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Quiver representations and dimension reduction in dynamical systems
Nijholt, Eddie; Rink, Bob W.; Schwenker, Sören
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SIAM Journal on Applied Dynamical Systems 2020
DOI (link to publisher) 10.1137/20M1345670 document version
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Nijholt, E., Rink, B. W., & Schwenker, S. (2020). Quiver representations and dimension reduction in dynamical systems. SIAM Journal on Applied Dynamical Systems, 19(4), 2428-2468. https://doi.org/10.1137/20M1345670
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SIAM J. APPLIEDDYNAMICALSYSTEMS ©2020 Society for Industrial and Applied Mathematics Vol. 19, No. 4, pp. 2428--2468
Quiver Representations and Dimension Reduction in Dynamical Systems\ast
Eddie Nijholt\dagger , Bob W. Rink\ddagger , and S\"oren Schwenker\S
Abstract. Dynamical systems often admit geometric properties that must be taken into account when studying their behavior. We show that many such properties can be encoded by means of quiver represen-tations. These properties include classical symmetry, hidden symmetry, and feedforward structure, as well as subnetwork and quotient relations in network dynamical systems. A quiver equivariant dynamical system consists of a collection of dynamical systems with maps between them that send solutions to solutions. We prove that such quiver structures are preserved under Lyapunov--Schmidt reduction, center manifold reduction, and normal form reduction.
Key words. coupled networks, symmetry, bifurcation theory, quiver representations, normal forms AMS subject classification. 37C81
DOI. 10.1137/20M1345670
1. Introduction. In this paper we show that various structural properties of dynamical systems (ODEs and iterated maps) can be encoded using the language of quiver representa-tions. These structural properties include classical symmetry, but also feedforward structure, subnetwork and quotient relations in network dynamical systems, and so-called hidden sym-metry, including interior symmetry and quotient symmetry. This paper aims to provide a unifying framework for studying dynamical systems with quiver symmetry.
A quiver representation consists of a collection of vector spaces with linear maps between them. A simple example is the representation of a group (where there is only one vector space). We shall speak of a dynamical system with quiver symmetry when a dynamical system is defined on each of the vector spaces of the representation, and so that all the linear maps in the representation send the orbits of one dynamical system to orbits of another one. We will argue that quiver symmetry is quite prevalent in dynamical systems that have the structure of an interacting network. Essentially, this insight can already be found in the work of Golubitsky, Stewart, et al. [7], [10], [12], [24], [25], who realized that every trajectory of a so-called quotient of a network gives rise to a trajectory in the original network system. DeVille
\ast
Received by the editors June 15, 2020; accepted for publication (in revised form) by M. Golubitsky September 17, 2020; published electronically November 2, 2020.
https://doi.org/10.1137/20M1345670
Funding:This work was partially supported by the Dutch Research Council (NWO) via the first author's research program ``Designing Network Dynamical Systems through Algebra."" The work of the second author was supported by the Sydney Mathematical Research Institute.
\dagger
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA (eddie.nijholt@ gmail.com).
\ddagger
Department of Mathematics, Vrije Universiteit Amsterdam, Amsterdam, 1081 HV, The Netherlands (b.w.rink@vu.nl).
\S
Department of Mathematics, Universi\"at Hamburg, Hamburg, 22111, Germany (soeren.schwenker@uni-hamburg.de).
and Lerman [1] later generalized this result and formulated it in the language that we shall use in this paper. Inspired by these ideas, we shall define two distinct quiver representations for each network dynamical system---the quiver of subnetworks and the quiver of quotient networks---and we will investigate how these quivers impact the dynamics of a network.
The advantage of interpreting a property of a dynamical system as a quiver symmetry, lies in the fact that quiver symmetry is an intrinsic property of a dynamical system. It is, for example, preserved under composition of maps and Lie brackets of vector fields. Unlike, for example, network structure, which is generally destroyed when a coordinate transformation is applied, quiver symmetry is thus defined in a coordinate-invariant manner. This motivates us to start developing a theory for dynamical systems with quiver symmetry.
As a consequence of its intrinsic definition, quiver symmetry can be incorporated quite easily in many of the tools that are available for the analysis of dynamical systems. In this paper, we focus on the impact of quiver symmetry on local dimension reduction techniques. It is well-known that classical symmetry (of compact group actions) is preserved by various of these reduction techniques---see [2], [3], [4], [6], [9], [11] and references therein for an overview of results. In this paper, we will generalize these results by proving that quiver symmetry can be preserved in Lyapunov--Schmidt reduction, center manifold reduction, and normal form reduction. More precisely, we show that the dynamical systems that result after applying these reduction techniques inherit the quiver symmetry of the original dynamical system. Partial results in this direction were obtained by the authors in earlier papers [16], [17], [18], [19], [20], [21], [23]. This paper provides a unifying context for these earlier results. Because the results in this paper apply to any (finite) quiver, we shall not yet try to use any of the more involved results from the theory of quiver representations, such as Gabriel's classification theorem [5].
We will start this paper with a simple illustrative example of a dynamical system with quiver symmetry in section 2. We then define quiver representations and quiver equivariant maps in section3. In sections4and5we discuss two natural examples of quiver representations that one encounters in the study of network dynamical sytems. In section 6 we gather some properties of endomorphisms of quiver representations. This prepares us to prove the results on Lyapunov--Schmidt reduction, center manifold reduction, and normal forms in sections 7, 8, and 9. We finish the paper with an example in section 10.
2. A simple feedforward system. Before describing our results in more generality, let us start with a simple example. To this end, let E1 and E2 be finite dimensional real vector
spaces and consider a differential equation of the feedforward form \Biggl\{ dx
dt = f (x) , dy
dt = g(x, y) ,
(2.1)
where x \in E1and y \in E2. We will show that such a feedforward system can in fact be thought
of as a system with quiver symmetry. To explain this, let us (artificially) replace (2.1) by two separate systems of differential equations:
\Biggl\{ dx
dt = f (x) , dy
dt = g(x, y)
for (x, y) \in E1\times E2,
(2.2)
\biggl\{ dX
dt = f (X) for X \in E1. (2.3)
This unconventional step allows us to formulate the following simple lemma. It states that a feedforward system can be thought of as a system with symmetry.
Lemma 2.1. A pair of (systems of ) differential equations \Biggl\{ dx dt = F (x, y) , dy dt = G(x, y) , (2.4) \biggl\{ dX dt = H(X) (2.5)
is of the feedforward form (2.2), (2.3) if and only if the map R : E1\times E2\rightarrow E1 defined by R(x, y) := x
sends every solution of (2.4) to a solution of (2.5).
Proof. First, assume that (2.4) and (2.5) are actually of the form (2.2), (2.3), respectively. This means that
F (x, y) = f (x), G(x, y) = g(x, y), H(X) = f (X) .
Assume now that (x(t), y(t)) solves (2.4). Then dxdt(t) = F (x(t), y(t)) and hence X(t) := R(x(t), y(t)) = x(t) satisfies
dX dt (t) =
dx
dt(t) = F (x(t), y(t)) = f (x(t)) = f (X(t)) = H(X(t)) . So R sends solutions of (2.4) to solutions of (2.5).
For the other direction, assume that for every solution (x(t), y(t)) of (2.4) the curve X(t) := R(x(t), y(t)) = x(t) is a solution of (2.5). Let (x, y) \in E1\times E2be arbitrary, and let (x(t), y(t))
be the solution of (2.4) with (x(0), y(0)) = (x, y). Define X := R(x, y) = x and X(t) := R(x(t), y(t)) = x(t). Then X(0) = X and dXdt(t) = H(X(t)). It follows that F (x, y) =
dx dt(0) =
dX
dt (0) = H(X(0)) = H(X) = H(x). So F (x, y) = H(x) for all x, y. If we now define
f (X) := H(X) and g(x, y) := G(x, y), then obviously F (x, y) = H(x) = f (x). In other words, (2.2) coincides with (2.4) and (2.3) coincides with (2.5).
Lemma2.1translates the property that an ODE has feedforward structure into a somewhat unconventional symmetry property. We will see many more examples of this phenomenon later. It is important to note that the symmetry in Lemma2.1is a noninvertible map between two different vector spaces. We are thus not in the classical setting where the symmetries form a group. Instead, they form a (rather simple) quiver.
The statement of the following lemma is not new (see [7]), but we provide a new proof that is based on the observation in Lemma 2.1. This proof nicely illustrates how quiver symmetry can be taken into account when we analyze a dynamical system. Moreover, the proof below easily generalizes to dynamical systems with more complicated quiver symmetries; see Theorem 8.2below.
Lemma 2.2. Let (x0, y0) be an equilibrium point of the feedforward system
\Biggl\{ dx
dt = f (x) , dy
dt = g(x, y)
for (x, y) \in E1\times E2.
