 2.1.
Suppose that inequalities (3) and (4) hold for each k , ≤ k ≤ p and thateach unstable manifold f W u ( O k ) is contained in a compact forward invariant set as specifiedin Lemma 3.10 (see also Remark 3.11). When p is even, the union of unstable manifolds Γ is homeomorphic to a cylinder. When p is odd, Γ is homeomorphic to a M¨obius strip The proof of this theorem, which is slightly involved, is left for the appendix.Consider the following definition.
Definition 2.2.
Let Σ u and Σ s be the set of stable and unstable eigenvalues, respectively,of the linearization of a vector field at a saddle equilibrium, i.e., max Re (Σ s ) < and min Re (Σ u ) > . We say that the saddle is dissipative if max Re (Σ u ) < − max Re (Σ s ) . In other words, the weakest stable eigenvalue is stronger than the strongest unstable eigen-value. (See [4].)
In terms of the specific vector field under study all the eigenvalues in question are real andfor each k we have:max i =1 , { σ k + i − ρ k + i,k σ k } < min j = k,k +1 ,k +2 {| σ j − ρ jk σ k | , σ k } (indices mod p ) . (5)The main goal of this manuscript is to show that, under the condition that each saddleequilibrium is dissipative, then Γ is asymptotically stable. Specifically, our main theoremis: Theorem 2.3.
Suppose that inequalities (3) , (4) and (5) hold for each k , ≤ k ≤ p andthat each unstable manifold f W u ( O k ) is contained in a compact forward invariant set asspecified in Section 3.3. Then Γ is asymptotically stable. ǫ -close to Γ, but is distant from eachof the fixed points O i . In such a case, the dynamics of the system are controlled largelyby three consecutive saddles O i , O i +1 , and O i +2 . In Section 3, we consider the restrictionof (1) to three consecutive dimensions; we will gain information on the full-dimensionalsystem by viewing it as a perturbation of this restriction. The second part of the proofis to consider the dynamics as a trajectory passes near a fixed point; we consider this inSection 4.In Section 4.2 we prove the main theorem. In Section 4.3 we show that there is a non-emptyparameter set for which the conditions of the theorem are satisfied. We begin our proof of Theorems 2.1 and 2.3 with a study of the restriction of the systemto three dimensional sub-spaces corresponding to three consecutive coordinate directions.Through a series of geometric lemmas, we prove the main result of the section, Theo-rem 3.12, which provides information on the behavior of trajectories inside this invariantsubspace. This will be used in later sections to complete the proofs of the main results byproviding information on those parts of the full phase space where all but three coordinatesare small.
Let O i , O i +1 , and O i +2 be any three consecutive equilibria and restrict the system (1) to thethree dimensions spanned by consecutive coordinates x i , x i +1 , and x i +2 . For convenience,we will refer to these three variables as x , x , and x . Our goal in three dimensions is toshow the existence of a compact, forward invariant set containing O , O , and O such thatany trajectory with an initial value in the interior of that set converges to O .Restricted to three dimensions, the equations (1) are reduced to˙ x = x ( σ − x − ρ x − ρ x ) , ˙ x = x ( σ − x − ρ x − ρ x ) , ˙ x = x ( σ − x − ρ x − ρ x ) . (6)The restriction of the dimension of the unstable manifolds via the eigenvalue conditions,and the positivity conditions on σ , σ , and σ yield the following inequalities: − σ < < σ j − ρ j σ , j = 2 , , (7) − σ < < σ − ρ σ , (8) σ − ρ σ < , (9) σ − ρ σ < , (10) σ − ρ σ < . (11)5f those inequalities, (7) controls the behavior of the system at O , (8) - (9) control thebehavior of the system at O , and (10) - (11) at O . The point ( σ , ,
0) is a saddle witha two-dimensional unstable manifold, (0 , σ ,
0) is a saddle with a one-dimensional unstablemanifold, and (0 , , σ ) is a sink for the system (6). We remark that although (6) has theform of the May-Leonard model, the particular parameter restrictions under considerationyield simple dynamics (see Theorem 3.12), and prevent the more complex behavior usuallystudied in that context. In particular, they are inconsistent with the symmetric May-Leonard model as it was introduced in [19]. σ σ σ Figure 2: Illustration of dynamics of the system projected onto the first three coordinates.All consecutive triplets ( x i , x i +1 , x i +2 ) (where x p + i = x i ) of coordinates possess the samequalitative dynamics.For each i , we will be interested in the points where ˙ x i = 0; each such set is the union ofthe plane x i = 0 and some “nontrivial” plane. We designate those planes: P := { σ − ρ x − ρ x − x = 0 } , (12) P := { σ − ρ x − ρ x − x = 0 } , (13) P := { σ − ρ x − ρ x − x = 0 } , (14)where ˙ x i = 0 on P i .We will also refer to the plane passing through the points ( σ , , , σ , , , σ ),which we denote by Σ. Observe that Σ is given by the equationΣ := x σ + x σ + x σ = 1 . For ease of discussion, we will also name the coordinate planes: P := { ( x , x , x ) : x = 0 } ,P := { ( x , x , x ) : x = 0 } ,P := { ( x , x , x ) : x = 0 } . P , P , P , and Σ as shorthand for the intersection of those planes with the first octant withoutthe risk of confusion. We observe that the intersection of Σ and of each P i with R n + is acompact triangle, a fact we will use repeatedly in the following section.In order to describe the dynamics of orbits, we are interested in when one plane lies “above”another in the first octant. Consider the following definition: Definition 3.1.
Observe that each plane P i can be written as the graph of a function z i ( x , x ) . A plane P i dominates a plane P j if ( x , x , z i ( x , x )) ∈ P i and ( x , x , z j ( x , x )) ∈ P j implies that z j ( x , x ) < z i ( x , x ) for all x , x > . A plane P i is dominated by aplane P j if P j dominates P i . Each plane P i divides R into two regions, one where ˙ x i is positive and another where it isnegative. This allows information about ˙ x i to be gained from purely geometric information.For example, ˙ x is positive below P , and negative above it. Since P dominates P (i.e.is always above it), we instantly see that ˙ x | P < x , x , and x axes. Denote the compact triangle thus formed by a plane S as T S and the non-zero component of its vertex on the i -th axis as S i . Then a plane S dominates a plane R if R i ≤ S i for i = 1 , ,
3, with at least one of those a strict equality,i.e. if its vertices are farther from the origin.
