Asymptotic Dynamics of Hamiltonian Polymatrix Replicators
AASYMPTOTIC DYNAMICS OF HAMILTONIANPOLYMATRIX REPLICATORS
HASSAN NAJAFI ALISHAH, PEDRO DUARTE, AND TELMO PEIXE
Abstract.
In a previous paper [3] we have studied flows on poly-topes presenting a method to encapsulate its asymptotic dynamicsalong the heteroclinic network formed by the polytope’s edges andvertices. These results apply to the class of polymatrix replicatorsystems, which contains several important models in EvolutionaryGame Theory. Here we establish the Hamiltonian character of theasymptotic dynamics of Hamiltonian polymatrix replicators.
Contents
1. Introduction 12. Outline of the construction 23. Poisson Poincaré maps 64. Hamiltonian polymatrix replicators 115. Asymptotic dynamics of polymatrix replicators 146. Hamiltonian character of the asymptotic dynamics 227. Example 32Acknowledgements 39References 391.
Introduction
A new method to study the asymptotic dynamics of flows defined onpolytopes is presented in [3]. This method allows us to study the as-ymptotic dynamics of flows defined on polytopes along the heteroclinicnetwork formed out of the polytope’s vertices and edges. Examplesof such dynamical systems arise naturally in the context of evolution-ary game theory (EGT) developed by J. Maynard Smith and G. R.Price [18].One such example is the polymatrix replicator , introduced in [2, 4],that is a system of ordinary differential equations developed to studythe dynamics of the designated polymatrix game . This game modelsthe time evolution of the strategies that individuals from a stratifiedpopulation choose to interact with each other.The polymatrix replicator induces a flow in a polytope defined bya finite product of simplices. These systems extend the class of the a r X i v : . [ m a t h . D S ] F e b ALISHAH, DUARTE, AND PEIXE replicator and the bimatrix replicator equations studied in [19] andin [16, 17], respectively.In [2] the authors have introduced the subclass of conservative poly-matrix replicators (see Definition 4.4) which are Hamiltonian systemsw.r.t. appropriate Poisson structures. Moreover, in [4] the study ofthis subclass is developed toghether with the subclass of dissipativepolymatrix replicators.In this paper we will study the asymptotic dynamics of the conser-vative polymatrix replicators, studying in particular its Hamiltoniancharacter. Namely, the main result of this paper states that for con-servative polymatrix replicators a flow map defined in the dual of thephase space is Hamiltonian w.r.t. some appropriate Poisson structureon a system of cross sections (Theorem 6.19).For this, we need to remember the main strategy of the methoddeveloped in [3], trying to avoid as much as possible the formal defi-nitions as well the technical details that can be seen in more detail inthe referred paper.The paper is organized as follows. In Section 2 we establish the nec-essary background (based on [3]) to contextualize the results of thispaper. In particular we outline the construction of the asymptotic dy-namics for a large class of flows on polytopes that includes the polyma-trix replicators. In Section 3 we define a Poincaré map for Hamiltoniansystems on Poisson manifolds. In Section 4 we provide a short intro-duction to polymatrix replicators, following [2]. Namely, we state thebasic definitions and results for the class of conservative polymatrixreplicators, that, based on the main results of [2, 4], we designate as
Hamiltonian polymatrix replicators . In Section 5 we recall the tech-nique developed in [3] to analyze the asymptotic dynamics of a flowalong the heteroclinic network formed by the edges and vertices of thepolytope where it is defined. In particular we review the main defini-tions and results for the polymatrix replicator vector field. In Section 6we study the Poisson geometric properties of the Poincaré maps in thecase of Hamiltonian polymatrix replicators. Finally, in Section 7 wepresent an example of a Hamiltonian polymatrix replicator with a nontrivial dimension to provide an illustration of the concepts and mainresults of this paper. The graphics in this section were produced with
Wolfram Mathematica and
Geogebra software.2.
Outline of the construction
We now outline the construction of the asymptotic dynamics for alarge class of flows on polytopes that includes the polymatrix repli-cators. A polytope is a compact convex set in some Euclidean spaceobtained as the intersection of finitely many half-spaces. A polytope iscalled simple if the number of edges (or facets) incident with each vertexequals the polytope’s dimension. Prisms, phase spaces of polymatrix
AMILTONIAN POLYMATRIX REPLICATORS 3 replicators, are examples of simple polytopes. In [3] we consider andstudy analytic vector fields on simple polytopes which have the prop-erty of being tangent to every face of the polytope. Such vector fieldsinduce complete flows on the polytope which leave all faces invariant.Vertexes of the polytope are singularities of the vector field, while edgeswithout singularities, called flowing edges , consist of single orbits flow-ing between two end-point vertexes. The vertexes and flowing edgesform a heteroclinic network of the vector field. The purpose of thisconstruction is to analyse the asymptotic dynamics of the vector fieldalong this one-dimensional skeleton. Throughout the text we assumethat every vector field is regular . This means that the transversal de-rivative of the vector field is never identically zero along any facet ofthe polytope.The analysis of the vector field’s dynamics along its edge heteroclinicnetwork makes use of Poincaré maps between cross sections tranversalto the flowing edges. Any Poincaré map along a heteroclinic or ho-moclinic orbit is a composition of two types of maps, global and localPoincaré maps. A global map , denoted by P γ , is defined in a tubu-lar neighbourhood of any flowing-edge γ . It maps points between twocross sections Σ − γ and Σ + γ transversal to the flow along the edge γ . A local map , denoted by P v , is defined in a neighbourhood of any vertexsingularity v . For any pair of flowing-edges γ, γ (cid:48) such that v is both theending point of γ (cid:48) and the starting point of γ , the local map P v takespoints from Σ + γ (cid:48) to Σ − γ . See Figure 1. Figure 1.
Local and global Poincaré maps along a heteroclinicorbit.
Asymptotically, the nonlinear character of the global Poincaré mapsfade away as we approach a heteroclinic orbit. This means that thesenon-linearities are irrelevant for the asymptotic analysis. For regularvector fields, the skeleton character at a vertex, defined as the set ofeigenvalues of the tangent map along the edge eigen-directions, com-pletely determines the asymptotic behaviour of the local Poincaré mapat that vertex.
ALISHAH, DUARTE, AND PEIXE
To describe the limit dynamical behaviour we introduce the dual cone of a polytope where the asymptotic piecewise linear dynamics unfolds.This space lies inside R F , where F is the set of the polytope’s facets.The dual cone of a d -dimensional simple polytope Γ is the union C ∗ (Γ) := (cid:91) v ∈ V Π v , where for each vertex v , Π v is the d -dimensional sector consisting ofpoints y ∈ R F with non-negative coordinates such that y σ = 0 for everyfacet that does not contain v . See Figure 2. Figure 2.
Dual cone of a triangle in R F . Given a vector field X on a d dimensional polytope Γ ⊂ R d , we nowdescribe a rescaling change of coordinates Ψ X(cid:15) , depending on a blow upparameter (cid:15) . See Figure 3.
Figure 3.
Asymptotic linearisation on the dual cone. The leftimage represents an orbit on the simplex ∆ and the right one thecorresponding (nearly) piecewise linear image under the map Ψ X(cid:15) on the dual cone.
This change of coordinates maps tubular neighbourhoods of edgesand vertices to the dual cone C ∗ (Γ) . For instance, the tubular neigh-bourhood N v of a vertex v is defined as follows. Consider a system AMILTONIAN POLYMATRIX REPLICATORS 5 ( x , . . . , x d ) of affine coordinates around v , which assigns coordinates (0 , . . . , to v and such that the hyperplanes x j = 0 are precisely thefacets of the polytope through v . Then N v is defined by N v := { p ∈ Γ d : 0 ≤ x j ( p ) ≤ for ≤ j ≤ d } . The sets { x j = 0 } ∩ N v are called the outer facets of N v . The remainingfacets of N v , defined by equations like x i = 1 , are called the inner facetsof N v . The previous cross sections Σ ± γ can be chosen to match theseinner facets of the neighbourhoods N v .The rescaling change of coordinates Ψ X(cid:15) maps N v to the sector Π v .Enumerating F so that the facets through v are precisely σ , . . . , σ d ,the map Ψ X(cid:15) is defined on the neighbourhood N v by Ψ X(cid:15) ( q ) := ( − (cid:15) log x ( q ) , . . . , − (cid:15) log x d ( q ) , , . . . , . Similarly, given an edge γ , Ψ X(cid:15) maps a tubular neighbourhood N γ of γ to the facet sector Π γ := Π v ∩ Π v (cid:48) of Π v where v (cid:48) is the other end-pointof γ . The map Ψ X(cid:15) sends interior facets of N v and N γ respectively toboundary facets of Π v and Π γ while it maps outer facets of N v and N γ to infinity. As the rescaling parameter (cid:15) tends to , the rescaledpush-forward (cid:15) − (Ψ X(cid:15) ) ∗ X of the vector field X converges to a constantvector field χ v on each sector Π v . This means that asymptotically,as (cid:15) → , trajectories become lines in the coordinates ( y σ ) σ ∈ F = Ψ X(cid:15) .Given a flowing-edge γ between vertices v and v (cid:48) , the map Ψ X(cid:15) over N γ depends only on the coordinates transversal to γ . Moreover, as (cid:15) → the global Poincaré map P γ converges to the identity map in the coordi-nates ( y σ ) σ ∈ F = Ψ X(cid:15) . Hence the sector Π γ is naturally identified as thecommon facet between the sectors Π v and Π v (cid:48) . Hence the asymptoticdynamics along the vertex-edge heteroclinic network is completely de-termined by the vector field’s geometry at the vertex singularities andcan be described by a piecewise constant vector field χ on the dualcone, whose components are precisely those of the skeleton characterof X . We refer to this piecewise constant vector field as the skeletonvector field of X . This vector field χ induces a piecewise linear flow onthe dual cone whose dynamics can be computationally explored.We use Poincaré maps for a global analysis of the asymptotic dynam-ics of the flow of X . We consider a subset S of flowing-edges with theproperty that every heteroclinic cycle goes through at least one edge in S . Such sets are called structural sets. The flow of X induces a Poincarémap P S on the system of cross sections Σ S := ∪ γ ∈ S Σ + γ . Each branchof the Poincaré map P S is associated with a heteroclinic path startingwith an edge in S and ending at its first return to another edge in S .These heteroclinic paths are the branches of S . The flow of the skeletonvector field χ also induces a first return map π S : D S ⊂ Π S → Π S onthe system of cross sections Π S := ∪ γ ∈ S Π γ . This map π S , called the skeleton flow map , is piecewise linear and its domain is a finite union ALISHAH, DUARTE, AND PEIXE of open convex cones. In some cases, see Proposition 5.18, the map π S becomes a closed dynamical system.We can now recall the main result in [3], Theorem 5.20 below, whichsays that in the rescaling change of coordinates Ψ X(cid:15) , the Poincaré map P S converges in the C ∞ topology to the skeleton flow map π S , in thesense that the following limit holds lim (cid:15) → Ψ X(cid:15) ◦ P S ◦ (Ψ X(cid:15) ) − = π S with uniform convergence of the map and its derivatives over any com-pact set contained in the domain D S ⊂ Π S .Consider now, for each facet σ of the polytope, an affine function R d (cid:51) q (cid:55)→ x σ ( q ) ∈ R which vanishes on σ and is strictly positive onthe rest of the polytope. With this family of affine functions we canpresent the polytope as Γ d = ∩ σ ∈ F { x σ ≥ } . Any function function h : int(Γ d ) → R of the form h ( q ) = (cid:88) σ ∈ F c σ log x σ ( q ) ( c σ ∈ R ) rescales to the following piecewise linear function on the dual cone η ( y ) := (cid:88) σ ∈ F c σ y σ in the sense that η = lim (cid:15) → (cid:15) − ( h ◦ (Ψ X(cid:15) ) − ) . When all coefficients c σ have the same sign then η is a proper function on the dual cone andall levels of η are compact sets. If the function h is invariant under theflow of X , i.e. h ◦ P S = h , then the piecewise linear function η is alsoinvariant under the skeleton flow, i.e. η ◦ π S = η . Thus integrals ofmotion (of vector fields on polytopes) of the previous form give rise to(asymptotic) piecewise linear integrals of motion for the skeleton flow.3. Poisson Poincaré maps
In this section we will define Poincaré map for Hamiltonian systemson Poisson manifolds. For Hamiltonian vector fields on symplecticmanifolds it is well known that the Poincaré map preserves the in-duced symplectic structure on any transversal section (see [20, Theorem1.8.]). We extend this fact to Hamiltonian systems on Poisson mani-folds, showing that any transversal section inherits a Poisson structureand the Poincaré map preserves this structure.A Poisson manifold is a pair ( M, π ) where M is a smooth manifoldwithout boundary and π a Poisson structure on M . Recall that aPoisson structure is a smooth bivector field π with the property that [ π, π ] = 0 , where [ · , · ] is the Schouten bracket (cf. e.g. [10]). Thebivector field π defines a vector bundle map π (cid:93) : T ∗ M → T M by ξ → π ( ξ, . ) . (3.1) AMILTONIAN POLYMATRIX REPLICATORS 7
The image of this map is an integrable singular distribution whichintegrates to a symplectic foliation, i.e., a foliation whose leaves havea symplectic structure induced by the Poisson structure.Notice that a Poisson structure can also be defined as a Lie bracket {· , ·} on C ∞ ( M ) × C ∞ ( M ) satisfying the Leibniz rule { f, gh } = { f, g } h + g { f, h } , f, g, h ∈ C ∞ ( M ) . These two descriptions are related by π (d f, d g ) = { f, g } . In a localcoordinate chart ( U, x , .., x n ) , or equivalently when M = R n , a Poissonbracket takes the form { f, g } ( x ) = (d x f ) t [ π ij ( x )] ij d x g, where π ( x ) = [ π ij ( x )] ij = [ { x i , x j } ( x )] ij is a skew symmetric matrixvalued smooth function satisfying n (cid:88) l =1 ∂π ij ∂x l π lk + ∂π jk ∂x l π li + ∂π ki ∂x l π lj = 0 ∀ i, j, k . Definition 3.1.