(2.6)
Denote by L(x0,y0)the Jacobian of (2.6) at (x0, y0) and by E1\times E2= E
c\oplus Ehthe decomposition
into its center and hyperbolic subspaces. We denote by
\pi c: E1\times E2 = Ec\oplus Eh \rightarrow Ec , (x, y) \mapsto \rightarrow (xc, yc)
the projection onto Ecalong Eh. Assume that (2.6) admits a global center manifold at (x0, y0).
Then \pi c conjugates the dynamics on this center manifold to a dynamical system on Ec of the form \Biggl\{ dxc dt = f c(xc) , dyc dt = g c(xc, yc) . (2.7)
Proof. Let us define F : E1\times E2\rightarrow E1\times E2 by F (x, y) := (f (x), g(x, y)). Then (2.6) can
be written as dtd(x, y) = F (x, y). Recall from Lemma2.1that the feedforward structure of F implies that R : E1 \times E2 \rightarrow E1, (x, y) \mapsto \rightarrow x sends solutions of this ODE to solutions of the
ODE
dX
dt = f (X) . (2.8)
In other words, we have that
R \circ F = f \circ R .
This clearly implies that X0 := R(x0, y0) = x0 is an equilibrium point of (2.8), and if we write
LX0 = Df (X0) for the Jacobian of (2.8) at X0, then R \circ L(x0,y0)= LX0 \circ R . (2.9)
This follows from differentiating R(F (x, y)) = f (R(x, y)) at (x, y) = (x0, y0). Let us
decom-pose E1 = Ec
\prime
\oplus Eh\prime into the center and hyperbolic subspaces of L
X0. Then it follows from (2.9) that R maps any generalized eigenspace of L(x0,y0) into the generalized eigenspace of LX0 with the same eigenvalue. It follows that R(E
c) \subset Ec\prime and R(Eh) \subset Eh\prime . Denoting by
\Pi c: E1 = Ec
\prime \oplus Eh\prime
\rightarrow Ec\prime
the projection onto Ec\prime along Eh\prime , we conclude that \Pi c\circ R = R \circ \pi c.
The next step is to prove that R sends the global center manifold of (2.6) at (x0, y0) to the
global center manifold of (2.8) at X0. For this we recall that a solution (x(t), y(t)) lies in the
global center manifold of (2.6) if and only if sup
t\in \BbbR
| | \pi h(x(t), y(t))| | < \infty ,
where we write \pi h = 1 - \pi c. We similarly write \Pi h = 1 - \Pi c. If we define X(t) := R(x(t), y(t)) = x(t) for such a solution, then clearly \Pi h(X(t)) = \Pi h(R(x(t), y(t)) = R(\pi h(x(t),
y(t))), and so
sup
t\in \BbbR
| | \Pi h(X(t))| | \leq | | R| | \cdot sup
t\in \BbbR
| | \pi h(x(t), y(t))| | < \infty ,
where | | R| | denotes the operator norm of R (which equals 1 here). So X(t) lies in the global center manifold of (2.8). This proves that R maps the center manifold of (2.6) into the center manifold of (2.8).
Next, recall that the center manifolds of (2.6) and (2.8) are the graphs of certain (finitely many times continuously differentiable) functions \phi : Ec \rightarrow Eh and \psi : Ec\prime \rightarrow Eh\prime ,
respec-tively. In other words, for every (x, y) in the center manifold of (2.6) and X in the center manifold of (2.8), we have (x, y) = (xc, yc) \underbrace{} \underbrace{} \in Ec + \phi (xc, yc) \underbrace{} \underbrace{} \in Eh , (2.10) X = Xc \underbrace{} \underbrace{} \in Ec\prime + \psi (Xc) \underbrace{} \underbrace{} \in Eh\prime . (2.11)
Pick an (x, y) in the center manifold of (2.6). Applying R to (2.10) yields that x = xc+ R(\phi (xc, yc)) .
Note that this x lies in the center manifold of (2.8) by the result above. We also have that xc = R(xc, yc) \in Ec\prime because (xc, yc) \in Ec and R(Ec) \subset Ec\prime . Similarly, R(\phi (xc, yc)) \in Eh\prime because \phi (xc, yc) \in Eh and R(Eh) \subset Eh\prime . But this means that xc is the center part of x and R(\phi (xc, xh)) is its hyperbolic part. So (2.11) gives that R(\phi (xc, yc)) must be equal to \psi (xc) = \psi (R(xc, yc)). This proves that
R \circ \phi = \psi \circ R .
Next, let (x(t), y(t)) be an integral curve of F lying inside the center manifold of (2.6), and let us once again write
(x(t), y(t)) = (xc(t), yc(t)) + \phi (xc(t), yc(t)) . Because dtd(x(t), y(t)) = F (x(t), y(t)), it then follows that
d dt(x
c(t), yc(t)) = (\pi c\circ F )((xc(t), yc(t)) + \phi (xc(t), yc(t))) .
This shows that the restriction of \pi c to the center manifold sends integral curves of F to integral curves of the vector field Fc: Ec\rightarrow Ecdefined by
Fc(xc, yc) := (\pi c\circ F )((xc, yc) + \phi (xc, yc)) .
Similarly, the restriction of \Pi c to the center manifold of (2.8) sends integral curves of f to integral curves of fc: Ec\prime \rightarrow Ec\prime defined by
fc(Xc) := (\Pi c\circ f )(Xc+ \psi (Xc)) . Now we simply notice that
R(Fc(xc, yc)) = (R \circ \pi c\circ F )((xc, yc) + \phi (xc, yc)) = (\Pi c\circ R \circ F )((xc, yc) + \phi (xc, yc)) = (\Pi c\circ f \circ R)((xc, yc) + \phi (xc, yc)) = (\Pi c\circ f )(R(xc, yc) + R(\phi (xc, yc)))
= (\Pi c\circ f )(R(xc, yc) + \psi (R(xc, yc)))
= fc(R(xc, yc)) , which proves that
R \circ Fc= fc\circ R .
In this last formula, R in fact denotes the restriction R : Ec\rightarrow Ec\prime given by R(xc, yc) = xc.
Lemma2.1 thus guarantees that Fcis of the feedforward form Fc(xc, yc) = (fc(xc), gc(xc, yc)) for some function gc: Ec\rightarrow Ec. This finishes the proof.
3. Quiver equivariant dynamical systems. The pair of ODEs (2.2), (2.3) is a simple example of a quiver equivariant dynamical system. We shall now give the general definition. In this paper, we only consider quivers with finitely many vertices and arrows, because this simplifies our proofs.
Definition 3.1.
i) A quiver is a directed (multi)graph
Q = \{ A \rightrightarrows st V \}
consisting of a finite set of arrows A, a finite set of vertices V , a source map s : A \rightarrow V , and a target map t : A \rightarrow V .
ii) A representation (E, R) of a quiver Q consists of a set E of finite dimensional vector spaces Ev (one for each vertex v \in V ), and a set R of linear maps
Ra: Es(a)\rightarrow Et(a) (one for each arrow a \in A).
iii) A Q-equivariant map F of a representation (E, R) of a quiver Q consists of a collection of maps Fv : Ev \rightarrow Ev (one for each vertex v \in V ) so that
Ft(a)\circ Ra= Ra\circ Fs(a) for every arrow a \in A .
We shall write F \in C\infty (E, R) if Fv \in C\infty (Ev) for every v \in V . We shall
some-times refer to a Q-equivariant map as a Q-equivariant vector field or a Q-equivariant dynamical system.
The following simple proposition expresses that quiver-equivariance is an intrinsic prop-erty. Proposition 9.4formulates the corresponding result for the Lie bracket.
Proposition 3.2. Let (E, R) be a representation of a quiver Q = \{ A \rightrightarrows st V \} and let F, G \in
C\infty (E, R). Define the composition F \circ G to consist of the maps (Fv\circ Gv) : Ev \rightarrow Ev (for
v \in V ). Then F \circ G \in C\infty (E, R).
Proof. Smoothness of (Fv\circ Gv) is obvious. Now let a \in A be an arrow. Then
Ra\circ (Fs(a)\circ Gs(a)) = Ft(a)\circ Ra\circ Gs(a)= (Ft(a)\circ Gt(a)) \circ Ra.
The next example shows that the feedforward system of section 2 constitutes a quiver equivariant dynamical system.