Lemma 3.2.
The plane P is dominated by the plane Σ .Proof. We consider where each plane intersects each axis: • The planes P and Σ both intersect the x − axis at the point O . • The plane P intersects the x axis at (0 , σ ρ , O .We know from (9) that σ ρ < σ . • The plane P intersects the x − axis at (0 , , σ ρ ), while Σ intersects the axis at O .We know from (10), that σ ρ < σ .Since P i ≤ Σ i for all i , P is dominated by Σ. Lemma 3.3.
The plane Σ is dominated by P .Proof. We consider where each plane intersects each axis:7
The plane P intersects the x − axis at the point ( σ ρ , , O . We know from (7) that σ < σ ρ . • The plane P intersects the x axis at (0 , σ ρ , O .We know from (8) that σ < σ ρ . • The planes P and Σ both intersect the x axis at O .Since Σ i ≤ P i for all i , P dominates Σ.The property “is dominated by” is clearly transitive, so the following corollary holds. Corollary 3.4. P is dominated by P . Since ˙ x < P plane, the following corollary follows immediately. Corollary 3.5.
On the plane P , ˙ x < . If we further had that Σ dominates P , then the eigenvalue conditions introduced in [4] andsummarized as (3) - (8) would be sufficient to ensure the existence of a positively invariantregion. It happens, however, that this is not the case. Lemma 3.6.
The plane Σ neither dominates nor is dominated by P .Proof. We consider where each plane intersects each axis: • P intersects the x − axis at the point ( σ ρ , , O .We know from (7) that σ < σ ρ , and therefore Σ does not dominate P . • P intersects the x − axis at (0 , , σ ρ ), while Σ intersects the axis at O . We knowfrom (11) that σ ρ < σ , and therefore P does not dominate Σ.Thus, neither plane dominates the other.The situation is somewhat salvaged by the following. Lemma 3.7. If σ ρ ≤ σ ρ , then P is dominated by P .Proof. In the proof of Lemma 3.3 (second bullet point), we established that P ≤ P .In the proof of Lemma 3.6 (second bullet point), we established that P ≤ P . All thatremains for P to dominate P is for P ≤ P , which occurs if and only if σ ρ ≤ σ ρ .The parameter restriction in (3.7) written in terms of the general systems gives that foreach k , 1 ≤ k ≤ p , σ k +1 ρ k +1 ,k ≤ σ k +2 ρ k +2 ,k (indices mod p ) . (15)Similarly to Corollary 3.5, we have the following:8 orollary 3.8. In the region of parameter space where the hypotheses of Lemma 3.7 aresatisfied, ˙ x | P < . We note here one additional observation.
Lemma 3.9.
The rectangular box: B = { x : 0 ≤ x i ≤ σ i , i = 1 , . . . , n } is forward invariant with respect to the system (1) . Further, the unstable manifolds f W uk areall contained in B . Forward invariance follows immediately from the differential equations (1) and the assump-tion that ρ ii = 1. The conclusion that the unstable manifold at O is inside B follows easilyby noting that the unstable eigenspace at O (restricted to the first orthant) is strictlyinside B .Figure 3: The plane P (dotted) is dominated by Σ (solid) which is dominated by P (dashes). If the hypotheses of Lemma 3.7 are satisfied, then the plane P (dashes and dots)is dominated by P .Rather than requiring that P dominates P , we could require only that P dominates P inside the forward invariant box B . Since P is strictly outside of B by Lemma 3.3 (seeFigure 3.2), this produces a strictly greater set of allowable parameter values. We remarked that our goal in three dimensions was to show the existence of a compact,forward-invariant set whose trajectories converge to O . We now carry this out. Lemma 3.10.
In the region of parameter space where the inequalities (7) - (11) and thehypotheses of Lemma 3.7 are satisfied, the planes P , P , P , and P enclose a positivelyinvariant region, i.e. no trajectory leaves in positive time. roof. No trajectory can leave through any of the x i = 0 planes, since ˙ x i | x i -plane = 0. Theoutward normal vector to P is N = h ρ , ρ , i , and its scalar product with the vectorfield (1) is given by ¯ n · F = ρ ˙ x + ρ ˙ x when ˙ x = 0. This is negative by Corollary 3.5and Corollary 3.8. Remark 3.11.
Since the pair of inequalities ρ ˙ x < and ρ ˙ x < form a sufficient,but not necessary, condition for the inequality ρ ˙ x + ρ ˙ x < to be satisfied, a positivelyinvariant region might exist even if the hypothesis of Lemma 3.7 is not satisfied. Forinstance, since the box B (Lemma 3.9) is forward invariant in may be that ˙ x | P ∩ B isnegative even when the hypotheses of Lemma 3.7 fail. Lemma 3.10 can therefore be extendedto a region of parameter space that includes the region defined by Lemma 3.7 as a propersubset. Theorem 3.12.
