Coupled cell networks: semigroups, Lie algebras and normal forms

We introduce the concept of a semigroup coupled cell network and show that the collection of semigroup network vector fields forms a Lie algebra. This implies that near a dynamical equilibrium the local normal form of a semigroup network is a semigroup network itself. Networks without the semigroup property will support normal forms with a more general network architecture, but these normal forms nevertheless possess the same symmetries and synchronous solutions as the original network. We explain how to compute Lie brackets and normal forms of coupled cell networks and we characterize the SN-decomposition that determines the normal form symmetry. This paper concludes with a generalization to nonhomogeneous networks with the structure of a semigroupoid.


Introduction
Coupled cell networks appear in many of the sciences and range from crystal models and electrical circuits to numerical discretization schemes, Josephson junction arrays, power grids, the world wide web, ecology, neural networks and systems biology. Not surprisingly, there exists an overwhelming amount of literature on coupled cell networks.
The last decade has seen the development of an extensive mathematical theory of dynamical systems with a network structure, cf. [16], [24], [35], [48], [51]. In these network dynamical systems, the evolution of the state of a constituent or "cell" is determined by the states of certain particular other cells. It is generally believed that a network structure has an impact on the behavior of a dynamical system, but it is not always clear how and why.
As an example, let us mention a system of differential equations with a homogeneous coupled cell network structure of the forṁ xi = f (x σ 1 (i) , . . . , x σn(i) ) for 1 ≤ i ≤ N. (1.1) These differential equations generate a dynamical system in which the evolution of the variable xi is only determined by the values of x σ 1 (i) , . . . , x σn(i) . The functions can thus be thought of as a network that prescribes how which cells influence which cells. The literature on network dynamical systems focuses on the analysis of equilibria, periodic solutions, symmetry, synchrony, structural stability and bifurcations. As in the classical theory of dynamical systems, one often faces the task here of computing a local normal form near a dynamical equilibrium. These normal forms are obtained from coordinate transformations, and in their computation one calculates Lie brackets of vector fields, either implicitly or explicitly. It is here that one encounters an important technical problem: Differential equations of the form (1.1) in general do not form a Lie-algebra.
As a consequence one can not expect that the normal form of a coupled cell network is a coupled cell network as well. This complicates the local analysis and classification of network dynamical systems, because it means that one always has to compute the normal form of a network explicitly to understand its generic behavior -unless one is willing to assume that the network is given in normal form from the beginning, cf. [24], [29]. Normal form computations in [13], [21], [26] have revealed that a network structure can have a nontrivial impact on this generic behavior. One wants to understand and predict this.
In this paper, we will formulate an easily verifiable condition on a network structure under which the coupled cell network vector fields do form a Lie subalgebra of the Lie algebra of vector fields. Our main result is the following: If {σ1, . . . , σn} is a semigroup, then the differential equations (1.1) form a Lie algebra.
In this case, the local normal form of (1.1) is also of the form (1.1).
In addition, we show that the Lie bracket of semigroup coupled cell network vector fields can be lifted to a symbolic bracket that only involves the function f . Normal form calculations can be performed at this symbolic level and one only returns to the reality of the differential equation when one is done computing. We also show that the symbolic space carries a dynamics of its own, determined by a certain fundamental network. This situation is analogous to that of Hamiltonian vector fields, of which the Lie bracket is determined by the Poisson bracket of Hamiltonian functions. As a consequence, Hamiltonian normal forms are usually computed at the level of functions. Moreover, the symbolic dynamics of Hamiltonian functions is determined by a Poisson structure, cf. [41].
When σ1, . . . , σn do not form a semigroup, then we suggest that one simply completes them to the smallest collection The normal form of (1.1) will now lie within the extended class of semigroup coupled cell networks, there being no guarantee that it is again of the original form (1.1).
Thus one can choose: either to respect any given network structure as if it were a law of nature, so that no normal form can be computed, or to extend every network to a semigroup network and live with the consequences. One can object that even simple networks may need a lot of extension before they form a semigroup. But as an argument in favor of the semigroup approach, let us mention that the symmetries and synchrony spaces of a network are not at all affected by our semigroup extension. This implies in particular that these symmetries and synchrony spaces will also be present in the local normal form of the network. This latter property is both pleasant and important, if only in view of the large amount of research that has been devoted to symmetry [5], [6], [9], [18], [25], [31] and synchrony [2], [3], [4], [8], [10], [11], [27], [32], [34], [38], [49], [51], [53] in coupled cell networks. Semigroups may well be the natural invariants of coupled cell networks, even more than groups and symmetries.
Normal forms are computed by applying coordinate transformations [42], [45], [46], [47]. These transformations can be in the phase space of a differential equation, but in our case they take place in the space of functions f and have the form of a series expansion Here f is the function to be transformed and normalized, g generates the coordinate transformation and ad Σ denotes a representation, in this case the adjoint representation of the Lie algebra of f 's. Although at first sight this may seem a needlessly complicated way to describe coordinate transformations, this "Lie formalism" allows for a very flexible theory which streamlines both the theory and the computations. The actual computation of the normal form of the function f , and in particular the matter of solving homological equations, will not be entirely standard in the context of networks. Some things remain as in the theory of generic vector fields. For example, we show that the adjoint action of a linear element admits an SN-decomposition that determines a normal form symmetry. Other aspects may not carry through, such as the applicability of the Jacobson-Morozov lemma to characterize the complement of the image of the adjoint action of a nilpotent element [7]. This is because the Lie algebra of the linear coupled cell network vector fields need not be reductive.
This paper is organized as follows. After giving a formal definition of a homogeneous coupled cell network in Section 2, we show in Section 3 that semigroups arise naturally in the context of coupled cell networks. In Sections 4 and 5 we prove that semigroup network dynamical systems are closed under taking compositions and Lie brackets. Section 6 explains how to compute the normal form of a network dynamical system, while in Sections 7 and 8 we prove that this normal form inherits both the symmetries and the synchrony spaces of the original network. In Section 9 we investigate the SN-decomposition of a linear coupled cell network vector field. This decomposition determines the normal form symmetry. Section 10 describes the aforementioned fundamental network. In Section 11 we actually compute the normal forms of some simple but interesting coupled cell networks, thus demonstrating that a coupled cell network structure can force anomalous steady state bifurcations. Finally, we show in Section 12 that our theory is also applicable to non-homogeneous or "colored" networks that display the structure of a semigroupoid.
Issues that we do not touch in this paper but aim to treat in subsequent work include: 1. The development of a linear algebra of semigroup coupled cell systems in order to define for example a "semigroup network Jordan normal form".