Let ( M, { , } M ) and ( N, { ., . } N ) be two Poisson mani-folds. A smooth map ψ : M → N will be called a Poisson map iff { f ◦ ψ, h ◦ ψ } M = { f, h } N ◦ ψ ∀ f, h ∈ C ∞ ( N ) . (3.2)Using the map π (cid:93) , defined at (3.1), this condition reads as ( Dψ ) π (cid:93)M ( Dψ ) ∗ = π (cid:93)N ◦ ψ, (3.3)where we use the notation ( Dψ ) ∗ to denote the adjoint operator of Dψ .Notice that, if Dψ is the Jacobian matrix of ψ in local coordinates, thenthe matrix representation of the pullback will be ( Dψ ) t . Remark 3.2.
When ψ is a diffeomorphism and only one of the mani-folds M or N is Poisson manifold, Definition 3.1 can be used to push-forward or pullback the Poisson structure to the other manifold. Definition 3.3.
Let ( M, π ) be a Poisson manifold. The Hamiltonianvector field associated to a given function H : M → R is defined byderivation X H ( f ) := { H, f } for f ∈ C ∞ ( M ) , or equivalently X H := π (cid:93) ( dH ) .As in the symplectic case, to define the Poincaré map we will con-sider the traversal sections inside the level set of the Hamiltonian. Wewill show that such a transversal section is a cosymplectic submanifoldsof the ambient Poisson manifold and naturally inherits a Poisson struc-ture. For more details on cosymplectic submanifolds see [21, Section . ]. Definition 3.4. N ⊂ ( M, π ) is a cosymplectic submanifold if it isthe level set of second class constraints i.e., N = ∩ ki =1 G − i (0) where { G , ..., G k } are functions such that [ { G i , G j } ( x )] i,j is an invertiblematrix at all points x ∈ N . ALISHAH, DUARTE, AND PEIXE
Remark 3.5.
A constraint is called first class if it Poisson commuteswith other constraints of the system. Sometimes, in the literature, aconstraint that has non-zero Poisson bracket with at least one otherconstraint of the system is called a second class constraint. Defini-tion 3.4 demands a stronger condition, but the cosymplectic submani-folds that we will use have codimension , where having non-zero Pois-son bracket with the other constraint is the same as [ { G i , G j } ( x )] i,j =1 , being an invertible matrix.Every, cosympletic submanifold is naturally equipped with a Poissonbracket called Dirac bracket. Paul Dirac, [7], developed this bracketto treat classical systems with second class constraints in Hamiltonianmechanics. Definition 3.6.
For cosymplectic submanifold N ⊂ ( M, π ) , let G , .., G k : U → R be its second class constraints, where U is a small enough neighbour-hood of N in M such that the matrix [ { G i , G j } ( x )] i,j is invertible atall points x ∈ U . The Dirac bracket is defined on C ∞ ( U ) by { f, g } Dirac = { f, g } − [ { f, G i } ] t [ { G i , G j } ] − [ { G i , g } ] , (3.4)where [ { ., G i } ] is the column matrix with components { ., G i } i = 1 , . . . , k .Dirac bracket is actually a Poisson bracket on the open submanifold U , see [21]. It takes an easy calculation to see that constraint functions G i , i = 1 , . . . , k are Casimirs of Dirac bracket. This fact allows therestriction of Dirac bracket to the cosymplectic submanifold N . Notethat, in general, restricting (pulling back) a Poisson structure to anarbitrary submanifold is not straightforward. Actually, the decompo-sition π (cid:93) ( T x N ◦ ) ⊕ T x N = T x M (3.5)that holds for every point x ∈ N and a strait forward calculationyield the independence of the extension in the following definition. InEquation (3.5), the term T x N ◦ is the annihilator of T x N in T ∗ x M . Wewill use this notation in the rest of the paper. Equation (3.5) canbe used as Definition of a cosymplectic submanifold, see [21], but forour propose it suits better to use of second class constraints to definecosymplectic submanifolds. Definition 3.7.
The restricted Dirac bracket on cosymplectic subman-ifold N , which will be also referred to as Dirac bracket, is simply definedby extending in any arbitrary way functions on N to functions on U ,calculating their Dirac bracket on U and restricting the result back to N .We consider a Hamiltonian H on the m -dimensional Poisson man-ifold ( M, π ) and its associated Hamiltonian vector field defined by AMILTONIAN POLYMATRIX REPLICATORS 9 X H = { H, . } = π (cid:93) ( dH ) . For a given point x ∈ M let U be a neigh-bourhood around it such that X H ( x ) (cid:54) = 0 ∀ x ∈ U , and E x be theenergy surface passing through x , i.e., the connected component of H − ( H ( x )) containing x . We call level transversal section to X H ata regular point x ∈ M any ( m − -dimensional transversal section Σ ⊂ E x ∩ U through x .The following lemma shows that Σ is a cosymplectic submanifold. Lemma 3.8.
Every level transversal section Σ is a cosymplectic sub-manifold of M .Proof. Since d x H (cid:54) = 0 , there exist a function G locally defined in U (shrink U if necessary) and linearly independent from H such that Σ = E x ∩ U ∩ G − ( G ( x )) . Then, we have π (d H, d G ) = X H ( G ) = dG ( X H ) (cid:54) = 0 by transversality. This finishes the proof. (cid:3) Remark 3.9.
The second term in the right hand side of Equation (3.4)is (cid:2) { f, H } { f, G } (cid:3) (cid:20) { H, G }{ G, H } (cid:21) − (cid:20) { H, g }{ G, g } (cid:21) . Then, using extensions ˜ f and ˜ g of f, g ∈ C ∞ (Σ) such that at everypoint x ∈ Σ their differentials vanishes on X H , yields { f, g } Dirac = { ˜ f , ˜ g }| Σ . We will use this fact to simplify our proofs but arbitrary extensions aremore suitable for calculating the Dirac structure. This also means thatthe Poisson structure on the level transversal section Σ is independentof the choice we make for the second class constraint G .We will use the same notation f for arbitrary extension and reservethe notation ˜ f for extension that their differentials vanishes on X H atevery point x ∈ Σ . To avoid any possible confusion, we observe thatin [6, Section 8] and [6, Section 8] the notation ˜ f is used in a slightlydifferent sense. Remark 3.10.
Cosymplectic submanifolds are special examples of theso called Poisson-Dirac submanifolds, see [6, Section 8]. The inducedPoisson structure on a Poisson-Dirac submanifold is defined by usingextensions such that their differentials vanish on π (cid:93) ( T Σ ◦ ) . In [6, Section8] and [21, Lemma . ] the notation ˜ f is used for this type of extensions.For a cosymplectic submanifold Σ given by second class constraints G , ..., G k , we have π (cid:93) ( T Σ ◦ ) = ⊕ ki =1 R X G i , (3.6) and the Dirac bracket coincides with the bracket induced in this way,see [21, Section . ]. In our case, we only have two constraints H, G andrequiring the vanishing of the differential only on X H (or X G ) at everypoint x ∈ Σ is enough to obtain the same induced Poisson bracket.For a fixed time t , let x = φ H ( t , x ) , where φ H is the flow of theHamiltonian vector field X H , and Σ , Σ be level transversal sectionsat x and x , respectively. As usual, a Poincaré map P = φ H ( τ ( x ) , x ) can be defined from an appropriate neighbourhood of x in Σ to aneighborhood of x in Σ . The existence of the smooth function τ ( x ) is guaranteed by the Implicit Function Theorem. We replace Σ and Σ by the domain and the image of the Poincaré map P .By Lemma 3.8 both Σ i , i = 0 , , are cosymplectic submanifoldsequipped with Dirac brackets { ., . } Dirac i , i = 0 , . We will show thatthe Poincaré map P is a Poisson map (see Definition 3.1). Proposition 3.11.