Suppose that P , x = 0 , x = 0 , and x = 0 enclose a positively invariantregion. Any trajectory φ t ( x ) in the above-described region that is not contained in P (the x = 0 plane) goes to (0 , , σ ) as t → + ∞ .Proof. Any trajectory in a compact positively invariant region has a non-empty ω -limitset. Fix a trajectory with initial condition x in the interior of the region, and let q be anarbitrary point in the ω -limit set Ω x . The proof breaks into three parts: we prove that q lies on P , that it lies on P , and that it lies on P ; the intersection of these planes is(0 , , σ ). The proofs of the second and third statements are essentially the same as theproof of the first statement, which we treat in detail.By way of contradiction, suppose that q does not lie on P . First of all, note that q doesnot lie on P , because by assumption, the x / ∈ P , and since the trajectory increases inthe x variable, φ t ( x ) cannot approach P = { x = 0 } Since ˙ x is continuous, and q lies on neither P nor P , the regions where ˙ x = 0, we canfind a spherical neighborhood O r ( q ) centered at q with radius r such that ˙ x | O r ( q ) > ǫ > ǫ . It follows that q is not a fixed point, and Ω x is not the singleton { q } , since ω − limit sets are forward invariant [21].The x component of φ t ( x ) is non-decreasing in the positively invariant set and it is boundedabove, since it is bounded above by σ . Thus it has a limit x ∗ . It follows by continuity thatthe x component of any point in Ω x is also x ∗ . We have supposed that q ∈ Ω x but q / ∈ P ;now consider φ t ( q ). Since ˙ x > q , and thus in a neighborhood of q , the x componentof φ t ( q ) must be strictly increasing as a function of time along the forward solution near q .This contradicts that the x component is x ∗ everwhere on Ω x .We repeat the argument twice. First, q must lie on the plane P ; otherwise ˙ x ( q ) < P . We observe here, with reference to Lemma 3.7, that even if P is notdominated by P , it is dominated by it on the restriction to P , and the argument thereforegoes through. Thus q must lie on the single point, (0 , , σ ), and our proof is complete.10 The Dynamics in the Full Phase Space O k We assumed in (5) that the saddle points O k are dissipative. Denote by ν the ratio: ν ≡ min |{ Re (Σ s ) }| max { Re (Σ u ) } . (16)We call ν the minimal saddle value of the equilibrium (see [27]). The equilibrium is dissi-pative if ν > ǫ from the stable manifold of the saddle comes to a point of distance on theorder of ǫ ν from the unstable manifold after going through a neighborhood of the saddle.Formulating this result more strictly, we label the variables in a neighborhood of O k intothe 2-dimensional unstable subspace η = ( x k +1 , x k +2 ) and the ( n − ξ = ( x j ) , j = k +1 , k +2. Let |·| denote the sup norm in these local coordinates. Bythe Stable Manifold Theorem, for each k there exists δ k > δ k -neighborhoodof O k , the unstable manifold W u ( O k ) is the graph of a (smooth) function ξ = h uk ( η ). Let δ be the minimum of these δ k and consider the δ -neighborhood V k of each O k , k = 1 , . . . , p For fixed k , and 0 < ǫ k sufficiently small, define a pair of sections, S = { ( ξ, η ) : | ξ | = δ, | η | ≤ ǫ k } and S = { ( ξ, η ) : | ξ | ≤ δ, | η | = δ } . By the classical Shil’nikov variables technique (see [27]), there exists ǫ k sufficiently smallso that every forward solution starting at S will intersect S before leaving V k . Let ǫ bethe minimum of the ǫ k ’s needed in the neighborhood of each O k . Let T denote the time atwhich such a solution intersects S . Theorem 4.1. If δ and ǫ are sufficiently small and if ( ξ (0) , η (0)) ∈ S and ( ξ ( T ) , η ( T )) ∈ S then | ξ ( T ) | ≤ C | η (0) | ν − e , where C > is independent of the initial point and ν − e > . The sections S and S , together with Theorem 4.1, are illustrated in two dimensions inFigure 4.1 Proof.
For ease of notation, consider O . Note from (2) that the eigenvector correspondingto the first diagonal element, − σ , can have only one non-zero component and that is inthe x direction. The eigenvector corresponding to the j -th diagonal element, j = 1, hasnon-zero components in the x and x j directions only.Now take into account that the hyperplane { ( x = x = 0) } is invariant under the flow ofthe equations and tangent to the stable eigenspace of DF ( O ). It thus coincides locallywith and contains W s ( O ).Next note that the unstable eigenspace has no non-zero components in the coordinatedirections x , x , . . . , x n and that the hyperplane where they are zero is invariant under11igure 4: A schematic diagram of Theorem 4.1. Initial conditions on a section intersectinga stable manifold are mapped in finite time to a section intersecting an unstable manifold,resulting in a contraction.the flow. It thus follows that the x , x , . . . , x n coordinates of the unstable manifold areall zero. Thus the unstable manifold W u ( O ) is given locally in the δ -neighborhood of O as the graph x = σ + h u ( x , x ) where h (0 ,
0) = 0 and h is smooth. Consider the localchange of variables that “straightens out” the unstable manifold: X = x − σ − h ( x , x ) . (17)Under this smooth change of coordinates (1) becomes:˙ X = − σ X − f ( X , x , . . . , x n )˙ x = x ( σ − x − ρ ( X + σ + h ( x , x )) − ρ x − ρ x − · · · − ρ n x n )˙ x = x ( σ − x − ρ ( X + σ + h ( x , x )) − ρ x − ρ x − · · · − ρ n x n )˙ x = x ( σ − x − ρ ( X + σ + h ( x , x )) − ρ x − ρ x − · · · − ρ n x n )...˙ x n = x n ( σ n − x n − ρ n ( X + σ + h ( x , x )) − ρ n x − ρ n x − · · · − ρ n,n − x n − ) . (18)Since W s ( O ) is contained in the coordinate hyperplane, we do not need a change of vari-ables corresponding to W s . Noting the f (0 , x , . . . , x n ) ≡
0, we use the MVT to define anew function f ( X ,x ,...,x n ) − X − = f ( X , x , . . . , x n ), and rewrite the equations to obtain:˙ X = − σ X + f ( X , x , . . . , x n ) X ˙ x = x ( σ − ρ σ − x − ρ ( X + h ( x , x )) − ρ x − ρ x − · · · − ρ n x n )˙ x = x ( σ − ρ σ − x − ρ ( X + h ( x , x )) − ρ x − ρ x − · · · − ρ n x n )˙ x = x ( σ − ρ σ − x − ρ ( X + h ( x , x )) − ρ x − ρ x − · · · − ρ n x n )... = ... (19)12ow note that distances from W s ( O ) and W u ( O ) are not effected by the coordinatechange (17). If we now denote ξ = ( X , x , . . . , x n ) then | ξ | is the distance to the unstablemanifold and | η | is the distance to the stable manifold (in the sup norm).It now follows immediately from (19) that inside the δ neighborhood of O the unstabledirections satisfy the estimates:˙ x ≤ x ( σ − ρ σ − e u ) and ˙ x ≤ x ( σ − ρ σ − e u ) , where we can take e u > δ and ǫ small. Thus, by asimple application of Gronwall’s inequality, solutions starting on S and remaining in our δ neighborhood of O must satisfy: | η ( t ) | ≤ | η (0) | e ( λ uu − e u ) t , where λ uu is the maximal unstable eigenvalue, i.e., λ uu = max j =2 , { σ j − ρ j σ } . Thus if T is defined by | η ( T ) | = δ , then T ≥ λ uu − e u ln( δη (0) ) . Similarly, using Gronwall’s inequality once again we obtain: | ξ ( t ) | ≤ | ξ (0) | e ( λ ls + e s ) t , where λ ls is the “leading” stable eigenvalue, i.e. since the eigenvalues are real, the negativeeigenvalue with the smallest absolute value. In terms of our parameters: λ ls = max {− σ , { σ j − ρ j σ , j = 4 , . . . , n }} . Thus | ξ ( T ) | ≤ C ( δ ) | η (0) | ( ν − e ) where e can be taken arbitrarily small. By the assumption that O is dissipative, ν > ν − e > We are now in a position to prove that the union of the unstable manifolds of our system,restricted to the positive orthant, form an asymptotically stable forward invariant set underappropriate parameter restrictions.