2.
Application of the results in this paper to semigroup networks that arise in applications, such as feed-forward motifs.
3. Understanding the impact of "input symmetries" on bifurcations and normal forms.

Homogeneous coupled cell networks
We shall be interested in dynamical systems with a coupled cell network structure. Such a structure can be determined in various ways [16], [35], [38], [51], but we choose to describe it here by means of a collection of distinct maps The collection Σ has the interpretation of a network with 1 ≤ N < ∞ cells. Indeed, it defines a directed multigraph with N vertices and precisely n arrows pointing into each vertex, where the arrows pointing towards vertex 1 ≤ i ≤ N emanate from the vertices σ1(i), . . . , σn(i).
The number n of incoming arrows per vertex is sometimes called the valence of the network. In a network dynamical system we think of every vertex 1 ≤ i ≤ N in the network as a cell, of which the state is determined by a variable xi that takes values in a vector space V . Definition 2.1 Let Σ = {σ1, . . . , σn} be a collection of n distinct maps on N elements, V a finite dimensional real vector space and f : V n → V a smooth function. Then we define Depending on the context, we will say that γ f is a homogeneous coupled cell network map or a homogeneous coupled cell network vector field subject to Σ. △ In the literature, γ f is also called an admissible map/vector field for the network Σ. Dynamical systems with a coupled cell network structure arise when we iterate the map γ f or integrate the vector field that it defines. The iterative dynamics on V N has the special property that the state of cell i at time m + 1 depends only on the states of the cells σ1(i), . . . , σn(i) at time m: The continuous-time dynamical system on V N displays the same property infinitesimally: it is determined by the ordinary differential equationṡ x = γ f (x) if and only ifẋi = f (x σ 1 (i) , . . . , x σn(i) ) for 1 ≤ i ≤ N.
We aim to understand how the network structure of γ f impacts these dynamical systems.
Example 2.2 An example of a directed multigraph is shown in Figure 1, where the number of cells is N = 3 and valence is equal to n = 2. The maps σ1 and σ2 are given by x 1 x 2 x 3 x 1 x 2 x 3 A coupled cell network map/vector field subject to {σ1, σ2} is of the form This network has obtained some attention [13], [24], [26], [29], [40] because it supports an anomalous codimension-one nilpotent double Hopf bifurcation when dim V = 2. △ Example 2.3 In this example we let σ1, σ2 be as in Example 2.2 and we also define σ3 as The network defined by {σ1, σ2, σ3} is depicted in Figure 2.

Semigroups
A first and obvious difficulty that arises in the study of coupled cell network dynamical systems is that the composition γ f • γg of two coupled cell network maps with an identical network structure may not have that same network structure. Dynamically, this implies for example that the equation γ f (x) = x for the steady states of γ f and the equation (γ f ) m (x) = x for its periodic solutions may have quite a different nature. We illustrate this phenomenon in the following example: Example 3.1 Again, let N = 3 and let σ1, σ2, σ3 be defined as in Examples 2.2 and 2.3. If γg(x1, x2, x3) = (g(x1, x1), g(x2, x1), g(x3, x2)) , are coupled cell network maps subject to {σ1, σ2}, then the composition in general is not a coupled cell network map subject to {σ1, σ2}.
On the other hand, when γ f and γg are network maps subject to {σ1, σ2, σ3}, i.e.
It is clear that every coupled cell network map γ f subject to Σ is also a coupled cell network map subject to the semigroup Σ ′ . Indeed, if we define f ′ (X1, . . . , Xn, Xn+1, . . . , X n ′ ) := f (X1, . . . , Xn) then it obviously holds that We thus propose to augment Σ to the semigroup Σ ′ and to think of every coupled cell network map subject to Σ as a (special case of a) coupled cell network map subject to Σ ′ . Example 3.3 Again, let N = 3 and let σ1, σ2, σ3 be defined as in Examples 2.2 and 2.3. It holds that σ 2 2 = σ3, so the collection {σ1, σ2} is not a semigroup. On the other hand, one quickly computes that the composition table of {σ1, σ2, σ3} is given by • σ1 σ2 σ3 σ1 σ1 σ2 σ3 σ2 σ2 σ3 σ3 σ3 σ3 σ3 σ3 .