The Poincaré map P : (Σ , { ., . } Dirac ) → (Σ , { ., . } Dirac ) is a Poisson map.Proof. We define ˜ P : U → U by ˜ P ( x ) := φ H (˜ τ ( x ) , x ) , where ˜ τ is an extension of τ to a neighborhood U of x such that itsdifferential, d ˜ τ , vanishes on X H . Both neighborhood U and U can beshrunk, if necessary, in a way that both Dirac brackets around Σ and Σ are defined in U and U , respectively. A straightforward calculationshows that for every point x in the domain of ˜ τ , we have D x ˜ P = D x φ ˜ τ ( x ) H + ( d x ˜ τ ) X H ( φ ˜ τ ( x ) H ( x )) , where φ ¯ τ ( x ) H ( . ) = φ H (˜ τ ( x ) , . ) . Furthermore, for every x ∈ Σ , we have D x ˜ P ( X H ( x )) = D x φ ˜ τ ( x ) H ( X H ( x )) + ( d x ˜ τ ( X H ( x )) (cid:124) (cid:123)(cid:122) (cid:125) =0 ) X H ( φ ˜ τ ( x ) H ( x ))= D x φ ˜ τ ( x ) H ( X H ( x )) = X H ( φ ˜ τ ( x ) H ( x )) = X H ( ˜ P ( x )) . (3.7)Note that D x φ ˜ τ ( x ) H in Equation (3.7) is the derivative of time- ˜ τ ( x ) flow of X H and the fixed time flow maps of X H are Poisson maps, i.e. it sendsHamiltonian vector fields to Hamiltonian vector field. Furthermore,the flow of X H preserves H , this means that H ( ˜ P ( x )) = H ( φ ˜ τ ( x ) H ( x )) = H ( x ) . As we set in Remark 3.9, let ˜ f be an extension of a given f ∈ C ∞ (Σ ) such that d x ˜ f ( X H ) = 0 , ∀ x ∈ Σ , AMILTONIAN POLYMATRIX REPLICATORS 11 then for every x ∈ Σ , d x ( ˜ f ◦ ˜ P )( X H ) = d ˜ P ( x ) ˜ f ◦ D x ˜ P ( X H ) = d ˜ P ( x ) ˜ f ( X H ) = 0 . Now, for f, g ∈ C ∞ (Σ ) and x ∈ Σ we have { ˜ f ◦ ˜ P , ˜ g ◦ ˜ P } ( x ) = π x (cid:16) ( D x ˜ P ) ∗ d x ˜ f , ( D x ˜ P ) ∗ d x ˜ g, (cid:17) = π x (cid:16) ( D x φ ¯ τ ( x ) H ) ∗ d x ˜ f + ( d x ¯ τ X H ) ∗ d x ˜ f , ( D x φ ¯ τ ( x ) H ) ∗ d x ˜ g + ( d x ¯ τ X H ) ∗ d x ˜ g (cid:17) = π x (cid:16) ( D x φ ¯ τ ( x ) H ) ∗ d x ˜ f , ( D x φ ¯ τ ( x ) H ) ∗ d x ˜ g (cid:17) = π (d ˜ f , d˜ g )( ˜ P ( x )) = { ˜ f , ˜ g } ( ˜ P ( x )) , and consequently, { f ◦ P, g ◦ P } Dirac = { ˜ f ◦ ˜ P , ˜ g ◦ ˜ P }| Σ = { ˜ f , ˜ g } ◦ ˜ P | Σ = { f, g } Dirac ◦ P, where we used Remark 3.9. This finishes the proof. (cid:3) Hamiltonian polymatrix replicators
In this section we provide a short introduction to polymatrix repli-cators, following [2]. In particular we will focus on the class of conser-vative polymatrix replicators that we designate as
Hamiltonian poly-matrix replicators .Consider a population divided in p groups where each group is la-belled by an integer α ∈ { , . . . , p } , and the individuals of each group α have exactly n α strategies to interact with other members of the pop-ulation (including of the same group). In total we have n = (cid:80) pα =1 n α strategies that we label by the integers i ∈ { , . . . , n } , denoting by [ α ] := { n + · · · + n α − + 1 , . . . , n + · · · + n α } ⊂ N the set (interval) of strategies of group α .Given α, β ∈ { , . . . , p } , consider a real n α × n β matrix, say A α,β ,whose entries a α,βij , with i ∈ [ α ] and j ∈ [ β ] , represent the the averagepayoff of an individual of the group α using the i th strategy wheninteracting with an individual of the group β using the j th strategy.Thus the matrix A with entries a α,βij , where α, β ∈ { , . . . , p } , i ∈ [ α ] and j ∈ [ β ] , is a square matrix of order n = n + . . . + n p , consisting ofthe block matrices A α,β .Let n = ( n , . . . , n p ) . The state of the population is described by apoint x = ( x α ) ≤ α ≤ p in the polytope Γ n := ∆ n − × . . . × ∆ n p − ⊂ R n , where ∆ n α − = { x ∈ R [ α ]+ : (cid:80) i ∈ [ α ] x αi = 1 } , x α = ( x αi ) i ∈ [ α ] and the entry x αi represents the usage frequency of the i th strategy within the group α . We denote by ∂ Γ n the boundary of Γ n .Assuming random encounters between individuals, for each group α ∈ { , . . . , p } , the average payoff of a strategy i ∈ [ α ] within a popu-lation with state x is given by ( Ax ) i = p (cid:88) β =1 (cid:0) A α,β (cid:1) i x β = p (cid:88) β =1 (cid:88) k ∈ [ β ] a α,βik x βk , where the overall average payoff of group α is given by (cid:88) i ∈ [ α ] x αi ( Ax ) i . Demanding that the logarithmic growth rate of the frequency of eachstrategy i ∈ [ α ] , α ∈ { , . . . , p } , is equal to the payoff difference betweenstrategy i and the overall average payoff of group α yields the systemof ordinary differential equations defined on the polytope Γ n , dx αi dt = x αi ( Ax ) i − (cid:88) i ∈ [ α ] x αi ( Ax ) i , α ∈ { , . . . , p } , i ∈ [ α ] , (4.1) that will be designated as a polymatrix replicator .The flow φ tn,A of this equation leaves the polytope Γ n invariant. Theargument is analogous to that for the bimatrix replicator equation,see [13, Section 10.3]. Hence, by compactness of Γ n , the flow φ tn,A iscomplete. From now on we will call ‘polymatrix replicator’ to sys-tem (4.1), to the flow φ tn,A and also to the underlying vector field on Γ n which we denote by X n,A .If p = 1 equation (4.1) becomes the usual replicator equation withpayoff matrix A . When p = 2 and A = A = 0 are null matrices,equation (4.1) becomes the bimatrix replicator equation with payoffmatrices A and ( A ) t .Given n = ( n , . . . , n p ) , let I n := { I ⊂ { , . . . , n } : I ∩ [ α ]) ≥ , ∀ α = 1 , . . . , p } . A set I ∈ I n determines the face σ I := { x ∈ Γ n : x j = 0 , ∀ j / ∈ I } of Γ n . The correspondence between labels in I n and faces of Γ n isbijective. Remark 4.1.
The partition of Γ n into the interiors σ ◦ I := int( σ I ) ,with I ∈ I n , is a smooth stratification of Γ n with strata σ ◦ I . Everystratum σ ◦ I is a connected open submanifold and for any pair σ ◦ I , σ ◦ I if σ ◦ I ∩ σ I (cid:54) = ∅ then σ I ⊂ σ I . For more on smooth stratification see [12]and references therein.For a set I ∈ I n consider the pair ( n I , A I ) , where n I = ( n I , . . . , n Ip ) with n Iα = I ∩ [ α ]) , and A I = [ a ij ] i,j ∈ I . AMILTONIAN POLYMATRIX REPLICATORS 13
Proposition 4.2. [2, Proposition ] Given I ∈ I n , the face σ I of Γ n is invariant under the flow of X n,A and the restriction of (4.1) to σ I isthe polymatrix replicator X n I ,A I . For a fixed n = ( n , . . . , n p ) the correspondence A (cid:55)→ X n,A is linearand its kernel consists of the matrices C = (cid:0) C α,β (cid:1) ≤ α,β ≤ p where eachblock C α,β has equal rows, i.e., has the form C α,β = c α,β c α,β . . . c α,βn c α,β c α,β . . . c α,βn ... ... ... c α,β c α,β . . . c α,βn . Thus X n,A = X n,A if and only if for every α, β ∈ { , . . . , p } the matrix A α,β − A α,β has equal rows (see [2, Proposition 1]).We have now the following characterization of the interior equilibria. Proposition 4.3. [2, Proposition ] Given a polymatrix replicator X n,A , a point q ∈ int(Γ n ) is an equilibrium of X n,A iff ( A q ) i = ( A q ) j for all i, j ∈ [ α ] and α = 1 , . . . , p .In particular the set of interior equilibria of X n,A is the intersectionof some affine subspace with int(Γ n ) . Definition 4.4.
A polymatrix replicator X n,A is said to be conservative if there exists:(a) a point q ∈ R n , called formal equilibrium , such that ( A q ) i =( A q ) j for all i, j ∈ [ α ] , and all α = 1 , . . . , p and (cid:80) j ∈ [ α ] q j = 1 ;(b) matrices A , D ∈ Mat n × n ( R ) such that(i) X n,A D = X n,A ,(ii) A is a skew symmetric, and(iii) D = diag( λ I n , . . . , λ p I n p ) with λ α (cid:54) = 0 for all α ∈ { , . . . , p } .The matrix A will be referred to as a skew symmetric model for X n,A ,and ( λ , . . . , λ p ) ∈ ( R ∗ ) p as a scaling co-vector .In [4], another characterization of conservative polymatrix replica-tors, using quadratic forms, is provided. Furthermore, in [1] the conceptof conservative replicator equations (where p = 1 ) is generalized usingDirac structures.In what follows, the vectors in R n , or R [ α ] , are identified with col-umn vectors. Let n = (1 , .., t ∈ R n . We will omit the subscript n whenever the dimension of this vector is clear from the context. Simi-larly, we write I = I n for the n × n identity matrix. Given x ∈ R n , wedenote by D x the n × n diagonal matrix D x := diag( x , . . . , x n ) . Foreach α ∈ { , . . . , p } we define the n α × n α matrix T αx := x α t − I , and T x the n × n block diagonal matrix T x := diag( T x , . . . , T px ) . Given an anti-symmetric matrix A , we define the skew symmetricmatrix valued mapping π A : R n → Mat n × n ( R ) π A ( x ) := ( − T x D x A D x T tx . (4.2)The interior of the polytope Γ n , denoted by int(Γ n ) , equipped with π A is a Poisson manifold, see [2, Theorem 3.5]. Furthermore, Theorem 4.5. [2, Theorem . ] Consider a conservative polymatrixreplicator X n,A with formal equilibrium q , skew symmetric model A and scaling co-vector ( λ , . . . , λ p ) . Then X n,A , restricted to int(Γ n ) , isHamiltonian with Hamiltonian function h ( x ) = p (cid:88) β =1 λ β (cid:88) j ∈ [ β ] q βj log x βj . (4.3)5. Asymptotic dynamics of polymatrix replicators
Given a polymatrix replicator X n,A , the edges and vertices of thepolytope Γ n form a heteroclinic edge network for the associated flow.In this section we recall the technique developed in [3] to analyze theasymptotic dynamics of a flow on a polytope along its heteroclinic edgenetwork. In particular we review the main definitions and results forthe polymatrix replicator X n,A on the polytope Γ n .The affine support of Γ n is the smallest affine subspace of R n thatcontains Γ n . It is the subspace E = E × . . . × E p where for α = 1 , . . . , p , E α := x α ∈ R [ α ] : (cid:88) i ∈ [ α ] x αi = 1 . Following [3, Definition . ] we introduce a defining family for thepolytope Γ n . The affine functions { f i : E → R } ≤ i ≤ n where f i ( x ) = x i ,are a defining family for Γ n because they satisfy:(a) Γ n = (cid:92) i ∈ I f − i ([0 , + ∞ [) ,(b) Γ n ∩ f − i (0) (cid:54) = ∅ for all i ∈ { , . . . , n } , and(c) given J ⊆ { , . . . , n } such that Γ n ∩ (cid:32)(cid:92) j ∈ J f − j (0) (cid:33) (cid:54) = ∅ , thelinear -forms (d f j ) p are linearly independent at every point p ∈ ∩ j ∈ J f − j (0) .Next we introduce convenient labellings for vertexes, facets and edgesof Γ n . Let ( e , . . . , e n ) be the canonical basis of R n and denote by V n theCartesian product V n := (cid:81) pα =1 [ α ] which contains (cid:81) pα =1 n α elements.Each label j = ( j , . . . , j p ) ∈ V n determines the vertex v j := ( e j , ..., e j p ) of Γ n . This labelling is one-to-one. The set F n := { , , . . . , n } canbe used to label the n facets of Γ n . Each integer i ∈ F n labels thefacet σ i := Γ n ∩ { x i = 0 } of Γ n . Edges can be labelled by the set AMILTONIAN POLYMATRIX REPLICATORS 15 E n := { J ∈ I n : J = p + 1 } . Given J ∈ E n there exists a unique(unordered) pair of labels j , j ∈ V n such that J is the union of thestrategies in j and j . The label J determines the edge γ J := { tv j +(1 − t ) v j : 0 ≤ t ≤ } . Again the correspondence J (cid:55)→ γ J betweenlabels J ∈ E n and edges of Γ n is one-to-one.Given a vertex v of Γ n , we denote by F v and E v respectively the setsof facets and edges of Γ n that contain v . Given j = ( j , . . . , j p ) ∈ V n F v j = { σ i : i ∈ F n \ { j , . . . , j p }} and this set of facets contains exactly n − p = dim(Γ n ) elements.Triples in C := { ( v, γ, σ ) ∈ V × E × F : γ ∩ σ = { v } } , are called corners . Any pair of elements in a corner uniquely determinesthe third one. Therefore, sometimes we will shortly refer to a corner ( v, γ, σ ) as ( v, γ ) or ( v, σ ) . An edge γ with end-points v, v (cid:48) determinestwo corners ( v, γ, σ ) and ( v (cid:48) , γ, σ (cid:48) ) , called the end corners of γ . Thefacets σ, σ (cid:48) are referred to as the opposite facets of γ . Remark 5.1.