Example 3.3. Consider a quiver Q consisting of two vertices V = \{ v1, v2\} and three arrows
A = \{ a1, a2, a3\} , where s(a1) = t(a1) = v1 and s(a2) = v1 and t(a2) = v2 and s(a3) = t(a3) =
v2. Define
Ev1 = E1\times E2 and Ev2 = E1 with E1 and E2 vector spaces, and
Ra1(x, y) = (x, y) , Ra2(x, y) = x , Ra3(X) = X . Then the pair of maps
Fv1 = (F, G) : Ev1 = E1\times E2\rightarrow Ev1 = E1\times E2, Fv2 = H : Ev2 = E1 \rightarrow Ev2 = E1
is Q-equivariant if and only if Ra2 \circ Fv1 = Fv2\circ Ra2, that is, if H(x) = Fv2(x) = Fv2(Ra2(x, y))
= Ra2(Fv1(x, y)) = Ra2(F (x, y), G(x, y)) = F (x, y) . So Q-equivariance just means that Fv1 is of feedforward form.
Example 3.4. Let E be a representation of a finite group G. This means that E is a vector space and that for every g \in G there is a (necessarily invertible) linear map Rg : E \rightarrow E, such
that Re= IdE and Rg1\circ Rg2 = Rg1g2.
Such a group representation can be thought of as a representation of a quiver with one vertex, say, V = \{ v\} , and exactly one arrow a = a(g) (to and from v) for each g \in G. This is done by defining Ev := E and Ra(g):= Rg. The quiver equivariant maps are then simply the
maps F : E \rightarrow E with
F \circ Rg = Rg\circ F for all g \in G .
So the quiver equivariant maps coincide with the usual G-equivariant maps.
Example 3.5. As a straightforward generalization of the previous example, one may study linear maps that are not invertible. Consider, for example, the map
R : x \mapsto \rightarrow 0 from \BbbR to \BbbR .
This map defines a representation of a quiver with just one vertex v \in V and one arrow a \in A, where Ev := \BbbR and Ra:= R.
Note that an ODE dxdt = F (x) satisfies F \circ R = R \circ F if and only if F (0) = 0. So having this quiver symmetry is equivalent to having a steady state at the origin. Interestingly, this is the setting in which the transcritical bifurcation
dx
dt = \lambda x \pm x
2
is the typical one-parameter bifurcation. Curiously, this shows that the transcritical bifurca-tion is a generic quiver equivariant bifurcabifurca-tion.
Example 3.6. In [16] it turned out natural to study dynamical systems that are equivariant under the action of a finite monoid \Sigma . A monoid is a set \Sigma with an associative multiplication (\sigma 1, \sigma 2) \mapsto \rightarrow \sigma 1\sigma 2and a multiplicative unit \sigma 0. A representation of \Sigma consists of (not necessarily
invertible) linear maps R\sigma : E \rightarrow E on a vector space E, so that R\sigma 0 = IdE and R\sigma 1 \circ R\sigma 2 = R\sigma 1\sigma 2.
This setup arises, for example, when studying the network in Figure1. The figure displays a network with five nodes and a map F = F (x1, x2, x3, x4, x5) that is ``compatible"" with the
structure of this network.
It turns out that an F of this form always commutes with the maps R\sigma 0(x1, x2, x3, x4, x5) = (x1, x2, x3, x4, x5) , R\sigma 1(x1, x2, x3, x4, x5) = (x2, x4, x3, x4, x5) , R\sigma 2(x1, x2, x3, x4, x5) = (x3, x5, x3, x4, x5) , R\sigma 3(x1, x2, x3, x4, x5) = (x4, x4, x3, x4, x5) , R\sigma 4(x1, x2, x3, x4, x5) = (x5, x4, x3, x4, x5) . 1 2 3 4 5 F \left( x1 x2 x3 x4 x5 \right) = \left( f (x1,x2,x3) f (x2,x4,x3) f (x3,x5,x3) f (x4,x4,x3) f (x5,x4,x3) \right)
Figure 1. A network map with a monoid of five symmetries.
These maps together form a representation of a monoid \Sigma with five elements. In [16] this representation was used to classify the bifurcations that occur in the dynamics of the ODE
dx
dt = F (x). We will not discuss these results in any detail here. Just like for groups, one may
think of a representation of a monoid as a special case of a representation of a quiver. Remark 1. The notion of interior network symmetry was defined in [8]. We will not discuss interior symmetry in any detail here, but we would like to point out that interior symmetry is equivalent to a special type of quiver symmetry, for a quiver with two vertices and a possibly quite large number of arrows. This fact was proved in section 9 of [16].
In the coming sections we provide more examples of dynamical systems with quiver sym-metry. We start by generalizing Example 3.3to include more general network systems.
4. The quiver of subnetworks. In this section and the next we consider dynamical sys-tems with the structure of an interacting network. We apologize for the somewhat heavy notation in this section, which we found impossible to avoid.
We start by letting N = \{ E \rightrightarrows st N \} be a directed graph consisting of a finite number of
nodes n \in N and directed edges e \in E (the letter N stands for network). This N should not be thought of as a quiver (we shall use N to define a quiver SubQ(N) later) but as the network structure of an iterated map or ODE. More precisely, we assume that for each n \in N we are given a vector space En (the so-called internal phase space of this node) and a map
Fn:
\bigoplus
e\in E : t(e)=n
Es(e)\rightarrow En.
So Fn depends only on those xm for which there is an edge e from m to n. Together the Fn
define a network map F\bfN :\bigoplus
m\in NEm\rightarrow \bigoplus m\in NEm given by
Fn\bfN \Biggl( \bigoplus m\in N xm \Biggr) = Fn \left( \bigoplus e\in E : t(e)=n xs(e) \right) .
One could say that this F\bfN is ``compatible"" with the network N. We may use F\bfN to define a ``network dynamical system"" on the ``total phase space""\bigoplus
m\in NEm, for example, the iteration
x(n+1)= F\bfN (x(n)) or the flow of the ODE dxdt = F\bfN (x).
Example 4.1. The network N in Figure2consist of two nodes (labeled 1 and 2) and three arrows. The network maps compatible with this network are the maps of the form
F\bfN (x1, x2) = (F1(x1), F2(x1, x2)) .
These are precisely the feedforward maps of Example 3.3.
1 2
Figure 2. Feedforward network with two nodes.
1
3 4
2 5
Figure 3. A feedforward type network with five nodes.
Example 4.2. Let N be the network consisting of 5 nodes (labeled 1, . . . , 5) and 12 arrows as defined in Figure3.
Then any network map F\bfN takes the form
F\bfN \left( x1 x2 x3 x4 x5 \right) = \left( F1(x1) F2(x1, x2) F3(x1, x2, x3) F4(x1, x3, x4) F5(x3, x4, x5) \right) . (4.1)
Thus F\bfN has a rather particular feedforward structure. Note also that when F\bfN and G\bfN are
two such network maps, then their composition will have the form
\bigl( F\bfN \circ G\bfN \bigr) \left( x1 x2 x3 x4 x5 \right) = \left( F1(G1(x1)) F2(G1(x1), G2(x1, x2)) F3(G1(x1), G2(x1, x2), G3(x1, x2, x3)) F4(G1(x1), G3(x1, x2, x3), G4(x1, x3, x4)) F5(G3(x1, x2, x3), G4(x1, x3, x4), G5(x3, x4, x5)) \right) .
This shows that (F\bfN \circ G\bfN )
4(x) depends explicitly on x2, while F4(x) and G4(x) do not.
Similarly, (F\bfN \circ G\bfN )
5(x) depends explicitly on x1, x2, while F5(x) and G5(x) do not. So we
see that the network structure of F\bfN and G\bfN is destroyed when we compose them. On the other hand, we also observe that a large part of the network structure of F\bfN and G\bfN remains intact in F\bfN \circ G\bfN .
In the remainder of this section we will show that network maps admit a specific quiver symmetry. This will clarify which characteristics of the network structure will survive if we, for example, compose network maps. We start with the definition of a subnetwork.
Definition 4.3. Let N = \{ E \rightrightarrows st N \} be a network and let N\prime \subseteq N . Assume that for every
e \in E with t(e) \in N\prime it holds that s(e) \in N\prime . Define
E\prime := \{ e \in E : s(e), t(e) \in N\prime \} .
Then N\prime = \{ E\prime \rightrightarrows st N\prime \} is called a subnetwork of N. We shall write N\prime \sqsubseteq N.
Remark 2. The relation \sqsubseteq defines a partial order on the set of subnetworks of N. Indeed, N\prime \sqsubseteq N\prime for all N\prime \sqsubseteq N (reflexivity), N\prime \sqsubseteq N\prime \prime and N\prime \prime \sqsubseteq N\prime imply N\prime = N\prime \prime (antisymmetry), and N\prime \prime \prime \sqsubseteq N\prime \prime and N\prime \prime \sqsubseteq N\prime together imply that N\prime \prime \prime \sqsubseteq N\prime (transitivity).