Theorem (Theorem 2.3, restated) . Suppose that inequalities (3) , (4) and (5) hold for each k , ≤ k ≤ p and that each unstable manifold f W u ( O k ) is contained in a compact forwardinvariant set as in Section 3.3. Then Γ ≡ S pk =1 ( f W uk ∪ O k ) is asymptotically stable.
13e consider the role each inequality of the hypothesis plays in the theorem. The inequalities(3) and (4) state that0 < min i =1 , { σ k + i − ρ k + i,k σ k } (indices mod p ) , for each 1 ≤ k ≤ p, and λ j = σ j − ρ jk σ k < , for j = k, k + 1 , k + 2 mod p, for each 1 ≤ k ≤ p. These inequalities are fundamental to the problem we are considering. They ensure thatthere is a heteroclinic channel from each equilibrium O k to the equilibria O k +1 and O k +2 (the first inequality), and that there are no heteroclinic channels to the other equilibria(the second inequality).The inequality (5) states that for each k ,max i =1 , { σ k + i − ρ k + i,k σ k } < min j = k,k +1 ,k +2 {| σ j − ρ jk σ k | , σ k } (indices mod p ) . This is the dissipativity condition. Informally, it may be taken to say that when a trajectorythat is close to one of the unstable manifolds whose union is Γ passes near one of the saddlefixed points, the distance of the trajectory to Γ contracts exponentially.It will become clear, from the proof of Theorem 2.3, that (5) is a much stronger condi-tion than is necessary. It ensures not only asymptotic stability, but a sort of monotonicasymptotic stability, such that whenever the trajectory passes near some O k , its distanceto Γ contracts exponentially. If, on the other hand, (5) held for some, but not all, valuesof k , then the trajectory would at times contract exponentially towards Γ, and at othertimes drift away from Γ, and stability would depend on how these attractive and repulsiveforces average over time. Attempting to formulate a replacement condition for (5) that isnecessary as well as sufficient is extremely nontrivial.The final hypothesis, that each unstable manifold is contained in a compact forward in-variant set, is necessary. We have seen one set of inequalities that ensure that such setsexist, σ k +1 ρ k +1 ,k ≤ σ k +2 ρ k +2 ,k (indices mod p ) , which are sufficient but not necessary. In Section 4.3, we see that there is a nonempty openregion of parameter space where the hypotheses of Theorem 2.3 hold. From our currentdiscussion, we see that the theorem applies to a larger region.We recall that Γ is asymptotically stable if given any neighborhood U of Γ (restricted to R n + ) there exists an ǫ -neighborhood of Γ, say V ǫ (Γ) ⊂ R n + , such that if x ∈ V ǫ (Γ) then x ( t, x ) ∈ U for t > t →∞ dist( x ( t, x ) , Γ) = 0, where x ( t, x ) is the solution of theinitial value problem (1) with x (0 , x ) = x . When we speak of an open ǫ -neighborhood ofΓ, we are speaking of a set that is open in the subspace topology. That is, an ǫ -neighborhoodin R n + is an ǫ -neighborhood in R n intersected with R n + . Proof of Theorem 2.3.