In a small neighbourhood of a given vertex v = v j , where j = ( j , . . . , j p ) ∈ V n , the affine functions f k : Γ n → R , f k ( x ) := x k ,with k ∈ F n \ { j , . . . , j p } , can be used as a coordinate system for Γ n .Given a polymatrix replicator X n,A and a facet σ i with i ∈ [ α ] , α ∈ { , . . . , p } , the i th component of X n,A is given by d f i ( X n,A ) = x i (cid:32) ( A x ) i − p (cid:88) β =1 ( x α ) T A α,β x β (cid:33) . Definition 5.2.
A polymatrix replicator X n,A is called regular if forany i ∈ F n , the function H i : Γ n → R , H i ( x ) := f i ( x ) − d f i ( X n,A ( x )) = ( A x ) i − p (cid:88) β =1 ( x α ) T A α,β x β is not identically zero along σ i .Clearly generic polymatrix replicators are regular. Using the conceptof order of a vector field along a facet [3, Definition . ], X n,A is regularif and only if all facets of Γ n have order . For the sake of simplicitywe will assume from now on that all polymatrix replicator vector fieldsare regular. Definition 5.3.
The skeleton character of polymatrix replicator X n,A is defined to be the matrix χ := ( χ vσ ) ( v,σ kα ) ∈ V × F where χ vσ := (cid:26) − H σ ( v ) , v ∈ σ otherwisewhere H σ stands for H i when σ = σ i with i ∈ F n . For a fixed vertex v ,the vector χ v := ( χ vσ ) σ ∈ F is referred to as the skeleton character at v . Remark 5.4.
Given a corner ( v, γ, σ ) of Γ n , H σ ( v ) is the eigenvalueof the tangent map (d X n,A ) v along the eigen-direction parallel to γ . Proposition 5.5. If X n,A is a regular polymatrix replicator for everyvertex v = v j with label j = ( j , . . . , j p ) ∈ V n , and every facet σ = σ i with i ∈ F n and i ∈ [ α ] the skeleton character of X n,A is given by χ vσ = (cid:40) (cid:80) pβ =1 ( a j α j β − a ij β ) if v ∈ σ otherwise . Proof.
Straightforward calculation. (cid:3)
Remark 5.6.
For a given corner ( v, γ, σ ) of Γ n , • if χ vσ < then v is the α -limit of an orbit in γ , and • if χ vσ > then v is the ω -limit of an orbit in γ .Let γ be an edge with end-points v and v (cid:48) and opposite facets σ and σ (cid:48) , respectively. This means that ( v, γ, σ ) and ( v (cid:48) , γ, σ (cid:48) ) are corners of Γ n . If X n,A does not have singularities in int( γ ) , then int( γ ) consistsof a single heteroclinic orbit with α -limit v and ω -limit v (cid:48) if and onlyif χ vσ < and χ v (cid:48) σ (cid:48) > . This type of edges will be referred to as flowingedges . The vertices v = s ( γ ) and v (cid:48) = t ( γ ) are respectively called the source and target of the flowing edge γ and we will write v γ −→ v (cid:48) toexpress it.Given v = v j with j = ( j , . . . , j p ) ∈ V n consider the vertex neigh-bourhood N v := { q ∈ Γ n : 0 ≤ f k ( q ) ≤ , ∀ k ∈ F n \{ j , . . . , j p } } . Rescaling the defining functions f k we may assume these neighbour-hoods are pairwise disjoint. See Remark 5.1.For any edge γ with end-points v and v (cid:48) we define a tubular neigh-bourhood connecting N v to N v (cid:48) by N γ := { q ∈ Γ n \ ( N v ∪ N v (cid:48) ) : 0 ≤ f k ( q ) ≤ , ∀ k ∈ F n with γ ⊂ σ k } . Again we may assume that these neighbourhoods are pairwise disjointbetween themselves. Finally we define the edge skeleton’s tubular neigh-bourhood of Γ n to be N Γ n := ( ∪ v ∈ V N v ) ∪ ( ∪ γ ∈ E N γ ) . (5.1)The next step is to define the rescaling map Ψ n,A(cid:15) on N Γ n \ ∂ Γ n . See[3, Definition . ]. We will write f σ to denote the affine function f k associated with the facet σ = σ k with k ∈ F n . Definition 5.7.
Let (cid:15) > be a small parameter. The (cid:15) -rescalingcoordinate system Ψ n,A(cid:15) : N Γ n \ ∂ Γ n → R F maps q ∈ N Γ n to y := ( y σ ) σ ∈ F where AMILTONIAN POLYMATRIX REPLICATORS 17 • if q ∈ N v for some vertex v : y σ = (cid:26) − (cid:15) log f σ ( q ) if v ∈ σ if v / ∈ σ • if q ∈ N γ for some edge γ : y σ = (cid:26) − (cid:15) log f σ ( q ) if γ ⊂ σ if γ (cid:54)⊂ σ We now turn to the space where these rescaling coordinates takevalues. For a given vertex v ∈ V we define Π v := { ( y σ ) σ ∈ F ∈ R F + : y σ = 0 , ∀ σ / ∈ F v } (5.2)where R + = [0 , + ∞ ) . Since { f σ } σ ∈ F v is a coordinate system over N v and the function h : (0 , → [0 , + ∞ ) , h ( x ) := − log x , is a diffeomor-phism, the restriction Ψ n,A(cid:15) : N v \ ∂ Γ n → Π v is also a diffeomorphismdenoted by Ψ n,A(cid:15),v .If γ is an edge connecting two corners ( v, σ ) and ( v (cid:48) , σ (cid:48) ) , F v ∩ F v (cid:48) = { σ ∈ F : γ ⊂ σ } and we define Π γ := { ( y σ ) σ ∈ F ∈ R F + : y σ = 0 when γ (cid:54)⊂ σ } . (5.3)Then Ψ n,A(cid:15) ( N γ \ ∂ Γ n ) = Π γ = Π v ∩ Π v (cid:48) has dimension d − while Π v =Ψ n,A(cid:15).v ( N v \ ∂ Γ n ) has dimension d . In particular the map Ψ n,A(cid:15),v is notinjective over N γ . See Figure 4. Figure 4.
An edge connecting two corners.
Definition 5.8.
The dual cone of Γ n is defined to be C ∗ (Γ n ) := (cid:91) v ∈ V Π v , where Π v is the sector in (5.2). Hence Ψ n,A(cid:15) : N Γ n \ ∂ Γ n → C ∗ (Γ n ) .Denote by { ϕ tn,A : Γ n → Γ n } t ∈ R the flow of the vector field X n,A .Given a flowing edge γ with source v = s ( γ ) and target v (cid:48) = t ( γ ) weintroduce the cross-sections Σ − γ := (Ψ n,Av,(cid:15) ) − (int(Π γ )) and Σ + γ := (Ψ n,Av (cid:48) ,(cid:15) ) − (int(Π γ )) transversal to the flow ϕ tn,A . The sets Σ − γ and Σ + γ are inner facets ofthe tubular neighbourhoods N v and N v (cid:48) respectively. Let D γ be theset of points x ∈ Σ − γ such that the forward orbit { ϕ tn,A ( x ) : t > } hasa first transversal intersection with Σ + γ . The global Poincaré map P γ : D γ ⊂ Σ − γ → Σ + γ is defined by P γ ( x ) := ϕ τ ( x ) n,A ( x ) , where τ ( x ) = min { t > ϕ tn,A ( x ) ∈ Σ + γ } . To simplify some of the following convergence statements we use theterminology in [3, Definition . ]. Definition 5.9.
Suppose we are given a family of functions F (cid:15) withvarying domains U (cid:15) . Let F be another function with domain U . As-sume that all these functions have the same target and source spaces,which are assumed to be linear spaces. We will say that lim (cid:15) → + F (cid:15) = F in the C k topology , to mean that:(1) domain convergence : for every compact subset K ⊆ U , we have K ⊆ U (cid:15) for every small enough (cid:15) > , and(2) uniform convergence on compact sets : lim (cid:15) → + max ≤ i ≤ k sup u ∈ K (cid:12)(cid:12) D i [ F (cid:15) ( u ) − F ( u )] (cid:12)(cid:12) = 0 . Convergence in the C ∞ topology means convergence in the C k topologyfor all k ≥ . If F (cid:15) is a composition of two or more mappings then itsdomain should be understood as the composition domain.Let now Π γ ( (cid:15) ) := { y ∈ Π γ : y σ ≥ (cid:15) whenever γ ⊂ σ } , (5.4)and define F (cid:15)γ := Ψ n,Av (cid:48) ,(cid:15) ◦ P γ ◦ (Ψ n,Av,(cid:15) ) − . Notice that lim (cid:15) → Π γ ( (cid:15) ) = int(Π γ ) . Lemma 5.10.
For a given k ≥ , there exists a number r such thatthe following limit holds in the C k topology, lim (cid:15) → + F (cid:15)γ | U (cid:15)γ = id Π γ , where U (cid:15)γ ⊂ Π γ ( (cid:15) r ) is the domain of F (cid:15)γ .Proof. See [3, Lemma 7.2]. (cid:3)
AMILTONIAN POLYMATRIX REPLICATORS 19
Hence, since the global Poincaré maps converge towards the identitymap as we approach the heteroclinic orbit, the asymptotic behaviourof the flow is solely determined by local Poincaré maps.From Definition 5.3, for any vertex v , the vector χ v is tangent to Π v ,in the sense that χ v belongs to the linear span of the sector Π v . Let Π v ( (cid:15) ) := { y ∈ Π v : y σ ≥ (cid:15) for all σ ∈ F v } (5.5)Using the notation of Definition 5.3 we have Lemma 5.11.