We shall use the subnetworks of N to define a quiver as follows.
Definition 4.4. Let N be a network. The quiver SubQ(N) = \{ A \rightrightarrows st V \} of subnetworks of
N has as its vertices the nonempty subnetworks of N, i.e., V = \{ N\prime | \emptyset \not = N\prime \sqsubseteq N \} .
There is exactly one arrow a \in A with s(a) = N\prime and t(a) = N\prime \prime if N\prime \prime \sqsubseteq N\prime .
A representation of SubQ(N) can be constructed in the following straightforward manner. Recall that for every n \in N there is a vector space En. We now set
E\bfN \prime := \bigoplus
m\in N\prime Em
and we define, for the arrow a \in A from N\prime to N\prime \prime (so assuming that N\prime \prime \subseteq N\prime ),
Ra: E\bfN \prime \rightarrow E\bfN \prime \prime by Ra \Biggl( \bigoplus m\in N\prime xm \Biggr) := \bigoplus
m\in N\prime \prime xm.
(4.2)
So Ra ``forgets"" the states xm with m \in N\prime \setminus N\prime \prime . Before we continue to explain why these
definitions are useful, let us briefly return to our two examples.
Example 4.5. Let N be the network of Example 4.1 and Figure2. It has two nonempty subnetworks, which we call N1 and N2. Figure 4 depicts the quiver SubQ(N). The arrows
in the quiver that express the subnetwork relations N1 \sqsubseteq N1, N1 \sqsubseteq N2, and N2 \sqsubseteq N2 are
drawn as snaking arrows.
It should be clear that the linear maps defining the representation are given by Ra1(x1, x2) = (x1, x2), Ra2(x1, x2) = x1, and Ra3(x1) = x1.
Example 4.6. Let N be the network of Example 4.2 as depicted in Figure 3. It has five nonempty subnetworks, which we depict in Figure 5. The figure also depicts some (but not all) of the arrows in the subnetwork quiver.
1 2 1
\bfN
2\bfN
1a2
a1 a3
Figure 4. Subnetwork quiver for the feedforward network in Figure2.
1 3 4 2 5 1 1 3 4 2 1 2 1 3 2 N1 N4 N2 N5 N3 a2 a5 a4 a7 a3 a8 a1 a 6
Figure 5. The subnetwork quiver for the network in Figure3. Loops from \bfN ito \bfN iare not drawn. Neither
are the compositions of arrows that are already depicted.
The maps Rai are given by
Ra1(x1, x2, x3, x4, x5) = (x1, x2, x3, x4), Ra2(x1, x2, x3, x4, x5) = (x1, x2, x3), Ra3(x1, x2, x3, x4, x5) = (x1, x2), Ra4(x1, x2, x3, x4) = (x1, x2, x3), Ra5(x1, x2, x3) = (x1, x2), Ra6(x1, x2, x3, x4) = x1, Ra7(x1, x2, x3) = x1, Ra8(x1, x2) = x1.
The following result reveals the dynamical meaning of the quiver SubQ(N).
Lemma 4.7. Let N = \{ E \rightrightarrows st N \} be a network and F\bfN : E\bfN \rightarrow E\bfN a network map, i.e.,
it is of the form Fn\bfN \Biggl( \bigoplus m\in N xm \Biggr) = Fn \left( \bigoplus e\in E : t(e)=n xs(e) \right)
for all n \in N .
For any subnetwork N\prime \sqsubseteq N define F\bfN \prime : E
\bfN \prime \rightarrow E\bfN \prime by
Fn\bfN \prime \Biggl( \bigoplus m\in \bfN \prime xm \Biggr) := Fn \left( \bigoplus e\in E : t(e)=n xs(e) \right)
for all n \in N\prime .
Then these F\bfN \prime together define a SubQ(N)-equivariant map.
Proof. First of all, note that the maps F\bfN \prime are well-defined because we assumed that N\prime \sqsubseteq N, so that s(e) \in N\prime whenever t(e) \in N\prime .
To prove SubQ(N)-equivariance, assume that a \in A is the arrow from N\prime to N\prime \prime . It then holds that N\prime \prime \sqsubseteq N\prime \sqsubseteq N, so
Fn\bfN \prime \Biggl( \bigoplus m\in \bfN \prime xm \Biggr) = Fn \left( \bigoplus e\in E : t(e)=n xs(e) \right) = Fn\bfN \prime \prime \Biggl( \bigoplus
m\in \bfN \prime \prime xm
\Biggr)
for all n \in N\prime \prime .
But this implies that Ra \Biggl( F\bfN \prime \Biggl( \bigoplus m\in \bfN \prime xm \Biggr) \Biggr) = \bigoplus
n\in N\prime \prime Fn\bfN \prime \Biggl( \bigoplus m\in \bfN \prime xm \Biggr) = \bigoplus
n\in N\prime \prime Fn\bfN \prime \prime
\Biggl( \bigoplus
m\in \bfN \prime \prime xm \Biggr) = F\bfN \prime \prime \Biggl( Ra \Biggl( \bigoplus m\in \bfN \prime xm \Biggr) \Biggr) ,
which proves the lemma.
The next result is the converse of Lemma 4.7 and the natural generalization of Lemma 2.1. The proof is similar to that of Lemma 4.7.
Lemma 4.8. A collection of maps F\bfN \prime : E\bfN \prime \rightarrow E\bfN \prime (one for each N\prime \sqsubseteq N) is SubQ(N)-equivariant if and only if for all N\prime \prime \sqsubseteq N\prime \sqsubseteq N it holds that
Fn\bfN \prime \Biggl( \bigoplus m\in N\prime xm \Biggr) = Fn\bfN \prime \prime \Biggl( \bigoplus
m\in N\prime \prime xm
\Biggr)
for all n \in N\prime \prime (4.3)
(in other words, if the nth components of all the maps are equal and depend only on the variables xm with m in the smallest subnetwork of N containing n).
Proof. Let N\prime \prime \sqsubseteq N\prime and let a \in A be the arrow with s(a) = N\prime and t(a) = N\prime \prime . By definition of Ra, we have on the one hand that
Ra \Biggl( F\bfN \prime \Biggl( \bigoplus m\in \bfN \prime xm \Biggr) \Biggr) = \bigoplus
n\in N\prime \prime Fn\bfN \prime \Biggl( \bigoplus m\in \bfN \prime xm \Biggr) .
On the other hand, F\bfN \prime \prime \Biggl( Ra \Biggl( \bigoplus m\in \bfN \prime xm \Biggr) \Biggr) = \bigoplus
n\in N\prime \prime Fn\bfN \prime \prime
\Biggl( \bigoplus
m\in \bfN \prime \prime xm
\Biggr) .
So Ra\circ F\bfN
\prime
= F\bfN \prime \prime \circ Ra if and only if (4.3) holds.
Example 4.9. Let us investigate what Lemma4.8says for the network N in Examples4.2 and 4.6. So assume that F\bfN \bfone , . . . , F\bfN \bffive form an equivariant map for the quiver depicted in Figure 5. Observe that N5 = N and that the smallest subnetwork of N that contains node 1
is N1. Substituting n = 1, N\prime = N, and N\prime \prime = N1 in (4.3) yields
F1\bfN (x1, x2, x3, x4, x5) = F1\bfN 1(x1) .
This shows that F1\bfN (x) depends only on x1. Continuing in this way for the other nodes,
choosing each time for N\prime \prime the smallest subnetworks containing them, we find F2\bfN (x1, x2, x3, x4, x5) = F2\bfN 2(x1, x2) ,
F3\bfN (x1, x2, x3, x4, x5) = F3\bfN 3(x1, x2, x3) ,
F4\bfN (x1, x2, x3, x4, x5) = F4\bfN 4(x1, x2, x3, x4) ,
F5\bfN (x1, x2, x3, x4, x5) = F5\bfN 5(x1, x2, x3, x4, x5) .
We conclude that SubQ(N)-equivariance is equivalent to F\bfN being of the form
F\bfN \left( x1 x2 x3 x4 x5 \right) = \left( F\bfN 1 1 (x1) F\bfN 2 2 (x1, x2) F\bfN 3 3 (x1, x2, x3) F\bfN 4 4 (x1, x2, x3, x4) F\bfN 5 5 (x1, x2, x3, x4, x5) \right) (4.4)
for some functions F\bfN i
i depending on an appropriate number of variables.
Note that (4.4) is different from (4.1). In fact, any map of the form (4.1) is also of the form (4.4) but not vice versa. Hence (4.4) defines a more general class of maps than (4.1). Nevertheless, by construction (4.1) and (4.4) have exactly the same subnetworks, so a lot of the network structure of (4.1) is also present in (4.4). More importantly, the network structure of (4.4) remains intact when we compose network maps (because quiver symmetry remains intact under composition; see Proposition3.2). We already saw in Example4.2that network maps of the form (4.1) do not possess this nice property.
