Two dimensional heteroclinic attractor in the generalized Lotka-Volterra system.
We study a simple dynamical model exhibiting sequential dynamics. We show that in this model there exist sets of parameter values for which a cyclic chain of saddle equilibria, , , have two dimensional unstable manifolds that contain orbits connecting each to the next two equilibrium points and in the chain (). We show that the union of these equilibria and their unstable manifolds form a -dimensional surface with boundary that is homeomorphic to a cylinder if is even and a Möbius strip if is odd. If, further, each equilibrium in the chain satisfies a condition called “dissipativity,” then this surface is asymptotically stable.
In the last decade it became clear that typical processes in many neural and cognitive networks are realized in the form of sequential dynamics (see , , , , ,  and references therein). The dynamics are exhibited as sequential switching among metastable states, each of which represents a collection of simultaneously activated nodes in the network, so that at most instants of time a single state is activated. Such dynamics are consistent with the winner-less competition principle [26, 2]. In the phase space of a mathematical model of such a system each state corresponds to an invariant saddle set and the switchings are determined by trajectories joining these invariant sets. In the simplest case these invariant sets may be saddle equilibrium points coupled by heteroclinic trajectories, and they form a heteroclinic sequence (HS). This sequence can be stable if all saddle equilibria have one-dimensional unstable manifolds , in the sense that there is an open set of initial points such that trajectories going through them follow the heteroclinic ones 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  and . General results on the stability of heteroclinic sequences have been obtained by Krupa and Melbourne [17, 18] in the form of necessary conditions that may also be sufficient in the presence of certain algebraic conditions.
An instability of a HS may be caused if some initial conditions follow trajectories on the unstable manifold different from the heteroclinic ones. M. Rabinovich suggested  considering the case when all trajetories on the unstable manifold of any saddle in a HS are heteroclinic to saddles in the HS, which implies an assumption that the HS is, in fact, a heteroclinic cycle. In this case one can expect some kind of stability, not of the HS, of course, but of the object in the phase space formed by all heteroclinic trajectories of the saddles in the HS. We study in our paper the Rabinovich problem. We deal here with the generalized Lotka-Volterra model  that is a basic model of sequential dynamics for which unstable sets are realized as saddle equilibrium points. All variables and parameters may take only nonnegative values, so we work in the positive orthant of the phase space . We impose some restrictions on parameters under which all unstable manifolds of the saddle point are two-dimensional and all trajectories on them (in the positive orthant) are heteroclinic in some specific way (see below). We prove that they form a piece-wise smooth manifold homeomorphic to the cylinder if the number of the saddle points in the HS is even or to the Möbius band if it is odd. We prove also that under the additional assumption that each equilibrium is dissipative (see below), then this manifold is the maximal attractor for some absorbing region (in the positive orthant). Trajectories in this region may follow different heteroclinic trajectories and may manifest some kind of weak complex behavior.
2 Notation and Results
We begin our consideration of two-dimensional heteroclinic channels with the study of a series of Lotka-Volterra equations. Lotka-Volterra models are widely used in the context of heteroclinic sequences where all the unstable manifolds are one dimensional, so that each equilibrium is connected to exactly one subsequent equilibrium (e.g. [13, 28, 10, 9]). It has been recently shown that they ask provide a general method for embedding directed graphs into a system of ordinary differential equations . We remark that although  allows graphs of high valency to be modeled by heteroclinic networks, and some work has been done on the stability of such systems (e.g. , which considers the competing dynamics of the “overlapping” channels and ), study of heteroclinic networks has usually only considered one-dimensional unstable manifolds. In keeping with the discussion of the introduction, we consider the dynamics of a system such that initial conditions arbitrarily close to any of saddle nodes may be mapped into neighborhoods of either one of two other saddle nodes. In particular, a simple general model for heteroclinic sequential dynamics was given in  by
where all of the parameters are assumed to be positive. The constants have biological meaning, representing inhibition of mode by mode . For the sake of simplicity, we assume that for all . Further, since the variable is assumed to encode biological information that is necessarily non-negative, e.g. activation levels or chemical concentrations, we restrict the system to the first closed orthant, . The system is so constructed that it contains saddle points lying on the axes, with being the -th coordinate of the -th saddle along the -th axis (), i.e., the system (1) has equilibrium points of the form , for . There may be other equilibria as well, but they are not relevant for our purposes; we only study transitions between the equilibria just defined.