Composition of network maps
To understand better how network maps behave under composition and in order to simplify our notation, let us define the maps This definition allows us to write (2.2) simply as Expression (3.7) moreover turns into the formula The following technical result helps us write the right hand side of (4.9) in the form h • πi for some function h : V n → V , whenever Σ is a semigroup.  Moreover, it holds that Aσ j 1 • Aσ j 2 = Aσ j 1 •σ j 2 for all 1 ≤ j1, j2 ≤ n.
This proves the theorem.
The identity Aσ j 1 • Aσ j 2 = Aσ j 1 •σ j 2 expresses that the Aσ j form a representation of the semigroup Σ. Using this representation we obtain: . . , σn} be a semigroup. Define for f, g : V n → V the function Then Proof: From formula (4.9) and Theorem 4.1.
Theorem 4.2 reveals once more that if Σ is a semigroup, then the composition of two coupled cell network maps γ f and γg is again a coupled cell network map, namely γ f • Σ g . More importantly, it shows how to compute f •Σ g "symbolically", i.e. using only the functions f and g and a representation of the network semigroup. The final result of this section ensures that the "symbolic composition" •Σ makes the space C ∞ (V n , V ) into an associative algebra. Proof: With Lemma 4.3 at hand, Theorem 4.2 just means that the linear map is a homomorphism of associative algebras. Substitution in (4.13) therefore yields that We conclude that f •Σ g equals the function h found in Example 3.1.
moreover reveals that This means in particular that the collection { σ1, . . . , σn} is closed under composition. The maps σ1, . . . , σn will play an interesting role in this paper. In fact, we will show in Section 10 that they are themselves the network maps of a certain "fundamental network" that fully determines the fate of all network dynamical systems subject to Σ. △ This observation can be used give a proof of Theorem 4.1 that is free of coordinates: Proof [Of Theorem 4.1 without coordinates]: We observe that and hence that σ i • σ j = σ σ j (i) . Using this, we find that Similarly, the computation As a consequence, Unfortunately, this coordinate free proof of Theorem 4.1 is relatively long. △

A coupled cell network bracket
We will now think of γ f : V N → V N as a vector field that generates the differential equatioṅ We suggestively denote by e tγ f the time-t flow of the vector field γ f on V N and by (e tγg ) * γ f the pushforward of the vector field γ f under the time-t flow of γg. We recall that the Lie bracket of γ f and γg is then the vector field [γ f , γg] : 14) The main result of this section is that if Σ is a semigroup, then the collection of coupled cell network vector fields is closed under taking Lie brackets.
Theorem 5.1 Let Σ = {σ1, . . . , σn} be a semigroup and let the Aσ j : V n → V n be as in Theorem 4.1. Define, for f, g : Then Proof: We start by remarking that Differentiating this identity with respect to t and evaluating the result at t = 0 gives that This proves that (Dγ f · γg)i = n j=1 Djf · (g • Aσ j ) • πi, and hence that Dγ f · γg is a coupled cell network vector field. With a similar computation for Dγg · γ f , we thus find that the Lie bracket between γ f and γg is given by This proves the theorem. Lemma 5.2 below states that the "symbolic bracket" [·, ·]Σ is a Lie bracket.
Differentiating this identity with respect to t and evaluating the result at t = 0 yields that With this in mind, we now compute that Using the symmetry of the second derivatives, the Jacobi identity follows from cyclically permuting f, g and h in the above expression and summing the results. This proves that C ∞ (V n , V ) is a Lie algebra. Theorem 5.1 means that γ is a Lie algebra homomorphism.
Example 5.3 Again, let N = 3 and let σ1, σ2, σ3 be defined as in Examples 2.2 and 2.3. We recall that Aσ 1 , Aσ 2 and Aσ 3 were computed in Example 4.4. It follows that