5. The quiver of quotient networks. Quotient networks were introduced by Golubitsky and Stewart, et al. [7], [10], [12], [24], [25], to compute robust synchrony patterns in network dynamical systems. It was shown for the first time in [12] that every solution of any quotient network lifts to a solution of the original network, i.e., that there is a linear map between the phase spaces that sends solutions of the quotient system to solutions of the original system. More recently, DeVille and Lerman [1] generalized this result and reformulated it using the language of category theory and graph fibrations. The goal of this section is to translate all these observations into the language of quiver representations.
The first half of this section has been added for completeness. We do not aim to provide a comprehensive exposition on quotient networks. Instead, we shall give the basic definitions that allow us to define the quiver of quotient networks. The informed reader may want to skip the first half of this section and start reading from Theorem 5.5.
We start this section by generalizing the notion of a network that was introduced in the previous section.
Definition 5.1. A colored network is a network N = \{ E \rightrightarrows st N \} in which all nodes and
edges are assigned a color, in such a way that
1. if two edges e1, e2 \in E have the same color, then so do their sources s(e1) and s(e2),
and so do their targets t(e1) and t(e2);
2. if two nodes n1, n2 \in N have the same color, then there is at least one color preserving
bijection
\beta n2,n1 : t
- 1(n
1) \rightarrow t - 1(n2)
between the edges that target n1 and n2.
One should think of the networks of section4 as colored networks in which all nodes and edges have a different color, so that conditions 1 and 2 are automatically satisfied. We remark that the node and arrow colors in Definition 5.1 are the same as the cell and arrow types defined in [12]. The collection of color preserving bijections
\BbbG \bfN := \{ \beta n2,n1 : t
- 1
(n1) \rightarrow t - 1(n2) colour preserving bijection | n1, n2 \in N \}
is the so-called symmetry groupoid of Stewart, Golubitsky, and Pivato [25]. These authors also make the following definition, generalizing the network maps that we defined in section4.
Definition 5.2. Let N = \{ E \rightrightarrows st N \} be a coloured network and assume that F\bfN :
\bigoplus
m\in N
Em \rightarrow \bigoplus m\in NEm is a map of the form
Fn\bfN \Biggl( \bigoplus m\in N xm \Biggr) = Fn \left( \bigoplus e\in E : t(e)=n xs(e) \right) .
Assume moreover that
En1 = En2 whenever n1, n2 \in N have the same color
and that for every n1, n2 \in N of the same color and every color preserving bijection \beta n2,n1 \in \BbbG \bfN it holds that Fn1 \left( \bigoplus e\in E : t(e)=n1 xs(\beta n2,n1(e)) \right) = Fn2 \left( \bigoplus e\in E : t(e)=n2 xs(e) \right) .
Then we say that F\bfN is an admissible map for N.
Example 5.3. Figure 6 shows an example of a colored network with two node colors and three edge colors. The edges from a node to itself representing internal dynamics are not depicted. Note that each yellow node is targeted by two blue edges. Hence there are two color preserving bijections between the edges targeting any two yellows nodes. Similarly, each green node is targeted by one red and one orange edge, so there is exactly one color preserving bijection between the edges targeting any two green nodes.
An admissible map for this network is of the form
F\bfN \left( x1 x2 x3 y4 y5 \right) = \left( F (x1,x2,x3) F (x2,x2,x3) F (x3,x1,x2) G(y4,y4,x1) G(y5,y4,x3) \right)
1 2
3
4 5
N
Figure 6.An example of a network with two node colors and three edge colors. Self-loops describing internal dynamics are not shown.
for some functions F and G. The bar indicates that variables may be interchanged, i.e., it expresses that F (x,y,z) = F (x,z,y) for all x, y, z.
The next definition is due to DeVille and Lerman [1].
Definition 5.4. Let N = \{ E \rightrightarrows st N \} and N\prime = \{ E\prime \rightrightarrows st N\prime \} be colored networks and let
\phi : N \rightarrow N\prime . Assume that
i) this \phi sends edges to edges and nodes to nodes, preserves the colors of nodes and edges, and sends the head and tail of every edge e \in E to the head and tail of \phi (e) \in E\prime ; ii) for every node n \in N , the restriction \phi | t - 1(n): t - 1(n) \rightarrow t - 1(\phi (n)) is a color preserving
bijection.
Then \phi is called a graph fibration.
The key result in [1] is the following theorem.
Theorem 5.5 (DeVille and Lerman). Let N = \{ E \rightrightarrows st N \} and N\prime = \{ E\prime \rightrightarrows st N\prime \} be colored
networks, let \phi : N \rightarrow N\prime be a graph fibration, and let F\bfN and F\bfN \prime be admissible maps for N
and N\prime , respectively. In particular, they have the form
Fn\bfN \Biggl( \bigoplus m\in N xm \Biggr) = Fn \left( \bigoplus e\in E : t(e)=n xs(e) \right) and Fn\bfN \prime \prime \Biggl( \bigoplus m\in N\prime xm \Biggr) = Fn\prime \prime \left( \bigoplus
e\prime \in E\prime : t(e\prime )=n\prime xs(e\prime )
\right) .
Finally, assume that for every n \in N, n\prime \in N\prime of the same color and every color preserving bijection \beta n\prime ,n: t - 1(n) \rightarrow t - 1(n\prime ), it holds that
Fn
\left( \bigoplus
e\in E : t(e)=n
xs(\beta n\prime ,n(e))
\right) = Fn\prime \prime
\left(
\bigoplus
e\prime \in E\prime : t(e\prime )=n\prime xs(e\prime )
\right) .
Then the linear map R\phi :
\bigoplus
m\prime \in \bfN \prime
Em\prime \rightarrow \bigoplus m\in \bfN Em defined by R\phi \Biggl( \bigoplus
m\prime \in N\prime xm\prime \Biggr) := \bigoplus m\in N x\phi (m) satisfies R\phi \circ F\bfN \prime = F\bfN \circ R\phi .
The proof of the theorem is simple and consists of combining all the definitions that were made. It can be found in [1].
It is not hard to see that N\prime \sqsubseteq N is a subnetwork if and only if the inclusion i : N\prime \rightarrow N is an injective graph fibration. The map Ri:\bigoplus m\in NEm \rightarrow \bigoplus m\in N\prime Em is then given by
Ri \Biggl( \bigoplus m\in N xm \Biggr) = \bigoplus m\in N\prime xi(m) = \bigoplus m\in N\prime xm.
So we recover the linear maps of section 4. In this section we shall be interested in surjective graph fibrations instead.
Definition 5.6. When \phi : N \rightarrow N\prime is a surjective graph fibration, then we call N\prime a quotient of N.
We are now ready to define the quiver of quotient networks.
Definition 5.7. Let N be a colored network. The quiver QuoQ(N) = \{ A \rightrightarrows st V \} of quotient
networks of N has as its vertices the nonempty quotients of N, i.e., V = \{ N\prime \not = \emptyset | N\prime is a quotient of N\} .
There is exactly one arrow a \in A with s(a) = N\prime and t(a) = N\prime \prime for each distinct surjective graph fibration \phi from N\prime \prime to N\prime .
A representation of QuoQ(N) is defined in a straightforward manner. To each quotient N\prime of N (i.e., each vertex N\prime of the quiver QuoQ(N)), we assign the vector space
E\bfN \prime := \bigoplus
m\in N\prime Em,
and for each arrow a \in A from N\prime to N\prime \prime (corresponding to the graph fibration \phi : N\prime \prime \rightarrow N\prime ), we define Ra = R\phi , where R\phi is the linear map defined in Theorem 5.5. In other words,
Ra: E\bfN \prime \rightarrow E\bfN \prime \prime is defined by the formula Ra \Biggl( \bigoplus m\in N\prime xm \Biggr) := \bigoplus
m\in N\prime \prime x\phi (m). (5.1)
Theorem5.5 then trivially translates into the following result.
Corollary 5.8. Let N = \{ E \rightrightarrows st N \} be a colored network and let F\bfN : E\bfN \rightarrow E\bfN be an
admissible map, so that in particular it is of the form
Fn\bfN \Biggl( \bigoplus m\in N xm \Biggr) = Fn \left( \bigoplus e\in E : t(e)=n xs(e) \right) .
For each surjective graph fibration \phi : N \rightarrow N\prime , define F\bfN \prime : E\bfN \prime \rightarrow E\bfN \prime by
F\phi (n)\bfN \prime \Biggl( \bigoplus m\in N\prime xm \Biggr) := Fn \left( \bigoplus e\in E : t(e)=n xs(\phi (e)) \right) .