In the present work we suppose that the first equilibria points are sequentially connected by a set of -dimensional unstable manifolds. For each , , there will be a heteroclinic orbit connecting to and a heteroclinic orbit connecting to . Furthermore, the system is closed in the sense that , i.e. is the modulus of the subscript. In the following, we will consider the restrictions necessary to enforce such dynamics.
To ensure that there are heteroclinic trajectories between and both and , we apply eigenvalue conditions to the system. Consider first . The linearization of the vector field (1) at is given by the upper triangular matrix:
and so the eigenvalues appear on the diagonal. The matrices have a similar simple structure, zeros everywhere except on the diagonal and on the -th row. It is then easy to see that the eigenvalues of at are
In particular, the eigenvalues are all real. One can also see from the structure of (2) that has a full set of eigenvectors, even if some eigenvalues are repeated.
Note that because of the particular form of the equations, all coordinate axes, planes and hyperplanes are invariant. Thus, in order that trajectories can travel from to or , it is sufficient to put the restriction , and otherwise, to ensure that they can only go in those directions. In particular, for each , , we require that
Note also that . These inequalities guarantee that each equilibrium is a hyperbolic saddle with unstable directions and stable directions. Let be the unstable manifold of restricted to the positive orthant. We show below that
forms a piecewise smooth surface that we will classify topologically as follows.
Suppose that inequalities (3) and (4) hold for each , and that each unstable manifold is contained in a compact forward invariant set as specified in Lemma 3.10 (see also Remark 3.11). When is even, the union of unstable manifolds is homeomorphic to a cylinder. When is odd, is homeomorphic to a Möbius strip
The proof of this theorem, which is slightly involved, is left for the appendix.
Consider the following definition.
Let and be the set of stable and unstable eigenvalues, respectively, of the linearization of a vector field at a saddle equilibrium, i.e., and . We say that the saddle is dissipative if
In other words, the weakest stable eigenvalue is stronger than the strongest unstable eigenvalue. (See .)
In terms of the specific vector field under study all the eigenvalues in question are real and for each we have:
The main goal of this manuscript is to show that, under the condition that each saddle equilibrium is dissipative, then is asymptotically stable. Specifically, our main theorem is:
The proof of Theorem 2.3 breaks roughly into two independent pieces. We start by considering the trajectory of a representative point that is -close to , but is distant from each of the fixed points . In such a case, the dynamics of the system are controlled largely by three consecutive saddles , , and . In Section 3, we consider the restriction of (1) to three consecutive dimensions; we will gain information on the full-dimensional system by viewing it as a perturbation of this restriction. The second part of the proof is to consider the dynamics as a trajectory passes near a fixed point; we consider this in Section 4.
3 Three Dimensions
We begin our proof of Theorems 2.1 and 2.3 with a study of the restriction of the system to three dimensional sub-spaces corresponding to three consecutive coordinate directions. Through a series of geometric lemmas, we prove the main result of the section, Theorem 3.12, which provides information on the behavior of trajectories inside this invariant subspace. This will be used in later sections to complete the proofs of the main results by providing information on those parts of the full phase space where all but three coordinates are small.
Let , , and be any three consecutive equilibria and restrict the system (1) to the three dimensions spanned by consecutive coordinates , , and . For convenience, we will refer to these three variables as , , and . Our goal in three dimensions is to show the existence of a compact, forward invariant set containing , , and such that any trajectory with an initial value in the interior of that set converges to .
Restricted to three dimensions, the equations (1) are reduced to
The restriction of the dimension of the unstable manifolds via the eigenvalue conditions, and the positivity conditions on yield the following inequalities:
Of those inequalities, (7) controls the behavior of the system at , (8) - (9) control the behavior of the system at , and (10) - (11) at . The point is a saddle with a two-dimensional unstable manifold, is a saddle with a one-dimensional unstable manifold, and is a sink for the system (6). We remark that although (6) has the form of the May-Leonard model, the particular parameter restrictions under consideration yield simple dynamics (see Theorem 3.12), and prevent the more complex behavior usually studied in that context. In particular, they are inconsistent with the symmetric May-Leonard model as it was introduced in .