Coupled cell network normal forms
Normal forms are an essential tool in the study of the dynamics and bifurcations of maps and vector fields near equilibria, cf. [42], [47]. In this section we will show that it can be arranged that the normal form of a semigroup coupled cell network is a coupled cell network as well. This normal form can moreover be computed "symbolically", i.e. at the level of the function f . With Theorem 5.1 at hand, this result is perhaps to be expected. We nevertheless state two illustrative theorems in this section, and sketch their proofs. We start by making a few standard definitions. First of all, we define for Next, we define for every k = 0, 1, 2, . . . the finite dimensional subspace One can observe that P 0 = L(V N , V ) and that if f ∈ P k and g ∈ P l , then [f, g]Σ ∈ P k+l , as is obvious from formula (5.15). In particular, we have that With this in mind, we formulate the first main result of this section. It essentially states that one may restrict the study of semigroup coupled cell networks near local equilibria to semigroup coupled cell networks of a very specific "normal form".
and assume that f (0) = 0. We Taylor expand f as Let 1 ≤ r < ∞ and for every 1 ≤ k ≤ r, let N k ⊂ P k be a subspace such that Then there exists an analytic diffeomorphism Φ, sending an open neighborhood of 0 in V N to an open neighborhood of 0 in V N , that conjugates the coupled cell network vector field γ f to a coupled cell network vector field γ f with We only sketch a proof without estimates here, because the construction of a normal form by means of "Lie transformations" is very well-known.
For g ∈ C ∞ (V n , V ) with g(0) = 0, the time-t flow e tγg defines a diffeomorphism of some open neighborhood of 0 in V N to another open neighborhood of 0 in V N . Thus we can consider, for any f ∈ C ∞ (V n , V ), the curve t → (e tγg ) * γ f ∈ C ∞ (V N , V N ) of pushforward vector fields. This curve satisfies the linear differential equation where the second equality holds by definition of the Lie bracket of vector fields (5.14) and we have used the conventional definition of adγ g : Solving the linear differential equation (6.18) together with the initial condition (e 0γg ) * γ f = γ f , we find that the time-1 flow of γg transforms γ f into The main point of this proof is that by Theorem 5.1 the latter expression is also equal to .
The diffeomorphism Φ in the statement of the theorem is now constructed as the composition of a sequence of time-1 flows e γg k (1 ≤ k ≤ r) of coupled cell network vector fields γg k with g k ∈ P k . We first take g1 ∈ P 1 , so that γ f is transformed by e γg 1 into It is the fact that N 1 ⊕ im ad Σ f 0 P 1 = P 1 that allows us to choose a (not necessarily unique) g1 ∈ P 1 in such a way that We proceed by choosing g2 ∈ P 2 in such a way that (e γg 2 • e γg 1 ) Continuing in this way, after r steps we obtain that Being the composition of finitely many flows of polynomial coupled cell network vector fields, Φ is obviously analytic.
In applications, one is often interested in the bifurcations that occur in the dynamics of a map or differential equation under the variation of external parameters. In the case of coupled cell networks, we may for example assume that f ∈ C ∞ (V n × R p , V ) and let To formulate an appropriate normal form theorem for parameter families of coupled cell networks, we define for k ≥ −1 and l ≥ 0, P k,l := {f : V n × R p → V homogeneous polynomial of degree k + 1 in X and degree l in λ} .
We observe that [P k,l , P K,L ]Σ ⊂ P k+K,l+L , which leads to the following with f k,l ∈ P k,l .
Let 1 ≤ r1, r2 < ∞ and for every −1 ≤ k ≤ r1 and 0 ≤ l ≤ r2, let N k,l ⊂ P k,l be a subspace such that Then there exists a polynomial family Φ λ of analytic diffeomorphisms, defined for λ in an open neighborhood of 0 in R p and each sending an open neighborhood of 0 in V N to an open Proof: [Sketch] The procedure of normalization is similar as in the proof of Theorem 6.1.
Because [P k,l , P K,L ]Σ ⊂ P k+K,l+L , we see that once f k,l has been normalized to f k,l , it is not changed/affected anymore by any of the subsequent normalization transformations.
Of course, Theorem 5.1 implies that many other standard results from the theory of normal forms will have a counterpart in the context of semigroup coupled cell networks as well.
We will compute the normal forms of some network differential equations in Section 11. Example 6.3 Again, let N = 3 and let σ1, σ2, σ3 be defined as in Examples 2.2 and 2.3. If is a coupled cell network subject to {σ1, σ2}, then its normal form will in general be a network subject to {σ1, σ2, σ3}, i.e. x2, x1)) .
In this section, we relate some of the existing theory on symmetry and synchrony to the semigroup extension that we propose. More precisely, we show that the semigroup extension does not affect the symmetries or synchrony spaces of a network. This implies in particular that the symmetries and synchrony spaces of a network are also present its normal form. The semigroup extension is thus quite harmless and very natural.
To start, let us say that a permutation p : {1, . . . , N } → {1, . . . , N } of the cells is a network symmetry for Σ if it sends the inputs of a cell to the inputs of its image. That is, if The permutations with this property obviously form a group. More importantly, they are of dynamical interest because the corresponding representations λp : V N → V N , (x1, . . . , xN ) → (x p(1) , . . . , x p(N) ) conjugate every coupled cell network map γ f to itself: In turn, this means that when t → (x1(t), . . . , xN (t)) is a solution to the differential equationsẋ = γ f (x), then so is t → (x p(1) (t), . . . , x p(N) (t)). And similarly that when m → (x Then p is a network symmetry for Σ if and only if it is a network symmetry for the semigroup Σ ′ generated by Σ.