For each k = 1 , ..., p , let V ( O k ) be a sufficiently small δ neighbor-hood of O k , such that Theorem 4.1 can be applied within each V ( O k ) and δ does not dependon k . 14et z be a representative point at an initial condition ǫ -close to Γ. Choose ǫ such that ǫ < δ . Then we can classify the dynamics as either local if z ∈ V ( O k ) for some k , andglobal otherwise.We observe that if z = O k for any k , its behavior is trivial, and likewise, if z lies on acoordinate plane, it remains on that plane while converging exponentially to some O k . Wetherefore assume without loss of generality that neither of these cases hold.Suppose that z / ∈ V ( O k ) for any k . The point is ǫ -close to Γ, and since Γ is a finiteunion, we can say that z is ǫ -close to f W uα , where α is fixed and depends on z . Considerthe projection of the system onto the three-dimensional subspace spanned by the axes x α , x α +1 , and x α +2 . In three dimensions, the specific route a solution takes has not beenimportant; a trajectory in the invariant set may go straight to a neighborhood of O α +2 , orit may detour to O α +1 , but the net result is the same (Theorem 3.12). We now formallydifferentiate between these two cases.We consider two cases: either the positive semitrajectory of z intersects V ( O α +2 ) withoutfirst intersecting V ( O α +1 ) (case (i)), or the positive semitrajectory intersects V ( O α +1 ), thenintersects V ( O α +2 ) (case (ii)).Before proceeding, we recall the definitions of S and S given in Theorem 4.1, and similarlydefine such sections S q , S q for 1 ≤ q ≤ p . Without loss of generality, we assume that z ∈ S .Suppose that case (i) occurs. For each k = 1 , ..., p , let ˆ V ( O k ) be the projection of V ( O k )into the three dimensions spanned by x k , x k +1 , and x k +2 ; note that x k is negligable for k = α, α + 1 , and α + 2. Then the projection of the orbit onto R intersects ˆ V ( O α +2 ) beforeit can intersect ˆ V ( O α +1 ). In R , we know that all trajectories inside of f W uα that do notintersect ˆ V ( P α +1 ) come to a neighborhood of O α +2 in bounded time, where the bound doesnot depend on the initial condition. We may consider the non-projected, full-dimensionalspace as a “perturbation” of the projected space, and cite smooth dependence of initialconditions; the trajectory going through a slightly perturbed initial point corresponding tosuch a case must enter V ( O α +2 ) in a well-behaved way. In particular if z belongs to S α then a mapping from a neighborhood of z on S α to S α +20 is well defined and Lipschitz-continuous.Suppose that case (ii) occurs. Then once an orbit of z enters V ( O α +1 ), it starts to manifestthe dissipative behavior. In particular, if d ( x, W uα ) < ǫ , then after passing through V ( O α +1 ), d ( e x, W uα +1 ) < Cǫ ν , by Theorem 4.1. Once the representative point leaves V ( O α +1 ), we mayapply (i), viewing its position after leaving the neighborhood as an initial condition thatdoes not re-enter the neighborhood V ( O α +1 ). Thus when the representative point finallyenters V ( O α +2 ), its distance to Γ has been contracted by an order of Cǫ ν α +1 − e α +1 , where1 < ν α +1 − e α +1 and C absorbs both the constant C from Theorem 4.1 and a Lipschitzconstant.Suppose now that z ∈ V ( O k ), where k is now fixed. Then the trajectory leaves V ( O k )without increasing its distance from the unstable manifold, and passes into V ( O k +1 ) as justdescribed. We may then apply Theorem 4.1. As the trajectory passes through V ( O k ), itsdistance from the unstable manifold is contracted on an order of ǫ ν k − e k .15ince the mapping contracts in the global dynamics and is Lipschitz (or contracting) in thelocal dynamics, simple inductions yields that as a representative point moves through thesystem, its distance from the manifold changes from ǫ to c ǫ ν − e to c ǫ ( ν − e )( ν − e ) , andso on.For a fixed i , the value ν i , representing a ratio of eigenvalues, is likewise fixed. The value e i is not; it depends on the distance between the representative point and the stable manifoldas the trajectory enters V ( O i ), which changes from one instance to the next. For a given i , however, there is some maximal value that e i can take, since the system is constantlycontracting towards the manifold and e i goes to 0 along with that distance. Thus thereexists a global value, 1 < ν < ν i − e i for all i and all e i , such that passing from the first tothe p -th unstable manifold is a contraction of order cǫ ν p . Throughout the paper, we have put a number of restrictions on the parameters of thesystem. One must ask whether the specified inequalities may be satisfied.
Lemma 4.2.
There are sets of positive parameters values { σ i } , i = 1 , . . . , n , and { ρ jk } , j, k = 1 , . . . , n , with non-empty interior such that the inequalities (3) , (4) , (5) and (15) hold.Proof. First note that given { σ i } the inequalities (3) and (4) are completely uncoupled andall trivially have positive solutions { ρ jk } . For each k and i = 1 , ρ jk < σ j σ k (20)We note that with these inequalities the stable eigenvalues may be freely chosen to takeany negative value and the unstable eigenvalues any positive values. Since the restrictions(5) concern only relative orderings of those eigenvalues at each O k ( k fixed), it is clear that(5) is satisfied for open subsets of the previously chosen sets.The final remaining inequality (15) concerns on ρ k +1 ,k and ρ k +2 ,k and so to finish the proofwe need only to consider whether this restriction on those values is consistent with theprevious restrictions on those parameters, namely (3) and (5), but not (4). Note that theinequalities in (3) can be written as: σ k +1 ρ k +1 ,k > σ k and σ k +2 ρ k +2 ,k > σ k . The constraint (15) only requires that σ k +1 ρ k +1 ,k ≤ σ k +2 ρ k +2 ,k . There is clearly no inconsistency in these inequalities. The final inequalities (5) involveeach ρ k +1 ,k and ρ k +2 ,k independently of the others. First they require that for each k and i = 1 , σ k − ρ k +1 ,k σ k < σ k . σ k + i σ k − < ρ k +1 ,k which can clearly be satisfied along with (20). If { ρ k + i,k } , i = 1 , ρ jk can be chosen to satisfy (5) by simply choosingthem sufficiently large, i.e., for each j, k , j = k, k + 1 , k + 2, ρ jk must satisfy: σ k +1 − ρ k + i,k σ k < ρ jk σ k − σ j , i = 1 , . This can be rewritten as: ρ jk > σ k + i + σ j σ k − ρ k + i,k , i = 1 , . Thus ρ jk can take any value greater than the maximum of these two values. Previously wehad only required (in (4)) that these parameters satisfy: ρ jk > σ j σ k . Thus, with all other choices of consistent parameters, any large enough ρ jk , will also beconsistent.For example if σ i = σ are all the same, then (3), (4) and (15) are satisfied if simple :0 < ρ k +2 ,k ≤ ρ k +1 ,k < ρ jk > , for j = k, k + 1 , k + 2 . The dissipative requirement (5) will be satisfied if further:2 σ − ρ k + i,k < ρ jk , for i = 1 , j = k, k + 1 , k + 2 . For instance, the parameter values: σ i = 1, ρ k +1 ,k = . ρ k +2 ,k = . ρ jk = 1 . j = k, k + 1 , k + 2, (indices mod p ) strictly satisfy all of the inequalities. We proved in the paper that under some conditions the generalized Lotka-Volterra systemadmits a two-dimensional attractor that consists of saddles and the unstable manifoldsjoining them into a heteroclinic system. Thus, trajectories inside the attractor manifestcompletely regular features. However, behavior of wandering trajectories in the basin of theattractor could be treated as weakly chaotic. Indeed, one can introduce an oriented graphwith vertices identified with the saddle equilibrium points O k and edges identified withheteroclinic trajectories joining O k and O k +1 (belonging to the coordinate plane P k,k +1 ) or O k and O k +2 (belonging to the plane P k,k +2 ). It is possible to show that for each finitepath through this graph there exists an open set of initial points in the basin such that17he trajectory going through any of these points follows the corresponding heteroclinictrajectories. The number of paths grows exponentially with the length, so the number ofpieces of trajectories with different behavior (in fact all of them are ( ǫ , T ) separated forsome values of ǫ and T ) grows as T → ∞ , the metric complexity function grows with time.A similar effect has been observed in [1], where it was called weak transient chaos. Weintend to describe its properties in another publication.In the context of the motivating application, functional sequential dynamics in neural net-works, the two-dimensional heteroclinic attractor Γ in the phase space of dynamical system(1) may be thought of as a mathematical image of diverse sequential dynamics based onthe parallel performance of not one but two different modalities. Many interesting applica-tions of this may be found in cognitive science. For example, the learning and performing ofsensory-motor human behaviors in many situations require the integration or binding of thesequential stimuli of one modality with the sequential stimuli of another. This seems to bethe case in one of the most important cognitive functions: sequential working memory. Inperformance of a cross-modal working memory task, there may be two sequentially discreteneural processes (different chains of metastable states) that represent simultaneous neuralactivities corresponding to cross-modal transfer of information in the working memory (seee.g. [20]).ACKNOWLEDGEMENT. V.A. was partially supported by the grant RNF 14-41-00044 ofthe Russian Science Foundation during his stay at Nizhny Novgorod University, and by aGlidden Professorship Award during his stay at Ohio University. The authors thank M. I.Rabinovich for bringing to their attention the problems considered in this manuscript. Wethank the referees for their comments and insight. A The Topological Form of Γ Theorem 2.1 states that Γ depends on the parity of p ; this is because the number ofconnected components in the boundary of Γ depends on the parity of p . Proposition A.1. If p is even, then the boundary ∂ Γ has two connected components. If p is odd, then ∂ Γ has one component.Proof. If p is even, then one connected component of the boundary will include the trajec-tories connecting the saddles O n where n is even, and another will include the trajectoriesconnecting the saddles O m , where m is odd.If p is odd, then as in the previous case, the saddles O n where n is even are contained inthe same component, and the saddles O n where n is odd are likewise contained in a singlecomponent. Furthermore, O p − and O are contained in the same component, where p − O = O p +1 . Thus all saddles O n are contained in the same component.Figure 5 will help visualize the situation. 18igure 5: A diagram representing p = 6 (left) and p = 5 (right). We clearly observe twodistinct boundaries for p = 6, and one for p = 5.It appears from the diagram that when p is even, Γ is a cylinder, and when p is odd, Γis a M¨obius strip. We will formalize that intuition. We first observe that the componentswhose union is the unstable manifold Γ are not literally triangles, as they are depicted inFigure 5, but rather curved surfaces. We cite a definition from algebraic topology (e.g. [8]). Definition A.2.
Suppose X is a compact Hausdorff space. A curved triangle in X isa subspace A of X and a homeomorphism H : T → A where T is a closed triangularregion in the plane. A triangulation of X is a collection of curved triangles A , ..., A n in X whose union is X and such that for i = j , the intersection A i ∩ A j is either empty, or avertex of both A i and A j , or an edge of both. We also require, if h i is the homeomorphismassociated with A i , that when A i ∩ A j = e is an edge of both, then the map h − j h i is a linearhomeomorphism of the edge h − i ( e ) of T i with the edge h − j ( e ) of T j . All compact surfaces have a triangulation, but we will prove in particular that the decom-position of Γ into equilibria and the closure of their unstable manifolds, each restricted tothe first orthant, forms a triangulation of Γ.
Definition A.3.
Consider the system of differential equations (1) . For each α , we call theclosure of f W uα ∪ O α a heteroclinic triangle, T α . Note that: T α = f W uα ∪ O α ∪ O α +1 ∪ O α +2 ∪ Γ α +1 ,α +2 where Γ α +1 ,α +2 is the heteroclinic orbit from O α +1 to O α +2 . This follows from the invarianceof the ( α, α + 1 , α + 2)-plane and Theorem 3.12. Theorem A.4.
A heteroclinic triangle T α is homeomorphic to a closed triangle in theplane. Certainly the boundary of a heteroclinic triangle T α , that is O α , O α +1 , and O α +2 , andthe smooth paths connecting them, is homeomorphic to the boundary of a closed trianglein the plane. Further, the interior of T α is f W uα , by the unstable manifold theorem and19heorem 3.12, homeomorphic to an open disk in the plane (and so also to the interior ofa triangular region). The only complication may arise from the behavior of the unstablemanifold near the edges or vertices. For instance, the f W uα could be “folded” as it approachesthe edge connecting O α +1 and O α +2 so that a neighborhood of a point on the edge is notlocally homeomorphic to a point on an edge of a closed triangle in the plane.Consider the dynamics projected onto three consecutive dimensions; for simplicity of nota-tion, we will use the standard x, y, z coordinates, and label the saddle points on each axis O x , O y , and O z , where there are heteroclinic connections O x → O y → O z and O x → O z ,which we denote as Γ xy , Γ xz , and Γ yz . We denote the corresponding heteroclinic triangleby T x . Proposition A.5.