We have (Ψ n,Av,(cid:15) ) ∗ X n,A = (cid:15) (cid:16) ˜ X (cid:15)v,σ (cid:17) σ ∈ F , where ˜ X (cid:15)v,σ ( y ) := (cid:26) − H σ (cid:0) (Ψ n,Av,(cid:15) ) − ( y ) (cid:1) if σ ∈ F v if σ / ∈ F v , Moreover, given k ≥ there exists r > such that the following limitholds in the C k topology lim (cid:15) → ( ˜ X (cid:15)v ) | Π v ( (cid:15)r ) = χ v . Proof.
See [3, Lemma . ]. (cid:3) Consider a vertex v with an incoming flowing-edge v ∗ γ −→ v and anoutgoing flowing-edge v γ (cid:48) −→ v (cid:48) . Denote by σ ∗ the facet opposed to γ (cid:48) at v . We define the sector Π γ,γ (cid:48) := (cid:26) y ∈ int(Π γ ) : y σ − χ vσ χ vσ ∗ y σ ∗ > , ∀ σ ∈ F v , σ (cid:54) = σ ∗ (cid:27) (5.6)and the linear map L γ,γ (cid:48) : Π γ,γ (cid:48) → Π γ (cid:48) by L γ,γ (cid:48) ( y ) := (cid:18) y σ − χ vσ χ vσ ∗ y σ ∗ (cid:19) σ ∈ F . (5.7)Notice that Π γ (cid:48) = { y ∈ Π v : y σ ∗ = 0 } as well as Π γ are facets to Π v . Proposition 5.12.
The sector Π γ,γ (cid:48) consists of all points y ∈ int(Π γ ) which can be connected to some point y (cid:48) ∈ int(Π γ (cid:48) ) by a line segmentinside the ray { y + tχ v : t ≥ } . Moreover, if y ∈ Π γ,γ (cid:48) then the otherendpoint is y (cid:48) = L γ,γ (cid:48) ( y ) .Proof. See [3, Proposition . ]. (cid:3) Given flowing-edges γ and γ (cid:48) such that t ( γ ) = s ( γ (cid:48) ) = v we de-note by D γ,γ (cid:48) the set of points x ∈ Σ v,γ such that the forward orbit { ϕ tn,A ( x ) : t ≥ } has a first transversal intersection with Σ v,γ (cid:48) . Thelocal Poincaré map P γ,γ (cid:48) : D γ,γ (cid:48) ⊂ Σ + γ → Σ − γ (cid:48) is defined by P γ,γ (cid:48) ( x ) := ϕ τ ( x ) n,A ( x ) , where τ ( x ) := min { t > ϕ tn,A ( x ) ∈ Σ − γ (cid:48) } . Lemma 5.13.
Let U (cid:15)γ,γ (cid:48) ⊂ Π γ ( (cid:15) r ) be the domain of the map F (cid:15)γ,γ (cid:48) := Ψ n,Av,(cid:15) ◦ P γ,γ (cid:48) ◦ (Ψ n,Av,(cid:15) ) − . Then for a given k ≥ there exist r > such that lim (cid:15) → + (cid:0) F (cid:15)γ,γ (cid:48) (cid:1) | U (cid:15)γ,γ (cid:48) = L γ,γ (cid:48) in the C k topology.Proof. See [3, Lemma . ]. (cid:3) Given a chain of flowing-edges v γ −→ v γ −→ v −→ . . . −→ v m γ m −→ v m +1 the sequence ξ = ( γ , γ , . . . , γ m ) is called a heteroclinic path , or a heteroclinic cycle when γ m = γ . Definition 5.14.
Given a heteroclinic path ξ = ( γ , γ , . . . , γ m ) :1) The Poincaré map of a polymatrix replicator X n,A along ξ isthe composition P ξ := ( P γ m ◦ P γ m − ,γ m ) ◦ . . . ◦ ( P γ ◦ P γ ,γ ) , whose domain is denoted by U ξ .2) The skeleton flow map (of χ ) along ξ is the composition map π ξ : Π ξ → Π γ m defined by π ξ := L γ m − ,γ m ◦ . . . ◦ L γ ,γ , whose domain is Π ξ := int(Π γ ) ∩ m (cid:92) j =1 ( L γ ∗ ,γ j ◦ . . . ◦ L γ ,γ ) − int(Π γ j ) . The previous lemmas 5.10 and 5.13 imply that given a heteroclinicpath ξ , the asymptotic behaviour of the Poincaré map P ξ along ξ isgiven by the Poincaré map π ξ of χ . Proposition 5.15.
Let U (cid:15)ξ be the domain of the map F (cid:15)ξ := Ψ n,Av m ,(cid:15) ◦ P ξ ◦ (cid:0) Ψ n,Av ,(cid:15) (cid:1) − from Π γ ( (cid:15) r ) into Π γ m ( (cid:15) r ) . Then lim (cid:15) → + (cid:0) F (cid:15)ξ (cid:1) | U (cid:15)ξ = π ξ in the C k topology.Proof. See [3, Proposition . ]. (cid:3) To analyse the dynamics of the flow of the skeleton vector field χ weintroduce the concept of structural set and its associated skeleton flowmap. See [3, Definition . ]. AMILTONIAN POLYMATRIX REPLICATORS 21
Definition 5.16.
A non-empty set of flowing-edges S is said to be a structural set for χ if every heteroclinic cycle contains an edge in S .Structural sets are in general not unique. We say that a heteroclinicpath ξ = ( γ , . . . , γ m ) is an S -branch if(1) γ , γ m ∈ S ,(2) γ j / ∈ S for all j = 1 , . . . , m − .Denote by B S ( χ ) the set of all S -branches. Definition 5.17.
The skeleton flow map π S : D S → Π S is defined by π S ( y ) := π ξ ( y ) for all y ∈ Π ξ , where D S := ∪ ξ ∈ B S ( χ ) Π ξ and Π S := ∪ γ ∈ S Π γ . The reader should picture π S : D S → Π S as the first return map ofthe piecewise linear flow of χ on C ∗ (Γ n ) to the system of cross-sections Π S . The following, see [3, Proposition . ], provides a sufficient con-dition for the skeleton flow map π S to be a closed dynamical system. Proposition 5.18.
Given a skeleton vector field χ on C ∗ (Γ n ) with astructural set S , assume(1) every edge of Γ n is either neutral or a flowing-edge,(2) every vertex v is of saddle type, i.e., χ vσ χ vσ < for some facets σ , σ ∈ F v .Then ˆ D S := (cid:92) n ∈ Z ( π S ) − n ( D S ) is a Baire space with full Lebesgue measure in Π S and π S : ˆ D S → ˆ D S is a homeomorphism. Given a structural set S any orbit of the flow ϕ tn,A that shadowssome heteroclinic cycle must intersect the cross-sections ∪ γ ∈ S Σ + γ recur-rently. The following map encapsulates the semi-global dynamics ofthese orbits. Definition 5.19.
Given X n,A , let S be a structural set of its skeletonvector field. We define P S : U S ⊂ Σ S → Σ S setting Σ S := ∪ γ ∈ S Σ + γ , U S := ∪ ξ ∈ B S ( χ ) U ξ and P S ( p ) := P ξ ( p ) for all p ∈ U ξ . The domaincomponents U ξ and U ξ (cid:48) are disjoint for branches ξ (cid:54) = ξ (cid:48) in B S ( χ ) .Up to a time re-parametrization, the map P S : D S ⊂ Σ S → Σ S embeds in the flow ϕ t ( n,A ) . In this sense the dynamics of P S encapsulatesthe qualitative behaviour of the flow ϕ tX of X along the edges of Γ n . Theorem 5.20.
Let X n,A be a regular polymatrix replicator with skele-ton vector field χ . If S is a structural set of χ then lim (cid:15) → + Ψ (cid:15) ◦ P S ◦ (Ψ (cid:15) ) − = π S in the C ∞ topology, in the sense of Definition 5.9.Proof. See [3, Theorem . ]. (cid:3) Hamiltonian character of the asymptotic dynamics
In this section we discuss the Poisson geometric properties of thePoincaré maps π ξ in the case of Hamiltonian polymatrix replicatorequations.Given a generic Hamiltonian polymatrix replicator, X n,A , we studyits asymptotic Poincaré maps, proving that they are Poisson maps.Let X n,A be a conservative polymatrix replicator, q a formal equilib-rium, A and D as in Definition 4.4, and h ( x ) = p (cid:88) β =1 (cid:88) j ∈ [ β ] λ β q βj log x βj (6.1)its Hamiltonian function as in Theorem 4.5. The Hamiltonian (6.1) be-longs to a class of prospective constants of motion for vector fields onpolytopes discussed in [3, Section ]. Since the polymatrix replicator isfixed we drop superscript ( n, A ) and use Ψ v,(cid:15) for the rescaling coordi-nate systems defined in Definition 5.7. The following proposition givesus the asymptotic constant of motion, on the dual cone, associated to h. Proposition 6.1.
Let η : C ∗ (Γ n ) → R be defined by η ( y ) := p (cid:88) β =1 (cid:88) j ∈ [ β ] λ β q βj y βj . (6.2) (1) η = lim (cid:15) → + (cid:15) h ◦ (Ψ v,(cid:15) ) − over int(Π v ) for any vertex v , with con-vergence in the C ∞ topology.(2) dη = lim (cid:15) → + (cid:15) (cid:2) (Ψ v,(cid:15) ) − (cid:3) ∗ ( dh ) over int(Π v ) for any vertex v , withconvergence in the C ∞ topology.(3) Since h is invariant under the flow of X n,A , i.e., dh ( X n,A ) ≡ ,the function η is invariant under the skeleton flow of χ , i.e., dη ( χ ) ≡ .Proof. See [3, Proposition . ]. (cid:3) We will use the following family of coordinate charts for the Poissonmanifold (int(Γ n ) , π A ) where π A is defined at (4.2). Definition 6.2.
Given a vertex v = ( e j , ..., e j p ) of Γ n , we set ˆ x α :=( x α ˆ k ) ˆ k ∈ [ α ] \{ j α } and ˆ x := (ˆ x α ) α , and define the projection map P v : int( N v ) → ( R n − × . . . × R n p − ) , P v ( x ) := ˆ x. Clearly, P v is a diffeomorphism onto its image (0 , n − p . The inversemap ψ v := P − v can be regarded as a local chart for the manifold int(Γ n ) . AMILTONIAN POLYMATRIX REPLICATORS 23
Remark 6.3.
The projection map P v extends linearly to R n and it isrepresented by ( n − p ) × n block diagonal matrix P v = diag( P v , . . . , P pv ) where P αv , α = 1 , . . . , p , is the constant ( n α − × n α matrix obtainedremoving the row j α from the identity matrix. Lemma 6.4.
Consider the Poisson manifold (int(Γ n ) , π A ) where π A is defined at (4.2) . Then for any vertex v , the matrix representation of π A in the local chart ψ v is π (cid:93) v A (ˆ x ) = ( − P v T x D x A D x T tx P tv , (6.3) where D x and T x are as defined at (4.2) .Proof. Notice that π (cid:93) v A (ˆ x ) := [ { x α ˆ k , x β ˆ l } ] with α, β = 1 , . . . , p and ˆ k ∈ [ α ] , ˆ l ∈ [ β ] . (cid:3) We used the notation (cid:93) v instead of (cid:93) to make it clear that the repre-senting matrix is w.r.t. the local chart ψ v . The following trivial Lemmagives us the differential of the (cid:15) -rescaling map Ψ v,(cid:15) , which defined inDefinition 5.7, in the coordinate chart ψ v . Lemma 6.5.