Proof:
Elements of the semigroup Σ ′ are of the form Thus, the collection of network symmetries of Σ is the same as the collection of network symmetries of Σ ′ . Lemma 7.1 implies in particular that the composition γ f • γg = γ f • Σ g and the Lie bracket [γ f , γg] = γ [f,g] Σ will exhibit the same network symmetries as γ f and γg.
Though not much more complicated, the situation is slightly more interesting for the synchronous solutions of a network. We recall that a synchrony space of a coupled cell network is an invariant subspace in which certain of the xi (with 1 ≤ i ≤ N ) are equal. First of all, the following result is classical, see [24], [49]. i) For all 1 ≤ j ≤ n and all 1 ≤ k1 ≤ r there exists a 1 ≤ k2 ≤ r so that σj(P k 1 ) ⊂ P k 2 .
2) For every f ∈ C ∞ (V n , V ) the subspace Syn P := {x ∈ V N | xi 1 = xi 2 when i1 and i2 are in the same element of P } is an invariant submanifold for the dynamics of γ f . Proof: The subspace Syn P is invariant under the flow of the differential equationẋ = γ f (x) if and only if the vector field γ f is tangent to Syn P . Similarly, Syn P is invariant under the map x (m+1) = γ f (x (m) ) if and only if γ f sends Syn P to itself. Both properties just mean that for all x ∈ Syn P it holds that f (x σ 1 (i 1 ) , . . . , x σn(i 1 ) ) = f (x σ 1 (i 2 ) , . . . , x σn(i 2 ) ) for all i1, i2 in the same element of P .
The latter statement holds for all f ∈ C ∞ (V n , V ) if and only if for all x ∈ Syn P , x σ j (i 1 ) = x σ j (i 2 ) for all 1 ≤ j ≤ n and all i1, i2 in the same element of P .
It is not hard to see that this is true precisely when all σj ∈ Σ map the elements of P into elements of P .
A partition P of {1, . . . , N } with property i) is sometimes called a balanced partition or balanced coloring and a subspace Syn P satisfying property ii) a (robust) synchrony space.
The following result says that the synchrony spaces of a network do not change if one extends the network architecture to a semigroup: Then Syn P is a (robust) synchrony space for Σ if and only if it is a (robust) synchrony space for the semigroup Σ ′ generated by Σ.

Proof:
Elements of the semigroup Σ ′ are of the form σj 1 • . . . • σj l for certain σj k ∈ Σ. This implies that the elements of Σ send the elements of P inside elements of P if and only if the elements of Σ ′ do. In other words: that the collection of balanced partitions of Σ and of Σ ′ are the same. The result now follows from Proposition 7.2. Lemma 7.3 implies in particular that the composition γ f • γg = γ f • Σ g and the Lie bracket [γ f , γg] = γ [f,g] Σ will exhibit the same synchrony spaces as γ f and γg.
We conclude this section with the following simple but important observation: . . , σn} be a collection of maps, not necessarily forming a semigroup, and γ f a coupled cell network vector field subject to Σ. Then a local normal form γ f of γ f has the same network symmetries and the same synchrony spaces as γ f . Proof: γ f is a coupled cell network with respect to the semigroup Σ ′ generated by Σ. Thus, the result follows from Lemma 7.1 and Lemma 7.3. Recall that a coupled cell network differential equation subject to {σ1, σ2} is of the forṁ The input symmetries of f obviously form a group.
We do not aim here to give a full answer to the question under which conditions the invariance (8.19) can be preserved in a normal form, because this question is very delicate. The first problem is that the presence of an input symmetry makes that the maps σ1, . . . , σn are not uniquely determined. A second complication is that an input symmetry of f has a nontrivial impact on its robust synchrony spaces.
The following result explains that network symmetries are preserved under taking compositions and Lie brackets: Let Σ be a semigroup and assume p • σj = σ q(j) • p for all 1 ≤ j ≤ n. Then Proof: With slight abuse of notation, we write This proves that (f •Σ g) • λq = (f • λq) •Σ (g • λq) on each im π p(i) and hence, because p is invertible, on each im πi. The proof for the Lie bracket is similar.
Finally, we conclude Corollary 8.4 Let Σ = {σ1, . . . , σn} be a collection of maps, not necessarily forming a semigroup, and γ f a coupled cell network vector field subject to Σ. Then the local normal form γ f of γ f can be chosen to have the same dynamical input symmetries as γ f .

Proof:
Let G denote the group of dynamical input symmetries of f = f0 + f1 + f2 + . . . and let us define the set of G-invariant functions It clearly holds that [P k G , P l G ]Σ ⊂ P k+l G . As a consequence, we can repeat the proof of Theorem 6.1 by replacing P k by P k G and choosing the normal form spaces N k This produces a normal form f ∈ C ∞ G (V n , V ).
These differential equations have a semigroup coupled cell network structure with N = 2 and n = 4, see Figure 3.

SN-decomposition
We recall from the previous sections that when f0 ∈ L(V n , V ), then ad Σ f 0 : P k → P k . The operators ad Σ f 0 | P k are called "homological operators" and they play an important role in normal form theory. This is first of all because the normal form spaces N k ⊂ P k of Theorem 6.1 must be chosen complementary to their images, and secondly because in computing a normal form one needs to "invert" them when solving the homological equations ad Σ f 0 (g k ) − h k ∈ N k , see the proof of Theorem 6.1.
For this reason, it is convenient to have at one's disposal the "SN-decompositions" (also called "Jordan-Chevalley decompositions") of the homological operators [7], [42], [47]. We recall that, since P k is finite-dimensional, the map ad Σ f 0 | P k admits a unique SN-decomposition in which the map (ad Σ f 0 | P k ) S is semisimple, the map (ad Σ f 0 | P k ) N is nilpotent and the two maps (ad Σ f 0 | P k ) S and (ad Σ f 0 | P k ) N commute. The aim of this somewhat technical section is to characterize this SN-decomposition in an as simple as possible way. We will do this in a number of steps, starting from the following technical result: Proposition 9.1 Assume that Σ is a semigroup. Then the linear map

Proof:
The definition (γ f 0 )i := f0 •πi implies that when f0 is linear, then γ f 0 = 0 precisely when f0 vanishes on im π1 + . . . + im πN . Thus, all we need to show is that Equivalently, we will show by contradiction that the map must be surjective. To this end, let us define for 1 ≤ j ≤ n the maps π j : (V N ) N → V by π j (x (1) , . . . , x (N) ) = (π1(x (1) ) + . . . + πN (x (N) ))j = x (1) In other words, π j is "π1 + . . . + πN " followed by the projection to the j-th factor of V n . In particular, if "π1 + . . . + πN " is not surjective, then there is a relation of the form This means that It is clear that this can only be true if σj = σ k for some k = j. This is a contradiction, because we assumed that the elements of Σ are distinct.
Next, we recall that when f0 ∈ L(V n , V ), then γ f 0 ∈ L(V N , V N ) and thus also the latter has an SN-decomposition We are going to relate the SN-decomposition of γ f 0 to that of ad Σ f 0 | P k . But first we show that both γ S f 0 and γ N f 0 are coupled cell network maps: Proof: We recall -see for instance [39], pp. 17 -that both the semisimple part γ S f 0 and the nilpotent part γ N f 0 of γ f 0 are polynomial functions of γ f 0 . More precisely, is a polynomial with coefficients a0, . . . , a d ∈ C.
By Lemma 4.3 this f S 0 is well-defined, while by Proposition 9.1 it is unique. Similarly, and Before we come to the desired characterization of the SN-decomposition of ad Σ f 0 | P k , we need to make one simple observation. It concerns the fact that two functions f, g ∈ C ∞ (V n , V ) generate the same network map (in the sense that γ f = γg) if and only if f − g ∈ ker γ = {h | γ h = 0} ⊂ C ∞ (V n , V ). Thus, ker γ consists of those functions h : V n → V that are irrelevant for the dynamics of coupled cell networks.
One can remark that when f, g ∈ C ∞ (V n , V ) and f ∈ ker γ, then γ [f,g] Σ = [γ f , γg] = 0 and hence also [f, g]Σ ∈ ker γ. This means ker γ ⊂ C ∞ (V n , V ) is a Lie algebra ideal. In particular it holds for every f ∈ C ∞ (V n , V ) that the adjoint map ad Σ f : C ∞ (V n , V ) → C ∞ (V n , V ) sends ker γ to ker γ and hence that ad Σ f descends to a well-defined map on C ∞ (V n , V )/ ker γ. We can now formulate our result: Theorem 9.3 For every k = 0, 1, 2, . . . the maps descend to the same map on P k / ker γ.

Proof:
We start by repeating that ad Σ f 0 maps ker γ into itself and hence descends to a well-defined map on C ∞ (V n , V )/ ker γ that moreover sends P k / ker γ into itself. More interestingly, since ad Σ f 0 | P k S and ad Σ f 0 | P k N are polynomial functions of ad Σ f 0 | P k , also these latter maps send ker γ into itself and thus descend to P k / ker γ.
For the actual proof of the theorem, we define for k ≥ 0 the vector spaces It is clear that γ : P k / ker γ → P k is an injective linear map. Moreover, the computation reveals that the maps ad Σ f 0 : P k / ker γ → P k / ker γ and adγ f 0 : P k → P k are conjugate by the map γ : P k / ker γ → P k . Similarly, ad Σ . Now we recall the well-known fact that the SN-decomposition of adγ f 0 | P k is Because γ is injective, we have thus proved that : P k / ker γ → P k / ker γ is the SN-decomposition of the quotient map. Because the SN-decomposition of the quotient is the quotient of the SN-decomposition, this proves the theorem.
Because the elements of ker γ are dynamically completely irrelevant, for all practical purposes we can think of Theorem 9.3 as saying that This is very convenient, because it means that one can determine the SN-decompositions of all operators ad Σ f 0 P k simultaneously by simply determining the splitting f0 = f S 0 + f N 0 .
Example 9.4 Even though by Proposition 9.1 the restriction γ| L(V n ,V ) is injective, the full map γ : C ∞ (V n , V ) → C ∞ (V N , V N ) may fail to be so. This situation occurs when ∪ N i=1 im πi = V n , because γ f = 0 already when f vanishes on every im πi. This is the reason for the somewhat difficult formulation of Theorem 9.3.

△
We conclude this section with the following dynamical implication of Theorem 9.3: Then it can be arranged that the normal form f = f0 has the special property that the truncated normal form coupled cell map/vector field γ f 0 +f 1 +...+f r commutes with the continuous family of maps t → e tγ f S 0 .
Proof: Recall that each one of the normal form spaces N k ⊂ P k of Theorem 6.1 is required to have the property that N k ⊕ im ad Σ f 0 P k = P k . It is not hard to see that whenever then this condition is fulfilled. Thus, let us choose N k so that it satisfies (9.25). The fact that ad f S 0 | P k S and ad f S 0 P k descend to the same map on P k / ker γ implies in particular that for such N k it holds that ad f S 0 (N k ) ⊂ ker γ.
Let now f = f0 + f 1 + f 2 + . . . be any normal form of f of order r with respect to the N k , meaning that f k ∈ N k for all 1 ≤ k ≤ r. Such a normal form exists by Theorem 6.1. Then it holds that ad Σ f S 0 (f k ) ∈ ker γ and in view of Theorem 4.2 we therefore have Hence, d dt t=0 (e

The continuous family
t → e tγ f S 0 of transformations of V N is called a normal form symmetry. This symmetry is sometimes used to characterize vector fields that are in normal form. It also plays an important role in finding periodic solutions near equilibria of the vector field γ f , using for example the method of Lyapunov-Schmidt reduction [12], [23], [28], [33], [37].