1 2 3 4 5 1 2 3 4 1 2 3 4 1 2 3 1 2 3 1 2 N1 N2 N3 N5 N4 N6 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12
Figure 7. The quiver of quotient networks for the network in Figure 6. Loops from \bfN i to \bfN iare not drawn.
Then each F\bfN \prime is well-defined and admissible for N\prime . Together the F\bfN \prime (N\prime quotient of N) form a QuoQ(N)-equivariant map.
Example 5.9. The network N in Figure6 has six nonempty quotients (including N = N1
itself). Figure7 shows the quiver of quotient networks.
To illustrate, note that there is a graph fibration \phi 2 : N1 \rightarrow N2 which sends node 1 and
2 to node 1, node 3 to node 2, node 4 to node 3, and node 5 to node 4. The corresponding linear map in the representation given by formula (5.1) is
Ra2(x1, x2, y3, y4) = (x1, x1, x2, y3, y4) .
The admissible maps for N1 and N2 are given by
F\bfN 1 \left( x1 x2 x3 y4 y5 \right) = \left( F (x1,x2,x3) F (x2,x2,x3) F (x3,x1,x2) G(y4,y4,x1) G(y5,y4,x3) \right) and F\bfN 2 \left( x1 x2 y3 y4 \right) = \left( F (x 1,x1,x2) F (x2,x1,x1) G(y3,y3,x1) G(y4,y3,x2) \right) .
One verifies that indeed Ra2\circ F
\bfN 2 = F\bfN 1 \circ R
a2.
The full list of representation maps for the arrows in Figure 7is given by Ra1(x1, y2) = (x1, x1, x1, y2, y2) , Ra2(x1, x2, y3, y4) = (x1, x1, x2, y3, y4) , Ra3(x1, y2) = (x1, x1, y2, y2) , Ra4(x1, y2, y3) = (x1, x1, y2, y3) , Ra5(x1, y2, y3) = (x1, x1, x1, y2, y3) , Ra6(x1, y2) = (x1, y2, y2) , Ra7(x1, x2, y3, y4) = (x1, x2, x1, y3, y4) , Ra8(x1, y2, y3) = (x1, x1, y2, y3) , Ra9(x1, y2) = (x1, x1, y2, y2) , Ra10(x1, x2, y3) = (x1, x2, y3, y3) , Ra11(x1, x2, y3) = (x1, x2, x1, y3, y3) , Ra12(x1, y2) = (x1, x1, y2) .
6. Endomorphisms of quiver representations. In this section, we gather some basic prop-erties of endomorphisms of quiver representations that will be important in the remainder of this paper. An endomorphism is simply a linear equivariant map.
Definition 6.1. An endomorphism of a quiver representation (E, R) of a quiver Q = \{ A \rightrightarrows s t
V \} is a set L of linear maps Lv : Ev \rightarrow Ev (one for each v \in V ) such that
Lt(a)\circ Ra= Ra\circ Ls(a) for every arrow a \in A .
The collection of all endomorphisms is denoted by End(E, R).
Example 6.2. For any representation (E, R) of any quiver Q = \{ A \rightrightarrows st V \} , the identity
Id, consisting of the maps Idv : Ev \rightarrow Ev (v \in V ), is an example of an endomorphism. This
is simply because Idt(a)\circ Ra= Ra\circ Ids(a).
Example 6.3. If F \in C\infty (E, R) is a smooth equivariant map of a representation (E, R) and F(0) = 0 (meaning that Fv(0) = 0 for every v \in V ), then the derivative L = DF(0)
(consisting of the maps Lv := DFv(0) : Ev \rightarrow Ev) is an example of an endomorphism. This
follows from differentiating the identities Ft(a)\circ Ra= Ra\circ Fs(a)at 0 and noting that Ra(0) = 0.
Definition 6.4. A subrepresentation D of a representation (E, R) of a quiver Q = \{ A \rightrightarrows st
V \} is a set of linear subspaces Dv \subset Ev (v \in V ) such that
Ra(Ds(a)) \subset Dt(a) for every arrow a \in A .
In other words, D is a subrepresentation of (E, R) if the restriction (D, R| \bfD ) defines a
representation. Examples of subrepresentations are the eigenspaces of endomorphisms. In this paper, we will use generalized eigenspaces more often than eigenspaces, so we formulate the following as a separate proposition.
Proposition 6.5. Let L be an endomorphism of a representation (E, R) of a quiver Q = \{ A \rightrightarrows s
t V \} . We say that \lambda \in \BbbC is an eigenvalue of L if there is at least one v \in V such that
\lambda is an eigenvalue of Lv. We then call \lambda an eigenvalue of all the Lw (w \in V ) even if the
corresponding eigenspace of Lw is trivial.
i) For \lambda \in \BbbR , denote by E\lambda
v \subset Ev the generalized eigenspace of Lv : Ev \rightarrow Ev for the
eigenvalue \lambda . Then the Ev\lambda define a subrepresentation E\lambda of (E, R).
ii) For \mu \in \BbbC \setminus \BbbR , denote by Ev\mu ,\=\mu \subset Ev the real generalized eigenspace of Lv : Ev \rightarrow Ev
for the eigenvalue pair \mu , \=\mu . Then the Ev\mu ,\=\mu define a subrepresentation E\mu ,\=\mu of (E, R).
Proof. Recall that L consists of linear maps Lv: Ev \rightarrow Ev (v \in V ) for which Ra\circ Ls(a)=
Lt(a) \circ Ra for each a \in A. Choose \lambda \in \BbbR and assume that x \in E\lambda
s(a). This means that
(Ls(a) - \lambda Ids(a))N(x) = 0 for any N \geq dim Es(a). But then
(Lt(a) - \lambda Idt(a))N(Rax) = Ra(Ls(a) - \lambda Ids(a))N(x) = 0 .
So Ra(Es(a)\lambda ) \subset Et(a)\lambda . For \mu \in \BbbC \setminus \BbbR , Ev\mu ,\=\mu = ker((Lv - \mu Idv)(Lv - \mu Idv))N. So the proof is
completely analogous.
7. Lyapunov--Schmidt reduction and quivers. In this and the coming sections, we will show that quiver symmetry can be preserved in a number of well-known dimension reduction techniques. We start with the most straightforward result, which shows that quiver symmetry can be preserved in the process of Lyapunov--Schmidt reduction. We only prove this for steady state bifurcations at this point. How to preserve quiver symmetry in the Lyapunov--Schmidt reduction for periodic orbits is left as an open problem.
Let us start by reviewing the classical Lyapunov--Schmidt reduction process for steady state bifurcations (so without any quiver symmetry) to set the stage for the proof of Theorem 7.1below. We consider the differential equation
dx
dt = F (x; \lambda ) for x \in E and \lambda \in \Lambda \subset \BbbR
p,
where F : E \times \Lambda \rightarrow E is a smooth vector field defined on a finite dimensional vector space E, depending smoothly on parameters from an open set \Lambda \subset \BbbR p. We also assume that for some value of the parameters this differential equation admits a steady state. We assume without loss of generality that F (0; 0) = 0. The goal is to find all other steady states near (x; \lambda ) = (0; 0) by reducing the equation
F (x; \lambda ) = 0 on E \times \Lambda
to a simpler ``bifurcation equation"" with as few dimensions as possible.
To explain how this is done, denote by L = DxF (0; 0) : E \rightarrow E the derivative of F in the
direction of E at (0; 0). We shall denote by E\mathrm{k}\mathrm{e}\mathrm{r} \subset E the generalized kernel of L (i.e., the generalized eigenspace for the eigenvalue zero) and by E\mathrm{i}\mathrm{m} its reduced image (the sum of the remaining generalized eigenspaces). We write
\pi : E = E\mathrm{i}\mathrm{m}\oplus E\mathrm{k}\mathrm{e}\mathrm{r}\rightarrow E\mathrm{i}\mathrm{m}, x = x\mathrm{i}\mathrm{m}+ x\mathrm{k}\mathrm{e}\mathrm{r}\mapsto \rightarrow x\mathrm{i}\mathrm{m}
for the projection onto E\mathrm{i}\mathrm{m} along E\mathrm{k}\mathrm{e}\mathrm{r} (i.e., \pi has kernel E\mathrm{k}\mathrm{e}\mathrm{r} and is the identity on E\mathrm{i}\mathrm{m}). The derivative in the direction of E\mathrm{i}\mathrm{m} of
\pi \circ F : E\mathrm{i}\mathrm{m}\oplus E\mathrm{k}\mathrm{e}\mathrm{r}\times \Lambda \rightarrow E\mathrm{i}\mathrm{m} at (0; 0) is equal to
Dx\mathrm{i}\mathrm{m}(\pi \circ F )(0; 0) = \pi \circ L| E\mathrm{i}\mathrm{m} : E\mathrm{i}\mathrm{m}\rightarrow E\mathrm{i}\mathrm{m}.