The differential of the diffeomorphism Ψ v,(cid:15) ◦ ψ v : P v (int( N v )) → int(Π v ) is D (Ψ v,(cid:15) ◦ ψ v ) ˆ x = − (cid:15) diag( D (Ψ v,(cid:15) ◦ φ v ) ˆ x , . . . , D p (Ψ v,(cid:15) ◦ φ v ) ˆ x p ) , where D α (Ψ v,(cid:15) ◦ ψ v ) ˆ x α α = 1 , . . . , p is given by diag(( x α ) − ν α , . . . , ( x αj α − ) − ν jα − , ( x αj α +1 ) − ν jα +1 , . . . , ( x αn α ) − ν nα ) . We push forward, by the diffeomorphism Ψ v,(cid:15) ◦ ψ v , the Poisson struc-ture π (cid:93) v A defined on P v (int( N v )) to int(Π v ) . The following lemma pro-vides the matrix representation of the push forwarded Poisson struc-ture. In order to simplify the notation we set J (ˆ x ) := D (Ψ v,(cid:15) ◦ ψ v ) ˆ x P v T x D x (6.4)and for every α = 1 , . . . , p J α (ˆ x α ) := D α (Ψ v,(cid:15) ◦ ψ v ) ˆ x α P αv T αx D x α . (6.5)Notice that J (ˆ x ) = diag( J (ˆ x ) , . . . , J p (ˆ x p )) . Lemma 6.6.
The diffeomorphism Ψ v,(cid:15) ◦ ψ v pushes forward the Poissonstructure π (cid:93) v A to the Poisson structure π (cid:93) v A ,(cid:15) on int(Π v ) where π (cid:93) v A ,(cid:15) ( y ) = ( − J A J t ) ◦ (Ψ v,(cid:15) ◦ ψ v ) − ( y ) . (6.6) Proof.
See Definition 3.1 and Remark 3.2. (cid:3)
If all the faces σ ∈ F v have order one, the Poisson structure π (cid:93) v A ,(cid:15) isasymptotically equivalent to a linear Poisson structure. Let E v = diag( E v , . . . , E pv ) , (6.7)be (( n − p ) × n ) -matrix defined by diagonal blocks E αv , for α = 1 , .., p ,where α th block is the (( n α − × n α ) -matrices in which the column j α is equal to n α − and every other columns k α (cid:54) = j α is equal to − e k α ∈ R n α − . Lemma 6.7.
For a given vertex v = ( e j , ..., e j p ) , let E v be the matrixdefined at (6.7) and B v := E v A E tv . If ν σ = 1 for every σ ∈ F v , then lim (cid:15) → + − (cid:15) J ◦ (Ψ v,(cid:15) ◦ ψ v ) − ( y ) = E v , over int(Π v ) with convergence in C ∞ topology. Consequently, lim (cid:15) → + (cid:15) π (cid:93) v A ,(cid:15) ( y ) = B v , over int(Π v ) with convergence in C ∞ topology.Proof. A simple calculation shows that for every α = 1 , .., p − (cid:15) J α = D αx,ν ( x α − . . . x αj α − x αj α x αj α − . . . x αn α ... x α . . . ( x αj α − − x αj α x αj α +1 . . . x αn α x α , . . . x αj α − x αj α ( x αj α +1 − . . . x αn α ... x α . . . , x αj α − x αj α x αj α +1 . . . ( x αn α − , where D αx,ν := diag(( x α ) − ν α , . . . , ( x αj α − ) − ν jα − , ( x αj α +1 ) − ν jα +1 , . . . , ( x αn α ) − ν nα ) Since ν σ = 1 for every σ ∈ F v , for any ˆ k ∈ [ α ] we have lim (cid:15) → + x α ˆ k ◦ (Ψ v,(cid:15) ◦ ψ v ) − ( y ) = lim (cid:15) → + e − yα ˆ k(cid:15) = 0 . Considering that x αj α = 1 − (cid:88) ˆ k ∈ [ α ] x α ˆ k , we get the first claim of the lemmaand the second claim is an immediate consequence. (cid:3) Figure 5 explains the situation for
Γ = ∆ . Remark 6.8.
The same linear Poisson structure B v := E v A E tv ap-pears in [2, Theorem . ]. AMILTONIAN POLYMATRIX REPLICATORS 25 (0 , ,
1) (1 , R C ∗ (∆ ) (Π v , B v )(Π v , B v )(Π v , B v )∆ v v v ψ v ii =0 , , Ψ v i ,(cid:15) ( (cid:15) → i =0 , , Figure 5.
Poisson structures on the dual cone.
Lemma 6.9.
For a given vertex v = ( e j , ..., e j p ) , let χ v be the skeletoncharacter of X n,A , as in Definition 5.3. Then χ v = B v dη v , where η v is the restriction of function η (defined in (6.2) ) to int(Π v ) . Inother words, χ v restricted to int(Π v ) is Hamiltonian w.r.t. the constantPoisson structure B v having η v as a Hamiltonian function.Proof. We use the notation X v ( n,A ) (ˆ x ) := ( D x P v ) X n,A ( x ) for the localexpression of the replicator vector field X n,A in the local chart ψ v . Ifwe write the function h ( x ) , defined in (6.1), as h ( x ) = h ( ψ v ◦ P v ( x )) then d x h = ( P v ) t d ˆ x ( h ◦ ψ v )(ˆ x ) . Notice that DP v = P v . By Theorem 4.5, X n,A = π A dh . Locally, X v ( n,A ) (ˆ x ) = P v X n,A ( x ) = P v π A P tv d ˆ x ( h ◦ ψ v ) . Similarly, writing h ◦ ψ v (ˆ x ) = h ◦ ψ v ◦ (Ψ v,(cid:15) ◦ ψ v ) − ◦ (Ψ v,(cid:15) ◦ ψ v )(ˆ x ) wehave d ˆ x ( h ◦ ψ v ) = ( D ˆ x (Ψ v,(cid:15) ◦ ψ n )) t d y ( h ◦ (Ψ v,(cid:15) ) − ) . The vector field ˜ X (cid:15)v defined in Lemma 5.11 is ˜ X (cid:15)v = 1 (cid:15) ( D x (Ψ v,(cid:15) ) X n,A ) = 1 (cid:15) ( D ˆ x (Ψ v,(cid:15) ◦ ψ v ) X v ( n,A ) )= 1 (cid:15) ( D ˆ x (Ψ v,(cid:15) ◦ ψ v ) P v π A P tv D ˆ x (Ψ v,(cid:15) ◦ ψ n )) t d y ( h ◦ (Ψ v,(cid:15) ) − )= 1 (cid:15) π (cid:93) v A ,(cid:15) (cid:16) (cid:15) ((Ψ v,(cid:15) ) − ) ∗ d x h (cid:17) In the second equality we used ψ v ◦ P v = Id . Applying Lemma 5.11,Lemma 6.7 and Proposition 6.1 yields the result. Notice that Π v ( (cid:15) r ) ⊂ int(Π v ) . (cid:3) Our aim is to show that for a given heteroclinic path ξ = ( γ , γ , . . . , γ m ) ,the skeleton flow map of χ along ξ (see Definition 5.14), π ξ := L γ m − ,γ m ◦ . . . ◦ L γ ,γ , restricted to the level set of η , is a Poisson map. Notice that thePoisson structure B v is only defined in int(Π v ) and neither Π γ nor Π γ (cid:48) are submanifolds of int(Π v ) . So we need to define Poisson structureson the sections Π γ i ,γ i +1 for all i = 0 , ..., m .We start with a single flowing vertex v = ( e j , ..., e j p ) with an in-coming flowing-edge v γ −→ v and an outgoing flowing-edge v γ (cid:48) −→ v .By construction of Γ n , there exist ≤ α v , α v ≤ p , ˆ k ∈ [ α ] v and ˆ k ∈ [ α ] v such that v = v +(0 , . . . , e ˆ k − e j αv , . . . , and v = v +(0 , . . . , e ˆ k − e j αv , . . . , . Remark 6.10.
To simply notations, we omit the superscript v from α v , α v whenever there is no confusion. Also, denoting v = ( e j , ..., e j p ) ,clearly, ˆ k = j α v and similarly ˆ k = j α v . Again, in order to keep no-tations as simple as possible we use notations ˆ k and ˆ k .Clearly, γ (cid:48) = v + s (0 , . . . , e ˆ k − e j α , . . . , . The opposite facet to γ (cid:48) at v is σ ∗ := { y α ˆ k = 0 } . We also have Π v = { y ∈ R n + | y j = . . . = y pj p = 0 } , Π v r = { y ∈ R n + | y j = . . . = y α r − j αr − = y α r ˆ k r = y α r +1 j αr +1 = . . . = y pj p = 0 } , r = 0 , γ = { y ∈ R n + | y j = . . . = y α − j α − = y α j α = y α ˆ k = y α +1 j α = . . . = y pj p = 0 } , Π γ (cid:48) = { y ∈ R n + | y j = . . . = y α − j α − = y α j α = y α ˆ k = y α +1 j α = . . . = y pj p = 0 } . (6.8)The skeleton flow map of χ at vertex v is the linear map L γ,γ (cid:48) : Π γ,γ (cid:48) → Π γ (cid:48) defined by L γ,γ (cid:48) ( y ) := (cid:32) y α ˆ i α − ( χ v ) α ˆ i α ( χ v ) α ˆ k y α ˆ k (cid:33) α, ˆ i α , (6.9)where Π γ,γ (cid:48) := (cid:40) y ∈ int(Π γ ) : y α ˆ i α − ( χ v ) α ˆ i α ( χ v ) α ˆ k y α ˆ k > , ∀ α, ˆ i α (cid:54) = α , ˆ k (cid:41) . (6.10) Remark 6.11.
In [3], this map is referred to as
Asymptotic Poincarémap . However, it is not a Poincaré map in the conventional way, sincethe skeleton vector filed χ is only defined in int(Π v ) for all v ∈ F andits value on the sections Π γ is zero. For this reason, we cannot use, AMILTONIAN POLYMATRIX REPLICATORS 27 directly, the techniques introduced in Section 3. To solve this issue wewill use two cosymplectic foliations of int(Π v ) .In fact, we need to restrict ourself to a tubular neighborhood, seeDefinition 6.12, rather than the entire int(Π v ) . For the flowing vertex v γ −→ v γ (cid:48) −→ v , we define the following family of maps L δγ,γ (cid:48) ( y ) := (cid:32) y α ˆ i α − δ ( χ v ) α ˆ i α ( χ v ) α ˆ k y α ˆ k (cid:33) α, ˆ i α , δ ∈ (0 , . Notice that L γ,γ (cid:48) ( y ) = φ χ v ( t ( y ) , y ) where φ χ v ( t, y ) = y + tχ v , is theflow of the skeleton vector field χ v and t ( y ) := − y α k ( χ v ) α k . Definition 6.12.
We denote by T γ,γ (cid:48) := (cid:91) <δ< L δγ,γ (cid:48) (Π γ,γ (cid:48) ) , (6.11)the tubular neighborhood containing the segments of the flow of χ v connecting points in the domain of the asymptotic Poincaré map, L γ,γ (cid:48) ,to their images.We, now, describe the Poisson structures on Π γ,γ (cid:48) and L γ,γ (cid:48) (Π γ,γ (cid:48) ) . Lemma 6.13.