Each orbit in f W u ( O x ) is uniquely identified with an angle ≤ φ ≤ π .Proof. We define the usual δ -neighborhoods N δ ( O x ), N δ ( O y ), and N δ ( O z ). By the UnstableManifold Theorem, we may choose δ small enough that W u ( O x ) is the graph of a functionof ( y, z ) in N δ ( O k ). Using the monotonicity of the z coordinate inside the invariant region,we may also choose δ small enough such that once a trajectory leaves N δ ( O x ) it cannotreturn to it. By the previous observations, the other coordinates (other than x , y , and z )of f W u ( O x ) are all zero.Now consider that each orbit in f W u ( O x ) has a unique intersection point w with the bound-ary of N δ ( O x ). Since f W u ( O x ) is a graph over ( y, z ) consider the projection ¯ w of w onto( y, z )-plane. Let φ denote the angle of the ray through the origin and ¯ w with the z -axis.Note that φ lies between 0 and π/
2. The extreme angles φ = 0 and φ = π/ xz and Γ xy respectively. Definition A.6.
For a point p ∈ f W u ( O x ) ∪ O x define d ( p ) to be the distance from O x to p along the orbit containing p . For a point p ∈ f W s ( O z ) ∪ O z define e ( p ) to be the distancefrom p to O z along the orbit containing p . Proposition A.7.
The arc lengths d and e are finite and continuous where defined.Proof. By basic existence theory the orbits are smooth and therefore arc length is locallywell-defined on them.Denote the solution with initial value p as p ( t ). Recall from the Unstable Manifold Theoremthat p ( t ) → O x as t → −∞ and in fact | O x − p ( t ) | ≤ Ce ( λ lu − e u ) t , for t ≤ , (21)where λ lu is the leading unstable eigenvalue, i.e. in this case the minimum of σ − ρ σ and σ − ρ σ , and e u may be chosen to be arbitrarily small.20ow consider the length of the orbit: d ( p ) = Z −∞ p ˙ x + ˙ y + ˙ z dt = Z τ p ˙ x + ˙ y + ˙ z dt + Z τ −∞ p ˙ x + ˙ y + ˙ z dt (22)If we substitute the equations (6) into the integrals, then substitute (21) into the secondintegral one easily sees that this integral goes to zero as τ → −∞ . Thus d ( p ) is finite.Now consider the orbits starting at points q close to p . Fix ǫ > τ so thatthe remainder integral above is less than ǫ/
3. Since (21) implies that | q ( t ) − p ( t ) | goes tozero exponentially as t → −∞ , we can make the difference in the remainders less than2 ǫ/ q ( τ ) is sufficiently close to p ( τ ). This we can accomplish by requiring q (0) and p (0)sufficiently close (by continuous dependence on initial conditions). We can also make theintegrals from τ to 0 less than ǫ/ q (0) close to p (0). For such q , | d ( q ) − d ( p ) | < ǫ .The arc length e ( p ) is finite for any p not in O x ∪ Γ xy ∪ O y since O z is a stable node inthe xyz -subspace and all orbits approach O z exponentially in time and so the same typeestimates as above hold here. Continuity of this distance also follows by a similar proof asfor d . Corollary A.8.
All orbits on T x have finite length and this length is a continuous functionof an initial point on f W u ( O x ) \ (Γ xy ∪ O y ∪ Γ yz )This follows from the continuity of d and e where they are defined. Lemma A.9.
The arc length d can be extended continuously to the edge Γ yz . The arclength is finite and uniformly bounded for all orbits that make up T x .Proof. Denote by D φ the length of the orbit in f W u ( O x ) identified by the angle φ for0 ≤ φ < π/ xy and Γ yz . In fact since Γ xy ⊂ f W u ( O x ) ∩ f W s ( O y ), and Γ yz ⊂ f W u ( O y ) ∩ f W s ( O z ) the arguments above show that the lengths of thesetwo solutions arcs are finite. Let D xy denote the length of Γ xy and D yz the length of Γ yz .Set D π/ = D xy + D yz , i.e. the combined length of Γ xy and Γ yz .For p ∈ Γ yz set: d ( p ) = D π/ − e ( p ) , i.e. the length to p along Γ xy and Γ yz . We claim that d thus defined is continuous at p ∈ Γ yz .Fix ǫ >
0. Consider a δ (sup norm) neighborhood N of O y . For any φ sufficiently close to π/
2, the orbit on T x indexed by φ passes through N . Denote by P − the point where Γ xy N and by P + the point where Γ yz intersects N . By continuity of d and e we maychose such a δ sufficiently small that: D xy − d ( P − ) < ǫ/ D yz − e ( P + ) < ǫ/ . Further, require that δ be sufficiently small so that: δ (cid:16) λ ls + e s + λ u + e u (cid:17) < ǫ/ φ sufficiently close to π/
2, denote by Q − and Q + the points where the orbit withangle φ intersects N . Again by continuity of d and e for all φ sufficiently close to π/ | d ( Q − ) − d ( P − ) | < ǫ/ e ( Q + ) − e ( P + ) < ǫ/ . Let q be a point sufficiently close to p so that the above conditions on φ are satisfied andsuch that | e ( q ) − e ( p ) | < ǫ/ d ( q ) will be equal d ( Q − ) plus the arc length from Q − to Q + plus the arc length from Q + to q . Denote these arc lengths by ℓ ( Q − , Q + ) and ℓ ( Q + , q ) respectively.Within the neighborhood N we can transform coordinates to straighten the unstable man-ifold Γ yz . In this subspace this unstable manifold is the graph of a function of z only andhas that has the form: y = σ y + h ( z ) . We thus straighten f W u ( O y ) by the coordinate change Y = y − σ y + h ( z ).By the Shilnikov variables technique [27], we have that the stable variables satisfy: | ( x ( t ) , Y ( t )) | < δe ( λ ls + e s ) t , ≤ t ≤ τ, while the unstable variable z satisfies: | z ( t ) | < δe ( λ u − e u )( t − τ ) , ≤ t ≤ τ, where τ is the time of passage through N . We then obtain: ℓ ( Q − , Q + ) = Z τ p ˙ x + ˙ y + ˙ z dt< Z τ | ˙ x | + | ˙ y | + | ˙ z | dt< δ Z τ e ( λ ls + e s ) t + e ( λ u + e u )( t − τ ) < δ λ ls + e s + 1 λ u + e u < ǫ . (23)22e have that d ( q ) − d ( p ) = d ( Q − ) + ℓ ( Q − , Q + ) + ℓ ( Q + , q ) − ( D xy + D yz − e ( p ))= d ( Q − ) + ℓ ( Q − , Q + ) + e ( Q + ) − e ( q ) − D xy + d ( P − ) − d ( P − ) − D yz + e ( P + ) − e ( P + ) + e ( p )= d ( Q − ) − d ( P − ) + ℓ ( Q − , Q + ) + e ( Q + ) − e ( P + )+ d ( P − ) − D xy + e ( P + ) − D yz + e ( p ) − e ( q ) . (24)Combining the above estimates we have: | d ( q ) − d ( p ) | < ǫ .Now let D φ , 0 ≤ φ ≤ π/ φ . By thecontinuity of d on all of T x , except O x and the continuity of e in a neighborhood of O z , D φ depends continuously on φ . It is thus uniformly bounded. Proof of Theorem A.4.