By construction this map is invertible. By the implicit function theorem there is thus a unique smooth function
\phi : U \subset E\mathrm{k}\mathrm{e}\mathrm{r}\times \Lambda \rightarrow W \subset E\mathrm{i}\mathrm{m}
defined on some open neighborhood U of (0; 0) \in E\mathrm{k}\mathrm{e}\mathrm{r}\times \Lambda and mapping into an open neigh-borhood W of 0 \in E\mathrm{i}\mathrm{m} that satisfies
(\pi \circ F )(x\mathrm{k}\mathrm{e}\mathrm{r}+ \phi (x\mathrm{k}\mathrm{e}\mathrm{r}; \lambda ); \lambda ) = 0 .
We clearly have \phi (0; 0) = 0 because F (0; 0) = 0. To find all other solutions (x\mathrm{i}\mathrm{m}, x\mathrm{k}\mathrm{e}\mathrm{r}; \lambda ) \in W \times U to the equation F (x\mathrm{i}\mathrm{m}, x\mathrm{k}\mathrm{e}\mathrm{r}; \lambda ) = 0, it then remains to solve only the reduced bifurcation equation
f (x\mathrm{k}\mathrm{e}\mathrm{r}; \lambda ) := ((1 - \pi ) \circ F )(x\mathrm{k}\mathrm{e}\mathrm{r}+ \phi (x\mathrm{k}\mathrm{e}\mathrm{r}; \lambda ); \lambda ) = 0 , (7.1)
where
f : U \subset E\mathrm{k}\mathrm{e}\mathrm{r}\times \Lambda \rightarrow E\mathrm{k}\mathrm{e}\mathrm{r}.
This method to (locally) reduce the equation F (x; \lambda ) = 0 to the lower-dimensional equation f (x\mathrm{k}\mathrm{e}\mathrm{r}; \lambda ) = 0 is called Lyapunov--Schmidt reduction. The following theorem states that the
reduced equation inherits quiver symmetry if it is present in the original equation.
Theorem 7.1 (quiver equivariant Lyapunov--Schmidt theorem). Let (E, R) be a representa-tion of a quiver Q = \{ A \rightrightarrows st V \} and assume that F \in C\infty (E \times \Lambda , R) is a smooth
parameter-dependent Q-equivariant map, i.e., for every v \in V there is a smooth Fv : Ev \times \Lambda \rightarrow Ev
satisfying
Ra(Fs(a)(x; \lambda )) = Ft(a)(Ra(x); \lambda ) for all x \in Es(a), \lambda \in \Lambda , and a \in A .
Assume moreover that F(0; 0) = 0, i.e., Fv(0; 0) = 0 for all v \in V .
Then the reduced maps fv : Uv \subset Ev\mathrm{k}\mathrm{e}\mathrm{r}\times \Lambda \rightarrow Ev\mathrm{k}\mathrm{e}\mathrm{r} (v \in V ) defined in (7.1) satisfy
Ra(fs(a)(x\mathrm{k}\mathrm{e}\mathrm{r}; \lambda )) = ft(a)(Ra(x\mathrm{k}\mathrm{e}\mathrm{r}); \lambda ) for all a \in A
and for all (x\mathrm{k}\mathrm{e}\mathrm{r}; \lambda ) \in U
s(a) in some open neighborhood Us(a) of (0; 0).
This means that the fv (v \in V ) define a Q-equivariant map f on an open neighborhood of
(0; 0) of the subrepresentation E\mathrm{k}\mathrm{e}\mathrm{r}\times \Lambda of (E \times \Lambda , R).
Proof. Fix an a \in A and consider the map Ra : Es(a) \rightarrow Et(a). Recall that Ra\circ Ls(a) =
Lt(a) \circ Ra, where Lv = DxFv(0; 0), so that Ra(Es(a)\mathrm{k}\mathrm{e}\mathrm{r}) \subset Et(a)\mathrm{k}\mathrm{e}\mathrm{r} and Ra(Es(a)\mathrm{i}\mathrm{m} ) \subset Et(a)\mathrm{i}\mathrm{m} by
Proposition6.5. It follows in particular that
Ra\circ \pi s(a)= \pi t(a)\circ Ra.
Recall that by definition of \phi s(a): Us(a)\rightarrow Ws(a) it holds that
(\pi s(a)\circ Fs(a))(x\mathrm{k}\mathrm{e}\mathrm{r}+ \phi s(a)(x\mathrm{k}\mathrm{e}\mathrm{r}; \lambda ); \lambda ) = 0 for all (x\mathrm{k}\mathrm{e}\mathrm{r}; \lambda ) \in Us(a). It follows that
0 = (Ra\circ \pi s(a)\circ Fs(a))(x\mathrm{k}\mathrm{e}\mathrm{r}+ \phi s(a)(x\mathrm{k}\mathrm{e}\mathrm{r}; \lambda ); \lambda )
= (\pi t(a)\circ Ft(a))(Ra(x\mathrm{k}\mathrm{e}\mathrm{r}) + Ra(\phi s(a)(x\mathrm{k}\mathrm{e}\mathrm{r}; \lambda )); \lambda ) .
By definition of \phi t(a): Ut(a)\rightarrow Wt(a) it thus holds that
\phi t(a)(Ra(x\mathrm{k}\mathrm{e}\mathrm{r}); \lambda ) = Ra(\phi s(a)(x\mathrm{k}\mathrm{e}\mathrm{r}; \lambda ))
for all (x\mathrm{k}\mathrm{e}\mathrm{r}; \lambda ) \in Us(a) with (Ra(x\mathrm{k}\mathrm{e}\mathrm{r}); \lambda ) \in Ut(a) and Ra(\phi s(a)(x\mathrm{k}\mathrm{e}\mathrm{r}; \lambda )) \in Wt(a). The (x\mathrm{k}\mathrm{e}\mathrm{r}; \lambda )
for which these inclusions hold form an open neighborhood \~Ua of (0; 0). For (x\mathrm{k}\mathrm{e}\mathrm{r}; \lambda ) \in \~Ua we
then have that
Ra(fs(a)(x\mathrm{k}\mathrm{e}\mathrm{r}; \lambda )) = (Ra\circ (1 - \pi s(a)) \circ Fs(a))(x\mathrm{k}\mathrm{e}\mathrm{r}+ \phi s(a)(x\mathrm{k}\mathrm{e}\mathrm{r}; \lambda ); \lambda )
= ((1 - \pi t(a)) \circ Ft(a))(Ra(x\mathrm{k}\mathrm{e}\mathrm{r}) + \phi t(a)(Ra(x\mathrm{k}\mathrm{e}\mathrm{r}); \lambda ); \lambda ) = ft(a)(Ra(x\mathrm{k}\mathrm{e}\mathrm{r}); \lambda ) .
This would prove the theorem if for every vertex v \in V there was at most one arrow a \in A with s(a) = v. If there are more such arrows, then the finite intersection Uv :=\bigcap a:s(a)=vU\~a
will satisfy the requirements.
8. Center manifolds and quivers. In this section we show that quiver symmetry can be preserved in the process of center manifold reduction. The main result is Theorem8.2below, which is a Q-equivariant global center manifold theorem. We encountered various obstructions in trying to prove a fully general Q-equivariant local center manifold theorem. These will be discussed in Remark 4 below.
We start our analysis by recalling the classical global center manifold theorem [26]. We will not prove this classical theorem here, and for simplicity we only formulate a version of the theorem without parameters. To formulate the classical result, let E be a finite dimensional real vector space and L : E \rightarrow E a linear map. Let us denote by Ec the center subspace of L (the sum of the generalized eigenspaces of L for the eigenvalues on the imaginary axis) and by Eh the hyperbolic subspace of L (the sum of the generalized eigenspaces of L for the eigenvalues not lying on the imaginary axis). We shall denote by
\pi c: E = Ec\oplus Eh \rightarrow Ecand by \pi h := 1 - \pi c: E = Ec\oplus Eh \rightarrow Eh
the projections corresponding to the splitting E = Ec\oplus Eh. Now we can formulate the global
center manifold theorem, referring to [26] for a proof.
Theorem 8.1. Let L : E \rightarrow E be a linear map and k \in \{ 1, 2, 3, . . .\} . Then there is an \varepsilon = \varepsilon (L, k) > 0 for which the following holds.