A fundamental semigroup network
As a byproduct of Theorem 4.1, and perhaps as a curiosity, we will show in this section that the dynamics of γ f on V N is conjugate to the dynamics of a certain network Γ f on V n . We will argue that Γ f acts as a "fundamental network" for γ f . We recall that if Σ = {σ1, . . . , σn} is a semigroup, then every σj ∈ Σ induces a map σj : {1, . . . , n} → {1, . . . , n} via the formula σ σ j (k) = σ k • σj .
We saw that σj 1 • σj 2 = σj 1 • σj 2 and hence the collection Σ := { σ1, . . . , σn} is closed under composition. One can now study coupled cell networks subject to Σ. They have the form The following theorem demonstrates that γ f and Γ f are dynamically related: Proof: For x ∈ V N we have that Theorem 10.1 implies that every πi sends integral curves of γ f to integral curves of Γ f and discrete-time orbits of γ f to discrete-time orbits of Γ f . In addition, the dynamics of γ f can be reconstructed from the dynamics of Γ f . More precisely, when X (i) (t) are integral curves of Γ f with X (i) (0) = πi(x(0)), then an integral curve x(t) of γ f can simply be obtained by integration of the equationṡ Similarly, if X The transition from γ f to Γ f is thus reminiscent of the symmetry reduction of an equivariant dynamical system: the dynamics of γ f descends to the dynamics of Γ f and the dynamics of γ f can be reconstructed from that of Γ f by means of integration. Nevertheless, n can of course be both smaller and larger than N . In the latter case, the dynamics of Γ f may be much richer than that of γ f and it is confusing to speak of reduction. In either case, Γ f captures all the dynamics of γ f . This means that the network map Γ f is given by In this example, the conjugacies from γ f to Γ f are The conjugacy π3 is bijective, which explains that Figures 1 and 4  • σ1 σ2 σ3 σ4 σ1 σ1 σ1 σ1 σ1 σ2 σ2 σ2 σ2 σ2 σ3 σ2 σ1 σ4 σ3 σ4 σ1 σ2 σ3 σ4 .
We stress that Lemma 10.4 holds due to the definition (Γ f )j := f • Aσ j and the fact that σj → Aσ j is a homomorphism. Lemma 10.4 implies for example that the symbolic computation of the normal form of Γ f is the same as the symbolic computation of the normal form of γ f . We propose to call Γ f the fundamental network of γ f . Two properties make this fundamental network fundamental: first of all, the network architecture of the fundamental network only depends on the multiplicative structure of the semigroup Σ and not on the explicit realization of Σ itself -in particular, it does not depend on N . This means that two semigroup networks have isomorphic fundamental networks if and only if their semigroups are isomorphic. The second fundamental property of the fundamental network is that it is equal to its own fundamental network, if the latter is defined. This follows from Proposition 10.5 below, in which we call a homomorphism of semigroups faithful if it is injective.
Proposition 10.5 Assume that the homomorphism σj → σj is faithful. Then σj = σj and therefore Aσ j = A σ j for all 1 ≤ j ≤ n .
Proposition 10.5 brings up the question when the homomorphism σj → σj is faithful, i.e. under which conditions the elements of Σ all have different left-multiplicative behavior. We give a partial answer to this question in Remark 10.9 below. The upshot of this remark is that one may essentially always assume the homomorphism to be faithful.
We finish this section with a few simple observations on synchrony and symmetry for Γ f . First of all, a direct consequence of Theorem 10.1 is that each im πi ⊂ V n is an invariant subspace for the dynamics of Γ f . Interestingly, another way to see this is by the following Proposition 10.6 Every im πi ⊂ V n is a robust synchrony space for the Γ f 's.

Proof:
Let us define a partition P of {1, . . . , n} by letting 1 ≤ j1, j2 ≤ n be in the same element of P if and only if σj 1 (i) = σj 2 (i). Then It remains to show that the partition P is balanced for Σ. This is easy though: when 1 ≤ j1, j2 ≤ n are in the same element of P , then it holds for all 1 ≤ k ≤ n that where the middle equality holds because σj 1 (i) = σj 2 (i). This proves that also σ k (j1) and σ k (j2) are in the same element of P and hence that the elements of Σ preserve P .
Recall that Aσ j • πi = π σ j (i) . This implies that Aσ j sends the Γ f -invariant subspace im πi to the Γ f -invariant subspace im π σ j (i) . But much more is true: the following result shows that Aσ j sends all orbits of Γ f to orbits of Γ f , even though Aσ j may not be invertible. Proof: The final result of this section shows that Γ f may even have more symmetry: the dynamical input symmetries of γ f are true symmetries of Γ f . Proof: Recall that under the conditions of the proposition, it holds that πi • λp = λq • π p(i) and that from this it followed that λp • γ f = γ f • λp. As a consequence, Because p is a permutation, this means that γ f • λq = λq • γ f on every im πi.