As we adapted in Lemma 6.9, let η v be the restrictionof function η , defined in (6.2) , to int(Π v ) . Consider two functions G v r : T γ,γ (cid:48) → R , r = 0 , defined by G v r ( y ) = y α r ˆ k r , then1) Level sets of ( η v , G v r ) : T γ,γ (cid:48) → R partition T γ,γ (cid:48) into a cosym-plectic foliation F v r i.e. every leaf of F v r is a cosymplecticsubmanifold of ( T γ,γ (cid:48) , B v ) . Furthermore, every leaf Σ of this fo-liation is a level transversal section to χ v at every point x ∈ Σ .2) For a fixed r ∈ { , } the Poincaré map between two given leafsof F v r is a Poisson map.3) The Poincaré map from one given leaf of F v to a leaf of F v is a Poisson map.Proof. For r = 0 , , we have that { η v , G v r } = X η v ( dG v r ) = χ v ( dG v r ) = ( χ v ) α r ˆ k r . Since v is a flowing vertex both ( χ v ) α r ˆ k r , r = 0 , are nonzero. Thismeans that η v and G v r are second class constraints, s their level setsare cosymplectic submanifolds, see Definition 3.4. The fact that Σ isa level transversal section is clear. Proposition 3.11 yields the secondand third claims of the lemma. (cid:3) The Poincaré map mentioned in Item (2) of Lemma 6.13 is a transla-tion. Fixing r ∈ { , } , consider two level set ( η v , G v r ) − ( c, d j ) , j = 0 , , then the Poincaré map between them is the translation P ( y ) = φ χ v (cid:32) d − d ( χ v ) α r ˆ k r , y (cid:33) = (cid:32) d − d ( χ v ) α r ˆ k r (cid:33) χ v + y. (6.12)The Poincaré map between two level sets ( η v , G v ) − ( c, d ) and ( η v , G v ) − ( c, d ) is P ( y ) = φ χ v (cid:32) d − y α ˆ k ( χ v ) α ˆ k , y (cid:33) (6.13)Notice that the flow of χ v = X η v preserves η v . Clearly, thePoincaré maps can be considered between level set of functions G v r , r =0 , . So we define: Definition 6.14.
For r = 0 , , let ¯ F v r be the foliation constituted bythe level set of functions G v r ( y ) = y α r ˆ k r .Instead of two foliations F v r , r = 0 , , we will consider foliations ¯ F v r . Every leaf of ¯ F v r is equipped with a Poisson structure, π v r , whichhas η v as a Casimir and the level sets of this Casimir are leafs ofcosymplectic foliation F v r . Cleary, for a fixed r ∈ { , } , the leafsof ¯ F v r are Poisson diffeomorphic through Poincaré maps of χ v . Sincethese Poincaré maps are simply translations, see Equation (6.12), theseleafs are essentially same (as Poisson manifolds). Definition 6.15. By (Σ v r , π v r ) , r = 0 , we will denote a typical leafof Poisson foliation ¯ F v r .Notice that (Σ v r , π v r ) is a union of Poisson submanifolds equippedwith Dirac bracket. Proposition 6.16.
If we identify Π γ,γ (cid:48) with (Σ v , π v ) and L γ,γ (cid:48) (Π γ,γ (cid:48) ) with (Σ v , π v ) via Poincaré maps (translations) of type defined in Equa-tion (6.12) then L γ,γ (cid:48) is Poisson map, see Figure 6.Proof. We decompose L γ,γ (cid:48) into three Poincaré maps P v , P v and P v where P v is used to identify Π γ,γ (cid:48) with (Σ v , π v ) , P v is used to identify L γ,γ (cid:48) (Π γ,γ (cid:48) ) with (Σ v , π v ) and P v is the Poincaré map from (Σ v , π v ) to (Σ v , π v ) . P and P are Poisson since they are simple translations.Item (3) of Lemma 6.13 shows that P , defined in Equation (6.13), isa Poisson map. (cid:3) We describe now the matrix representation of the Dirac bracketstructure π v r , r = 0 , . Lemma 6.17.
The matrix representation of the Dirac bracket gener-ated in int(Π v ) by second class constrains η v and G v r , r = 0 , , is ( π v Dirac ,r ) (cid:93) = B v − C ( v ,r ) , (6.14) AMILTONIAN POLYMATRIX REPLICATORS 29 C ∗ (Γ) \ ∪ i =0 Π v i Π v Π v Π v Π γ Π γ (cid:48) Σ v Σ v Σ v Σ v φ χ v φ χ v φ χ v Figure 6.
Illustration of Proposition 6.16. where C ( v ,r ) = [ C α,β ( v ,r ) ] α,β with C α,βv ,r = [ c ˆ i ˆ j ( α, β, v , r )] (ˆ i, ˆ j ) ∈ [ α ] × [ β ] where c ˆ i ˆ j ( α, β, v , r ) = 1( χ v ) α r ˆ k r (cid:16) ( χ v ) α ˆ i b α r ,β ˆ k r ˆ j + b α,α r ˆ i ˆ k r ( χ v ) β ˆ j (cid:17) . In the matrix ( π v Dirac ,r ) (cid:93) the line and the column associated to G v r = y α r ˆ k r are null. Removing these line and column one obtains the matrixrepresentation of the Poisson structure π v r , r = 0 , .Proof. Notice that ( π v Dirac ,r ) (cid:93) = (cid:104) { y α ˆ i , y β ˆ j } (cid:105) (ˆ i, ˆ j ) ∈ [ α ] × [ β ] . A simple calculation yields (6.14), see Definition of { ., . } Dirac at Equa-tion (3.4). The rest of the proof is trivial considering that the function G v r = y α r ˆ k r is a Casimir of π v Dirac ,r and the submanifold Σ v r is a level set Σ v r . (cid:3) Remark 6.18.
The Poisson structure considered on domain of L γ,γ (cid:48) is π v and on its image is π v , see Proposition 6.16. Furthermore, thisPoisson structure only depends on γ and is independent of L γ,γ (cid:48) so wecan actually consider this structure on whole Π γ . From now on we willdo so.The main result of this manuscript is Theorem 6.19.
Let ξ : v γ −→ v γ −→ v −→ . . . −→ v m γ m −→ v m +1 (6.15) be a heteroclinic path. Then(1) For every l = 1 , . . . , m , two Poisson submanifolds, (Σ v l − , π v l − ) and (Σ v l , π v l ) (see Definition 6.15), are Poisson diffeomorphic.In other words, the Poisson structures considered on Π γ l − ,γ l from int(Π v l − , B v l − ) and int(Π v l , B v l ) are the same, see Fig-ure 6.(2) If we identify every Π γ l − ,γ l , l = 1 , . . . , m with Poisson subman-ifold (Σ v l − , π v l − ) (or equivalently with (Σ v l , π v l ) ), the skeletonflow map of χ along ξ (see Definition 5.14), π ξ := L γ m − ,γ m ◦ . . . ◦ L γ ,γ , is a Poisson map.Proof. Item (2) if the lemma is a consequence of Item (1) and Propo-sition 6.16.Notice that the two sectors Π v l − and Π v l are only different in thegroup α v l − = α v l , where y α v ˆ k = 0 for the elements of Π v l − and y α v j αv =0 for the elements Π v l , see Remark 6.10. Let P v l − ,v l = diag( P v l − ,v l , . . . , P pv l − ,v l ) + T l − ,l , be the map defined by following items1) For β (cid:54) = α v l the associated component P βv l − ,v l is the identitymap.2) ( P v l − ,v l ( y )) α vl ˆ i = y α vl ˆ i − y α vl j αvl if ˆ i (cid:54) = ˆ k − y α vl j αvl if ˆ i = ˆ k
3) The constant vector T l − ,l is determined such that P v ,v (Σ v l − ) = Σ v l . Notice that Σ v l − is a level set of the function G v l − = y α vl j αvl , Σ v l is alevel set of G v l = y α vl ˆ k and P v l − ,v l sends the level set of the function G v l − to level sets of the function G v l . This means that Item (3) aboveis feasible. Using Item (3) above we set P v l − ,v l : U Σ vl − → U Σ vl , where U Σ vl − and U Σ vl are neighborhoods of Σ v l − and U Σ vl , respectively,such that the associated Dirac brackets are defined.It takes a simple calculation to verify that ( DP v l − ,v l ) E v l − = E v l .This fact together with the condition (3.3) and the definitions of B v l − , B v l (see Lemma 6.7) yields that P v l − ,v l : ( U Σ vl − , B v l − ) → ( U Σ vl , B v l ) is a Poisson map, i.e. P v l − ,v l preserves the ambient Poisson structure. AMILTONIAN POLYMATRIX REPLICATORS 31
We have η v l ◦ P v l − ,v l ( y ) = (cid:88) β (cid:54) = α vl (cid:88) ˆ j ∈ [ β ] λ β q βj y βj + (cid:88) ˆ i (cid:54) =ˆ k λ α vl q α vl ˆ i ( y α vl i − y α vl j αvl ) − λ α vl q α vl ˆ k y α vl j αvl = (cid:88) β (cid:54) = α vl (cid:88) ˆ j ∈ [ β ] λ β q βj y βj + (cid:88) ˆ i (cid:54) =ˆ k λ α vl q α vl ˆ i y α vl i − λ α vl (cid:88) ˆ i q α vl ˆ i y α vl j αvl = (cid:88) β (cid:54) = α vl (cid:88) ˆ j ∈ [ β ] λ β q βj y βj + (cid:88) ˆ i λ α vl q α vl ˆ i y α vl i − η v l − − , (6.16)where we used the fact that − (cid:80) ˆ i q α vl ˆ i = q α vl j αvl − . In other word, P v l − ,v l sends the second class constraint of η v l − of Σ v l − to secondclass constraint η v l of Σ v l (ignoring addition of − which does not doany harm). Furthermore, G v l ◦ P v l − ,v l = − G v l + T l − ,l . Notice that, as mentioned in Remark 3.9, the Poisson structure π v l − is independent of the choice of second class constraint G v l . Since − G v l is a second class constraint as well, considering it as the second classconstraint , beside η v l − , as the constraint used to define π v l − , we haveshown P v l − ,v l sends the second class constraints of Σ v l − to second classconstraints of Σ v l . Consequently, the map ( P v l − ,v l ) | Σ vl − : (Σ v l − , π v l − ) → (Σ v l , π v l ) , is a Poisson map since P v l − ,v l preserves the ambient Poisson structureas well.We denote by P v l − the map used to identify Π γ l − ,γ l with (Σ v l − , π v l − ) via flow of χ v l − , see Equation (6.12), and by P v l the map used to iden-tify Π γ l − ,γ l with (Σ v l , π v l ) via flow of χ v l . Another consequence ofEquation 6.16 is that P v l − ,v l sends χ v l − to χ v l which means that thefollowing diagram Π γ l − ,γ l Σ v l − Σ v l P vl P vl − P vl − ,vl is commutative. This finishes the proof. (cid:3) The Poisson structures defined on the sectors associated to an edgedepend only on the edge and on the skeleton vector field, and is in-dependent of the the heteroclinic path containing the edge (see Re-mark 6.18). Let us denote this structure by {· , ·} γ . Our result holdsfor the skeleton flow map (see Definition 5.17) as an immediate conse-quence of Theorem 6.19. Theorem 6.20.
Let B S ( χ ) denote the set of all S -branches of theskeleton vector field ξ (see Definition 5.16) and set D S := ∪ ξ ∈ B S ( χ ) Π ξ to be the open submanifold of (Π S , {· , ·} S ) := ∪ γ ∈ S (Π γ , {· , ·} γ ) , with the same Poisson structure. Then the skeleton flow map π S :( D S , {· , ·} S ) → (Π S , {· , ·} S )) is Poisson. Example
We will now present an example of a Hamiltonian polymatrix repli-cator system with a non trivial dimension. This example was chosen toprovide an illustration of the concepts and main results of this paper.In particular it has a small structural set with a simple heteroclinicnetwork.7.1.