For each , we will be interested in the points where ; each such set is the union of the plane and some “nontrivial” plane. We designate those planes:
where on .
We will also refer to the plane passing through the points , , and , which we denote by . Observe that is given by the equation
For ease of discussion, we will also name the coordinate planes:
Because of the positivity conditions on the parameters and variables, we may use , , , and as shorthand for the intersection of those planes with the first octant without the risk of confusion. We observe that the intersection of and of each with is a compact 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:
Observe that each plane can be written as the graph of a function . A plane dominates a plane if and implies that for all . A plane is dominated by a plane if dominates .
3.2 Geometrical Lemmas
Each plane divides into two regions, one where is positive and another where it is negative. This allows information about to be gained from purely geometric information. For example, is positive below , and negative above it. Since dominates (i.e. is always above it), we instantly see that (see Corollary 3.5, given below). We introduce geometric lemmas giving the information we can gain in this way; the proofs of all of them are parallel to one another, and can be summarized as follows: since the system is restricted to the first octant, we compare two planes (triangles) by seeing where they intersect the , , and axes. Denote the compact triangle thus formed by a plane as and the non-zero component of its vertex on the -th axis as . Then a plane dominates a plane if for , with at least one of those a strict equality, i.e. if its vertices are farther from the origin.
The plane is dominated by the plane .
We consider where each plane intersects each axis:
The planes and both intersect the axis at the point .
The plane intersects the axis at , while intersects the axis at . We know from (9) that .
The plane intersects the axis at , while intersects the axis at . We know from (10), that .
Since for all , is dominated by . ∎
The plane is dominated by .
We consider where each plane intersects each axis:
The plane intersects the axis at the point , while intersects the axis at . We know from (7) that .
The plane intersects the axis at , while intersects the axis at . We know from (8) that .
The planes and both intersect the axis at .
Since for all , dominates . ∎
The property “is dominated by” is clearly transitive, so the following corollary holds.
is dominated by .
Since above the plane, the following corollary follows immediately.
On the plane , .
If we further had that dominates , then the eigenvalue conditions introduced in  and summarized as (3) - (8) would be sufficient to ensure the existence of a positively invariant region. It happens, however, that this is not the case.
The plane neither dominates nor is dominated by .
We consider where each plane intersects each axis:
intersects the axis at the point , while intersects the axis at . We know from (7) that , and therefore does not dominate .
intersects the axis at , while intersects the axis at . We know from (11) that , and therefore does not dominate .
Thus, neither plane dominates the other. ∎
The situation is somewhat salvaged by the following.
If , then is dominated by .
The parameter restriction in (3.7) written in terms of the general systems gives that for each , ,
Similarly to Corollary 3.5, we have the following:
In the region of parameter space where the hypotheses of Lemma 3.7 are satisfied, .
We note here one additional observation.
The rectangular box:
is forward invariant with respect to the system (1). Further, the unstable manifolds are all contained in .
Forward invariance follows immediately from the differential equations (1) and the assumption that . The conclusion that the unstable manifold at is inside follows easily by noting that the unstable eigenspace at (restricted to the first orthant) is strictly inside .
3.3 Existence of a Positively Invariant Region
We remarked that our goal in three dimensions was to show the existence of a compact, forward-invariant set whose trajectories converge to . We now carry this out.
Since the pair of inequalities and form a sufficient, but not necessary, condition for the inequality to be satisfied, a positively invariant region might exist even if the hypothesis of Lemma 3.7 is not satisfied. For instance, since the box (Lemma 3.9) is forward invariant in may be that is negative even when the hypotheses of Lemma 3.7 fail. Lemma 3.10 can therefore be extended to a region of parameter space that includes the region defined by Lemma 3.7 as a proper subset.
Suppose that , , , and enclose a positively invariant region. Any trajectory in the above-described region that is not contained in (the plane) goes to as .
Any trajectory in a compact positively invariant region has a non-empty -limit set. Fix a trajectory with initial condition in the interior of the region, and let be an arbitrary point in the -limit set . The proof breaks into three parts: we prove that lies on , that it lies on , and that it lies on ; the intersection of these planes is . The proofs of the second and third statements are essentially the same as the proof of the first statement, which we treat in detail.