For each coordinate pair p = ( φ ( p ) , d ( p )), we normalize the ar-clength d by leting u = d/D φ Note that in this normalized distance u ( O z ) = 1 and so u iscontinuous on T x .We will show that the heteroclinic triangle T x with corners O x , O y , and O z is homeomorphicto the triangle ABC in the plane with corners at A = (0 , B = ( b, /
2) and C = (1 , w on T identified by ( φ, u ), consider the map: H : w ( u, v ) = ( u, h ( u, φ )) . where v = h ( u, φ ) is given by the continuous map: v = ( u b tan( φ/
2) if 0 ≤ u ≤ b − u − b ) tan( φ/
2) if b < u ≤ . The map H is a homeomorphism. It is one-to-one at all points. It maps O x , O y and O z onto A , B and C respectively. At interior points it is a local homeomorphism because itis a composition of local homeomorphisms. At the corners O x and O z , considering themappings as polar coordinates make it clear that H is a local homeomorphism there. It isalso clear that H is one-to-one along the edges Γ xz , Γ xy and Γ yz .Thus Figure 5 represents not merely an easy-to-understand representation of Γ, but a tri-angulation. We now investigate the orientibility of Γ. The most common definition oforientibility, in terms of normal vectors, is not useful in this situation, but having triangu-lated Γ, we may use instead a less common, but still standard definition (see e.g. [7]). Definition A.10.
Consider some arbitrary triangle in the triangulation of a manifold.Assign to each triangle in the triangulation a value of clockwise. This assigns correspondingdirections to each side of each of the triangles. Now let ∆ i and ∆ j be triangles sharing aside; observe that the common side has been assigned two different directions, i.e. twoadjacent triangles with the same orientation conflict on their shared side. If this can bedone without contradiction, then the surface is orientable. If a contradiction is reached, thesurface is non-orientable.
23t is a standard result of algebraic topology that orientibility is independent of the specifictriangulation used.Using this definition and the specific triangulation we have defined, the following lemmacan be easily proven.
Lemma A.11. If p is odd, Γ is non-orientable. If p is even, Γ is orientable.Proof. Let p be odd. We consider the triangulation by heteroclinic triangles. Start with∆ , the triangle defined by O , O , and O and without loss of generality assign it theclockwise orientation. Thus the direction on the O − O side of the triangle is given thedirection O → O . Likewise, the triangle ∆ i defined by O p , O , and O is assigned theclockwise orientation, and the O − O side of the triangle is given the direction O → O .But ∆ i and ∆ i are adjacent triangles, and the fact that they give the same direction to the O − O side means that the surface is non-orientable.Let p be even. We consider the triangulation by heteroclinic triangles. Giving the triangledefined by O , O , and O the clockwise orientation, and thus giving the O − O edge the O → O direction, we clearly observe that making the O p , O , O triangle clockwise givesthe O − O boundary, the only place where orientibility could be broken, the conflicting O → O direction.We cite one more result, a classification theorem [8]. Theorem A.12.
Given a compact connected triangulable -manifold Y with boundary suchthat ∂Y has k components, Y is homeomorphic to X -with- k -holes, where X is S or the n -fold torus T n or the m -fold projective plane P m . A “hole” in the sense of the theorem is a set homeomorphic to an small ǫ open ball.Our extensive build-up makes the proof of the theorem almost trivial. Theorem A.13. If p is odd, then Γ is homeomorphic to a M¨obius strip. If p is even, Γ ishomeomorphic to a cylinder.Proof. Let p be odd. Of the possible homeomorphic images named in Theorem A.12, onlythe projective plane is non-orientable; since we know from Lemma A.1 that Γ has onlyone boundary component, Γ is homeomorphic to the projective plane with a ball removed,which is homeomorphic to the M¨obius strip.Let p be even. Then Γ is a rectangle with two of its edges identified. Since the “right-hand”point of the bottom edge is identified with the “right-hand” point of the top edge, it is acylinder. 24 eferences [1] Afraimovich V S, Cuevas D and Young T 2013 Sequential dynamics of master-slavesystems Dynamical Systems: an International Journal (2) 154 - 172[2] Afraimovich V S, Tristan I, Huerta R and Rabinovich M 2008 Winnerless competitionprinciple and prediction of the transient dynamics in a Lotka-Volterra model CHAOS Discontinuity, Nonlinearity and Complexity (1) 21 - 41[4] Afraimovich V S, Zhigulin V and Rabinovich M 2014 On the origin of the reproducibleactivity in neural circuits CHAOS The Journal of Mathematical Neuroscience (13)[6] Ashwin P and Postlethwaite C M 2013 On designing heteroclinic networks from graphs Physica D
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