The fish example.
Consider the polymatrix replicator systemdefined by matrix A = − − − − − − . We denote by X A the vector field associated to this polymatrix repli-cator that is defined on the polytope Γ (5 , := ∆ × ∆ . The point q = (cid:18) , , , , , , (cid:19) ∈ Γ (5 , satisfies(1) Aq = (0 , , , , , , ;(2) q + q + q + q + q = 1 and q + q = 1 ,where q i stands for i -th component of the vector q , and hence is an equi-librium of X A (see Proposition 4.3). Since matrix A is skew-symmetric,the associated polymatrix replicator is conservative (see Definition 4.4). AMILTONIAN POLYMATRIX REPLICATORS 33
The polytope Γ (5 , has seven faces labelled by an index j rangingfrom to , and designated by σ , . . . , σ . The vertices of the phasespace Γ (5 , are also labelled by i ∈ { , . . . , } , and designated by v , . . . , v , as described in Table 1.Vertex Γ (5 , v = (1 ,
6) (1 , , , , , , v = (1 ,
7) (1 , , , , , , v = (2 ,
6) (0 , , , , , , v = (2 ,
7) (0 , , , , , , v = (3 ,
6) (0 , , , , , , Vertex Γ (5 , v = (3 ,
7) (0 , , , , , , v = (4 ,
6) (0 , , , , , , v = (4 ,
7) (0 , , , , , , v = (5 ,
6) (0 , , , , , , v = (5 ,
7) (0 , , , , , , Table 1.
Identification of the ten vertices of the polytope, v , . . . , v in Γ (5 , . The skeleton character χ A of X A is displayed in Table 2. (See Defi-nition 5.3 and Proposition 5.5.) χ vσ σ σ σ σ σ σ σ v ∗ ∗ − v ∗ − − − ∗ v − ∗ ∗ v ∗ − ∗ v − ∗ ∗ v − ∗ − ∗ v − ∗ ∗ v − ∗ ∗ v − ∗ ∗ v ∗ − ∗ Table 2.
The skeleton character χ A of X A , where the symbol ∗ in the i -th line and j -th column of the table means that the vertex v i does not belong to the face σ j of the polytope Γ (5 , . The edges of Γ (5 , are designated by γ , . . . , γ , according to Table 3,where we write γ = ( i, j ) to mean that γ is an edge connecting thevertices v i and v j . This model has edges: neutral edges, γ , γ , γ , γ , γ , γ , γ , γ , γ , γ , γ , γ , γ , and flowing-edges, γ , γ , γ γ , γ , γ , γ , γ , γ , γ , γ , γ , γ . The flowing-edge directed graph of χ A is depicted in Figure 7. γ = (1 , γ = (3 , γ = (2 , γ = (3 , γ = (8 , γ = (3 , γ = (2 , γ = (1 , γ = (4 , γ = (5 , γ = (5 , γ = (1 , γ = (2 , γ = (3 , γ = (6 , γ = (7 , γ = (2 , γ = (5 , γ = (4 , γ = (9 , γ = (10 , γ = (1 , γ = (6 , γ = (7 , γ = (8 , Table 3.
Edge labels.
From this graph we can see that S = { γ = (1 , } is a structural set for χ A (see Definition 5.16) whose S -branches denotedby ξ , . . . , ξ are displayed in Table 4, where we write ξ i = ( j, k, l, . . . ) toindicate that ξ i is a path from vertex v j passing along vertices v k , v l , . . . . Figure 7.
The oriented graph of χ A . From\To γ = (1 , γ = (1 , ξ = (1 , , , , , , , , ξ = (1 , , , , , , , , , ξ = (1 , , , , , , , , , , ξ = (1 , , , , , , , , , , ξ = (1 , , , , , , , , , , , Table 4. S -branches of χ A . Considering the vertex v , which has the incoming edge v γ −→ v andthe outgoing edge v γ −→ v , we will now illustrate Proposition 6.16. AMILTONIAN POLYMATRIX REPLICATORS 35 C ∗ (Γ) \ ∪ i =1 Π v i (Π v , B ) (Π v , B )(Π v , B ) Π γ Π γ (Σ v , π )(Σ v , π ) (Σ v , π )(Σ v π ) L γ γ Figure 8.
Illustration of Proposition 6.16 for the example.
By straightforward calculations we obtain B = − − − − − − − − − , B = − − − − − − − − −
21 1 1 2 0 and B = − − − −
12 0 2 1 01 − − − − , and by (6.14) we get ( π v Dirac , ) (cid:93) = ( π v Dirac , ) (cid:93) = − − − − and ( π v Dirac , ) (cid:93) = ( π v Dirac , ) (cid:93) = − − − − . The matrix ( π v Dirac , ) (cid:93) is the representation of the Poisson structureon Π γ in the coordinates ( y , y , y , y , y ) . Notice that y = 0 on Π γ . Similarly, the matrix ( π v Dirac , ) (cid:93) is the representation of the Poisson structure on Π γ in the same coordinates ( y , y , y , y , y ) . Notice againthat y = 0 on Π γ . Now the matrix representation of L γ γ in thecoordinates ( y , y , y , y , y ) is L γ γ = . A simple calculation shows that L γ γ ( π v Dirac , ) (cid:93) ( L γ γ ) t = ( π v Dirac , ) (cid:93) , confirming the fact that the asymptotic Poincaré map L γ γ is Poisson(see(3.3) in Definition 3.1).Consider now the subspaces of R H = (cid:40) ( x , . . . , x ) ∈ R : (cid:88) i =1 x i = 1 , (cid:88) i =6 x i = 1 (cid:41) and H = (cid:40) ( x , . . . , x ) ∈ R : (cid:88) i =1 x i = 0 , (cid:88) i =6 x i = 0 (cid:41) . For the given matrix A , its null space Ker( A ) has dimension . Takea non-zero vector w ∈ Ker( A ) ∩ H . For example, w = ( − , , − , , − , − , . The set of equilibria of the natural extension of X A to the affine hy-perplane H is Eq( X A ) = Ker( A ) ∩ H = { q + tw : t ∈ R } . The Hamiltonian of X A is the function h q : Γ (5 , → R h q ( x ) := (cid:88) i =1 q i log x i , where q i is the i -th component of the equilibrium point q (see Theo-rem 4.5). Another integral of motion of X A is the function h w : Γ (5 , → R h w ( x ) := (cid:88) i =1 w i log x i , where w i is the i -th component of w , which is a Casimir of the under-lying Poisson structure.The skeletons of h q and h w are respectively η q , η w : C ∗ (Γ (5 , ) → R , η q ( y ) := (cid:88) i =1 q i y i and η w ( y ) := (cid:88) i =1 w i y i , AMILTONIAN POLYMATRIX REPLICATORS 37 (see Proposition 6.1), which we use to define η : C ∗ (Γ (5 , ) → R , η ( y ) := ( η q ( y ) , η w ( y )) . Consider the skeleton flow map π S : Π S → Π S of χ A (see Def-inition 5.17). Notice that Π S = Π γ , where by Proposition 5.18, Π γ = (cid:83) i =1 Π ξ i (mod 0) . By Proposition 6.1 the function η is in-variant under π S . Moreover, the skeleton flow map π S is Hamiltonianw.r.t. a Poisson structure on the system of cross sections Π S (see The-orem 6.19).For all i = 1 , . . . , , the polyhedral cone Π ξ i has dimension . Hence,each polytope ∆ ξ i ,c := Π ξ i ∩ η − ( c ) is a -dimensional polygon. Remark 7.1.
We came from dimension to . This will happen forany other conservative polymatrix replicator with the same number ofgroups and the same number of strategies per group. In fact when n − p is odd, where n is the total number of strategies in the popula-tion and p is the number of groups, we will have a minimum drop of dimension. The reason is that a Poisson manifold with odd dimen-sion (in this example is ) has at least one Casimir, and consideringthe transversal section we drop two dimensions from the symplecticpart (not from the Casimir). So in total we drop a minimum of threedimensions. If the original Poisson structure has more Casimirs, theinvariant submanifolds yielded geometrically, are going to have evenless dimensions, which is good as long as it not zero. In the case of aneven dimension, the drop will be at least of two dimensions.By invariance of η , the set ∆ S,c is also invariant under π S . Considernow the restriction π S | ∆ S,c of π S to ∆ S,c . This is a piecewise affine areapreserving map. Figure 9 shows the domain ∆ S,c and
20 000 iteratesby π S of a point in ∆ S,c . Following the itinerary of a random point wehave picked the following heteroclinic cycle consisting of S -branches ξ := ( ξ , ξ , ξ , ξ ) . The map π ξ is represented by the matrix M ξ = − − −
01 0 1 − − − −
10 0 0 1 0 1 00 0 0 0 0 0 00 0 0 0 0 0 0 . The eigenvalues of M ξ , besides and (with geometric multiplicity and , respectively), are λ u = 5 . ..., and λ s = λ − u . Remark 7.2.
The determinant of ( π v Dirac , ) (cid:93) is zero which means thatthe Poisson structure on Π γ is non-degenerate. So, Π has a two di-mensional symplectic foliation invariant under the asymptotic Poincarémap. The leaf of this foliation are affine spaces parallel to the kernelof ( π v Dirac , ) (cid:93) | Π γ = − − − − , i.e. the set of the form { ( q , q , q , q ) + ( s, t, − t, − s ) | ( q , q , q , q ) ∈ Π γ s, t ∈ R } ∩ Π γ . The restriction of the asymptotic Poincaré map to these leafs is a sym-plectic map. One important consequence is that its eigenvalues are ofthe form λ and λ . Figure 9.
Plot of
20 000 iterates (in orange) by π S of a point in ∆ S,c , with c = (cid:0) , − . (cid:1) , the iterates of the periodic point p (ingreen) of the skeleton flow map π S with period , and the iteratesof another periodic point of the skeleton flow map π S with period (in blue). An eigenvector associated to the eigenvalue is p = (0 ., . , ., ., ., ., . ) . We have chosen c := ( c , c ) = (cid:0) , − . (cid:1) so that η ( p ) = c , i . e ., p ∈ ∆ S,c . In fact we have p ∈ ∆ ξ ,c ⊂ ∆ γ ,c . Hence p is a periodic AMILTONIAN POLYMATRIX REPLICATORS 39 point of the skeleton flow map π S with period (whose iterates arerepresented by the green dots in Figure 9).Figure 9 also depicts the polygons ∆ ξ ,c , ∆ ξ ,c , ∆ ξ ,c , ∆ ξ ,c , ∆ ξ ,c con-tained in ∆ γ , and the orbit of another periodic point of the skeletonflow map π S with period (represented by the blue dots in Figure 9).By [3, Theorem . ] we can deduce the existence of chaotic behaviourfor the flow of X A in some level set h − q ( c /(cid:15) ) ∩ h − w ( c /(cid:15) ) , with c =( c , c ) chosen above and for all small enough (cid:15) > . Acknowledgements
The first author was supported by mathematics department of UFMG.The second author was supported by FCT - Fundação para a Ciência e aTecnologia, under the projects UIDB/04561/2020 and UIDP/04561/2020.The third author was supported by FCT - Fundação para a Ciência e aTecnologia, under the Project CEMAPRE - UID/MULTI/00491/2019financed by FCT/MCTES through national funds
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Email address : [email protected] Departamento de Matemática and CMAF, Faculdade de Ciências,Universidade de Lisboa, Campo Grande, Edificio C6, Piso 2, 1749-016Lisboa, Portugal
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