By way of contradiction, suppose that does not lie on . First of all, note that does not lie on , because by assumption, the , and since the trajectory increases in the variable, cannot approach
Since is continuous, and lies on neither nor , the regions where , we can find a spherical neighborhood centered at with radius such that for some . It follows that is not a fixed point, and is not the singleton , since limit sets are forward invariant .
The component of is non-decreasing in the positively invariant set and it is bounded above, since it is bounded above by . Thus it has a limit . It follows by continuity that the component of any point in is also . We have supposed that but ; now consider . Since at , and thus in a neighborhood of , the component of must be strictly increasing as a function of time along the forward solution near . This contradicts that the component is everwhere on .
We repeat the argument twice. First, must lie on the plane ; otherwise , and the same contradiction will be obtained. Then, using the same argument, we see that it must lie on . We observe here, with reference to Lemma 3.7, that even if is not dominated by , it is dominated by it on the restriction to , and the argument therefore goes through. Thus must lie on the single point, , and our proof is complete. ∎
4 The Dynamics in the Full Phase Space
4.1 Local Dynamics Near an Equilibrium
We assumed in (5) that the saddle points are dissipative. Denote by the ratio:
We call the minimal saddle value of the equilibrium (see ). The equilibrium is dissipative if .
The main implication of this assumption is that a trajectory starting at an initial value at a distance from the stable manifold of the saddle comes to a point of distance on the order of 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 into the -dimensional unstable subspace and the -dimensional stable subspace . Let denote the sup norm in these local coordinates. By the Stable Manifold Theorem, for each there exists such that in a -neighborhood of , the unstable manifold is the graph of a (smooth) function . Let be the minimum of these and consider the -neighborhood of each ,
For fixed , and sufficiently small, define a pair of sections,
By the classical Shil’nikov variables technique (see ), there exists sufficiently small so that every forward solution starting at will intersect before leaving . Let be the minimum of the ’s needed in the neighborhood of each . Let denote the time at which such a solution intersects .
If and are sufficiently small and if and then , where is independent of the initial point and .
For ease of notation, consider . Note from (2) that the eigenvector corresponding to the first diagonal element, , can have only one non-zero component and that is in the direction. The eigenvector corresponding to the -th diagonal element, , has non-zero components in the and directions only.
Now take into account that the hyperplane is invariant under the flow of the equations and tangent to the stable eigenspace of . It thus coincides locally with and contains .
Next note that the unstable eigenspace has no non-zero components in the coordinate directions and that the hyperplane where they are zero is invariant under the flow. It thus follows that the coordinates of the unstable manifold are all zero. Thus the unstable manifold is given locally in the -neighborhood of as the graph where and is smooth. Consider the local change of variables that “straightens out” the unstable manifold:
Under this smooth change of coordinates (1) becomes:
Since is contained in the coordinate hyperplane, we do not need a change of variables corresponding to . Noting the , we use the MVT to define a new function , and rewrite the equations to obtain:
Now note that distances from and are not effected by the coordinate change (17). If we now denote then is the distance to the unstable manifold and is the distance to the stable manifold (in the sup norm).
It now follows immediately from (19) that inside the neighborhood of the unstable directions satisfy the estimates:
where we can take arbitrarily small by starting with and small. Thus, by a simple application of Gronwall’s inequality, solutions starting on and remaining in our neighborhood of must satisfy:
where is the maximal unstable eigenvalue, i.e.,
Thus if is defined by , then
Similarly, using Gronwall’s inequality once again we obtain:
where is the “leading” stable eigenvalue, i.e. since the eigenvalues are real, the negative eigenvalue with the smallest absolute value. In terms of our parameters:
where can be taken arbitrarily small. By the assumption that is dissipative, and so we can make .
4.2 Stable Heteroclinic Surface
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 under appropriate parameter restrictions.
Theorem (Theorem 2.3, restated).
These inequalities are fundamental to the problem we are considering. They ensure that there is a heteroclinic channel from each equilibrium to the equilibria and (the first inequality), and that there are no heteroclinic channels to the other equilibria (the second inequality).
The inequality (5) states that for each ,
This is the dissipativity condition. Informally, it may be taken to say that when a trajectory that is close to one of the unstable manifolds whose union is passes near one of the saddle fixed points, the distance of the trajectory to contracts exponentially.