Remark 10.9
To explain when the homomorphism σj → σj is faithful, we can make the following definition: we say that 1 ≤ i ≤ N is a slave for the network Σ if there are no 1 ≤ j ≤ n and 1 ≤ k ≤ N so that σj(k) = i. Thus, a slave is a cell that does not act as input for any other cell, not even for itself. The point of this definition is the following: Proposition 10.10 If Σ has no slaves, then σj → σj is a faithful homomorphism.
If a network has slaves, then we can reduce it until no slaves remain. This works as follows: first of all, we remove any slave from the network. Because slaves do not affect the dynamics of other cells, this can be done without any effect on the network dynamics. Removing slaves may create new slaves: these are the cells that acted as inputs only for the original slaves. These new slaves can also be removed, etc. until a network free of slaves remains.
The remaining network may not be defined unambiguously, because some of the maps in Σ may coincide after the removal of the slaves. This happens when distinct maps in Σ differ only at slaves. Such maps can be identified though, while f must be redefined. In this way, we produce an unambiguous network that is free of slaves. For such a network γ f , the corresponding Γ f is a true fundamental network. △

Some examples and their normal forms
In this section we illustrate the methods and results of this paper by computing the normal forms of two coupled cell networks. Keeping things simple, we restrict our attention to synchrony breaking steady state bifurcations in one-parameter families of networks with one-dimensional cells.

A skew product network
In the first example, we consider the homogeneous skew product differential equationṡ Here x1, x2 ∈ R and f : R 2 × R → R. As usual, we will denote the right hand side of (11.26) by γ f (x1, x2; λ) and we will henceforth assume that γ f (0, 0; 0) = 0 and Dxγ f (0, 0; 0) is not invertible.
We depicted this network in Figure 6.  of synchronous steady states. A straightforward stability analysis reveals that one of these branches consists of equilibria that are linearly stable in the direction of the synchrony space, while the other branch consists of unstable points. We remark that the saddle node bifurcation is also generic in codimension one in the context of vector fields without any special structure.
These branches exchange stability when they cross. This means that the normal form displays a synchrony breaking transcritical bifurcation. Such a bifurcation is not generic in codimension one in the context of vector fields without any special structure, and is hence forced by the network structure. More precisely, it follows from the presence of the invariant synchrony space.
Then it holds that mat Dxγ f (0, 0, 0; 0) = This shows that a steady state bifurcation takes place when either a1 +a2 +a3 = 0 or a1 = 0. Moreover, the linearization matrix is not semisimple. In fact, its SN-decomposition reads   a1 + a2 + a3 As a consequence, we should accordingly decompose f0,0 as Recalling that for this network the expression for the symbolic bracket is given in Example 5.3, it again requires a little computation to find that and similarly that Once more, we now consider the two codimension one cases: 1. If a1 + a2 + a3 = 0 and a1 = 0, then the kernel of ad Σ f 0,0 is spanned by terms (X1 − X3)X γ 3 , (X2 − X3)X γ 3 and X γ 3 with γ ≥ 0 . One checks that ad f N  Again, one of these branches is stable and the other one is unstable in the direction of the maximal synchrony space.

2.
When a1 = 0, a2 = 0 and a1 + a2 + a3 = 0, then ker ad f S This means that the general normal form of f is 2 and A(0) = a2, B(0) = a1 + a2 + a3, C(0) = 0. This gives the equations of motioṅ Under the generic assumption that C ′ (0), D(0) = 0, we now find three branches of steady states: This means that our normal form equations undergo a very particular synchrony breaking steady state bifurcation that comprises a fully synchronous trivial branch, a partially synchronous transcritical branch and fully nonsynchronous saddle-node branches. The solutions on these branches exchange stability in a specific way, as for example depicted in Figure 7.

Colored coupled cell networks
In this final section, we describe how our results on homogeneous coupled cell networks generalize to certain non-homogeneous coupled cell networks. So let us imagine a coupled cell network with cells of different types. We will refer to the different types of cells as colors.
More precisely, let us assume that there are 1 ≤ C < ∞ colors and that for every color 1 ≤ c ≤ C there are precisely Nc cells of color c. We label the cells of color c by 1 ≤ i ≤ Nc and assume that the state of the i-th cell of color c is described by x (c) i ∈ Vc, where Vc is a linear space that depends on c.
We furthermore assume that the discrete-or continuous-time evolution of x (c) i is determined by precisely n (1,c) cells of color 1, by n (2,c) cells of color 2, etc. This assumption is made precise in Definition 12.1 below that, although lengthy, is a straightforward generalization of Definition 2.1. Then we define γ f : for all 1 ≤ c ≤ C and 1 ≤ i ≤ Nc.
Again, one quickly checks that Σ is a semigroupoid. See Figure 9. △ Under the condition that Σ is a semigroupoid, all results of this paper on Lie algebras and normal forms can be generalized to colored coupled cell networks. As an illustration, we state a few facts here without proof. Theorem 12.7 If Σ is a semigroupoid, then n (C,c) . Theorem 12.8 If Σ is a semigroupoid, then .
In turn, Theorem 12.7 can be used to prove normal form theorems for colored coupled cell networks. That is, the theorems of Sections 6 and 9 remain true with the word "semigroup" replaced by "semigroupoid". We conclude with two results that say that the network symmetries and the robust synchrony spaces of a network remain unchanged by the semigroupoid extension. This means that λp sends orbits of γ f to orbits of γ f .
Then the collection of network symmetries of Σ is the same as the collection of network symmetries of the semigroupoid Σ ′ generated by Σ. The collection of robust synchrony spaces of Σ is the same as the collection of robust synchrony spaces of the semigroupoid Σ ′ generated by Σ.