It will become clear, from the proof of Theorem 2.3, that (5) is a much stronger condition than is necessary. It ensures not only asymptotic stability, but a sort of monotonic asymptotic stability, such that whenever the trajectory passes near some , its distance to contracts exponentially. If, on the other hand, (5) held for some, but not all, values of , then the trajectory would at times contract exponentially towards , and at other times drift away from , and stability would depend on how these attractive and repulsive forces average over time. Attempting to formulate a replacement condition for (5) that is necessary as well as sufficient is extremely nontrivial.
The final hypothesis, that each unstable manifold is contained in a compact forward invariant set, is necessary. We have seen one set of inequalities that ensure that such sets exist,
which are sufficient but not necessary. In Section 4.3, we see that there is a nonempty open region of parameter space where the hypotheses of Theorem 2.3 hold. From our current discussion, we see that the theorem applies to a larger region.
We recall that is asymptotically stable if given any neighborhood of (restricted to there exists an -neighborhood of , say , such that if then for and , where is the solution of the initial value problem (1) with . When we speak of an open -neighborhood of , we are speaking of a set that is open in the subspace topology. That is, an -neighborhood in is an -neighborhood in intersected with .
Proof of Theorem 2.3.
For each , let be a sufficiently small neighborhood of , such that Theorem 4.1 can be applied within each and does not depend on .
Let be a representative point at an initial condition -close to . Choose such that . Then we can classify the dynamics as either local if for some , and global otherwise.
We observe that if for any , its behavior is trivial, and likewise, if lies on a coordinate plane, it remains on that plane while converging exponentially to some . We therefore assume without loss of generality that neither of these cases hold.
Suppose that for any . The point is -close to , and since is a finite union, we can say that is -close to , where is fixed and depends on . Consider the projection of the system onto the three-dimensional subspace spanned by the axes , , and . In three dimensions, the specific route a solution takes has not been important; a trajectory in the invariant set may go straight to a neighborhood of , or it may detour to , but the net result is the same (Theorem 3.12). We now formally differentiate between these two cases.
We consider two cases: either the positive semitrajectory of intersects without first intersecting ) (case (i)), or the positive semitrajectory intersects , then intersects (case (ii)).
Before proceeding, we recall the definitions of and given in Theorem 4.1, and similarly define such sections for . Without loss of generality, we assume that .
Suppose that case (i) occurs. For each , let be the projection of into the three dimensions spanned by ; note that is negligable for . Then the projection of the orbit onto intersects before it can intersect . In , we know that all trajectories inside of that do not intersect come to a neighborhood of in bounded time, where the bound does not depend on the initial condition. We may consider the non-projected, full-dimensional space as a “perturbation” of the projected space, and cite smooth dependence of initial conditions; the trajectory going through a slightly perturbed initial point corresponding to such a case must enter in a well-behaved way. In particular if belongs to then a mapping from a neighborhood of on to is well defined and Lipschitz-continuous.
Suppose that case (ii) occurs. Then once an orbit of enters , it starts to manifest the dissipative behavior. In particular, if , then after passing through , , by Theorem 4.1. Once the representative point leaves , we may apply (i), viewing its position after leaving the neighborhood as an initial condition that does not re-enter the neighborhood . Thus when the representative point finally enters , its distance to has been contracted by an order of , where and C absorbs both the constant from Theorem 4.1 and a Lipschitz constant.
Suppose now that , where is now fixed. Then the trajectory leaves without increasing its distance from the unstable manifold, and passes into as just described. We may then apply Theorem 4.1. As the trajectory passes through , its distance from the unstable manifold is contracted on an order of .
Since the mapping contracts in the global dynamics and is Lipschitz (or contracting) in the local dynamics, simple inductions yields that as a representative point moves through the system, its distance from the manifold changes from to to , and so on.
For a fixed , the value , representing a ratio of eigenvalues, is likewise fixed. The value is not; it depends on the distance between the representative point and the stable manifold as the trajectory enters , which changes from one instance to the next. For a given , however, there is some maximal value that can take, since the system is constantly contracting towards the manifold and goes to along with that distance. Thus there exists a global value, for all and all , such that passing from the first to the -th unstable manifold is a contraction of order . ∎
4.3 Existence of Parameter Sets
Throughout the paper, we have put a number of restrictions on the parameters of the system. One must ask whether the specified inequalities may be satisfied.