If F : E \rightarrow E is a Ck vector field that satisfies F (0) = 0, DF (0) = L, sup
x\in E
| | D\alpha (F (x) - L)| | < \infty for all | \alpha | \leq k and sup
x\in E
| | DF (x) - L| | < \varepsilon ,
then there exists a Ck map \phi : Ec \rightarrow Eh, satisfying \phi (0) = 0 and D\phi (0) = 0, of which the
graph
Mc:= \{ xc+ \phi (xc) | xc\in Ec\} \subset E
is an invariant manifold for the flow of the differential equation dxdt = F (x). Moreover, if we denote this flow by etF, then
Mc= \biggl\{ x \in E \bigm| \bigm| \bigm| \bigm| sup t\in \BbbR
| | (\pi h\circ etF)(x)| | < \infty \biggr\}
. We call Mc the global center manifold of F .
Remark 3. Let x(t) be an integral curve of F , i.e., dx(t)dt = F (x(t)), and let us write xc(t) := \pi c(x(t)). Then
dxc(t) dt = (\pi
c\circ F )(x(t)) .
If x(t) happens to lie inside Mc, then by definition of \phi we moreover have that x(t) = xc(t) + \phi (xc(t)). So then
dxc(t) dt = (\pi
c\circ F )(xc(t) + \phi (xc(t))) .
This proves that the restriction of \pi cto Mcsends integral curves of F in Mcto integral curves of the vector field Fc: Ec\rightarrow Ec defined by
Fc(xc) := (\pi c\circ F )(xc+ \phi (xc)) .
We shall call this vector field Fcon Ec the center manifold reduction of F .
We are now ready to formulate our result on quivers and center manifolds, remarking that its proof is more or less identical to that of Lemma 2.2.
Theorem 8.2 (quiver equivariant center manifold theorem). Let (E, R) be a representation of a quiver Q = \{ A \rightrightarrows st V \} and let L \in End(E, R) and F \in Ck(E, R) (k = 1, 2, . . .) with
F(0) = 0 and DF(0) = L.
So we assume that for every v \in V there is a linear map Lv : Ev \rightarrow Ev and a Ck smooth
map Fv : Ev\rightarrow Ev with Fv(0) = 0, DFv(0) = Lv, such that
Ra\circ Ls(a)= Lt(a)\circ Ra and Ra\circ Fs(a)= Ft(a)\circ Ra for all a \in A .
Assume moreover that each of the Lv and Fv (v \in V ) satisfy the bounds of Theorem 8.1, so
that each Fv admits a unique global center manifold Mvc.
Then Ra maps the global center manifold of Fs(a) into that of Ft(a), i.e.,
Ra(Ms(a)c ) \subset M c t(a).
Moreover, the center manifold reductions Fvc: Evc\rightarrow Ec
v (v \in V ) satisfy
Ra\circ Fs(a)c = F c
t(a)\circ Ra for all a \in A .
So the Fvc define a Q-equivariant vector field Fc on the subrepresentation Ec of (E, R) con-sisting of the center subspaces Ec
v (v \in V ).
Proof. Fix an a \in A. By Proposition 6.5we have that Ra(Es(a)c ) \subset Et(a)c and Ra(Es(a)h ) \subset
Et(a)h , so in particular it holds that
Ra\circ \pi s(a)c = \pi ct(a)\circ Ra and Ra\circ \pi hs(a)= \pi t(a)h \circ Ra.
Next, choose an x \in Ms(a)c and recall that for such x we have sup
t\in \BbbR
| | \Bigl( \pi hs(a)\circ etFs(a) \Bigr)
(x)| | < \infty .
Because Ra\circ etFs(a) = etFt(a)\circ Ra and Ra\circ \pi s(a)h = \pi ht(a)\circ Ra, this implies that
sup
t\in \BbbR
| | (\pi t(a)h \circ etFt(a))(R
a(x))| | = sup t\in \BbbR
| | Ra(\pi s(a)h \circ etFs(a))(x))| |
\leq | | Ra| | \cdot sup t\in \BbbR
| | (\pi s(a)h \circ etFs(a))(x)| | < \infty ,
where | | Ra| | is the operator norm of Ra. We conclude that Ra(x) \in Mt(a)c . This proves that
Ra(Ms(a)c ) \subset Mt(a)c .
Next, recall that if x \in Mc
s(a), then it is of the form
x = xc \underbrace{} \underbrace{} \in Ec s(a) + \phi s(a)(xc) \underbrace{} \underbrace{} \in Eh s(a) ,
where \phi s(a) : Es(a)c \rightarrow Eh
s(a) is the C
k function whose graph is Mc
s(a). Applying Ra to this
equality we find that
Ra(x) = Ra(xc) \underbrace{} \underbrace{} \in Ec t(a) + Ra(\phi s(a)(xc)) \underbrace{} \underbrace{} \in Eh t(a) \in Mt(a)c .
But every X \in Mt(a)c can uniquely be written in the form X = Xc \underbrace{} \underbrace{} \in Ec t(a) + \phi t(a)(Xc) \underbrace{} \underbrace{} \in Eh t(a) ,
where \phi t(a) : Et(a)c \rightarrow Eh
t(a) is the C
k function whose graph is Mc
t(a). This proves that
Ra(\phi s(a)(xc)) = \phi t(a)(Ra(xc)), i.e., that
Ra\circ \phi s(a)= \phi t(a)\circ Ra.
Recalling the definition of the center manifold reductions Fvc: Evc\rightarrow Ec
v, we finish by noticing
that
Ra(Fs(a)c (xc)) = (Ra\circ \pi cs(a)\circ Fs(a))(xc+ \phi s(a)(xc))
= (\pi t(a)c \circ Ft(a)\circ Ra)(xc+ \phi s(a)(xc))
= (\pi t(a)c \circ Ft(a))(Ra(xc) + Ra(\phi s(a)(xc)))
= (\pi t(a)c \circ Ft(a))(Ra(xc) + \phi t(a)(Ra(xc)))
= Ft(a)c (Ra(xc)) ,
i.e., Ra\circ Fs(a)c = Ft(a)c \circ Ra. This finishes the proof.
Remark 4. Theorem 8.2 is a Q-equivariant global center manifold theorem. Assuming that the first derivatives of the nonlinearities Fv - Lv are globally small, and that their
higher derivatives are globally bounded, it guarantees the existence of a globally defined center manifold. The global conditions on the nonlinearities are rather unnatural though, as in practice the nonlinearities will only be small in a neighborhood of the equilibrium under consideration. The global center manifold theorem is a (very important) step in the proof of a local center manifold theorem---where the global bounds are not required and a center manifold is guaranteed in a small neighborhood of the equilibrium.
Although it is reasonable to assume that a local version of Theorem8.2holds as well, we were so far unable to prove such a theorem for general Q-equivariant systems. The problem arises from the way one usually makes the step from a global to a local center manifold theorem: one replaces the unbounded nonlinearities Fv - Lv by globally bounded nonlinearities, for
example, by replacing the ODE dxdt = Fv(x) by the ODE dxdt = \widetilde Fv(x) := Lvx + \zeta v(x)(Fv(x) -
Lvx), where \zeta v : Ev \rightarrow \BbbR is a smooth bump function with \zeta v(x) = 1 for small | | x| | . By
shrinking the support of \zeta v one can then satisfy the assumptions of Theorem8.1. The problem
that we encounter is that in general it is unclear how to choose the bump functions \zeta v(v \in V )
in such a way that Q-equivariance is preserved.
This problem can sometimes be circumvented if the Q-equivariant vector field happens to be an admissible vector field F\bfN for some network N. In that case one can multiply
the nonlinear parts of each of the separate components Fn\bfN of the vector field with a bump function, choosing the same bump function for nodes with the same color (more precisely, choosing bump functions that are invariant under the symmetry groupoid \BbbG \bfN ). The resulting
vector field \widetilde F\bfN will then have the same network structure as F\bfN and will hence admit, for example, the same quiver of subnetworks and quiver of quotient networks. In [17], [18] it was shown in detail how this works out for so-called fully homogeneous networks with asymmetric inputs. It is not hard to see that the same procedure can be applied to the admissible maps of any network; see Definition5.2.
On the other hand, quiver symmetry is not always the same as network structure. There-fore even proving an equivariant local center manifold theorem for specific quivers remains problematic. The mentioned fully homogeneous networks with asymmetric inputs are an ex-ception, as we proved in [20] that such networks admit a quiver symmetry that is equivalent to a particular network structure (which may be more general than the original network structure though). Such a result will not hold for other types of networks and quivers. For instance, it is not clear to us that equivariance of F under QuoQ(N) implies that F is an admissible vector field for some network that is somehow related to N. We therefore do not know at this moment how to prove a QuoQ(N)-equivariant local center manifold theorem.
9. Normal forms and quivers. The normal form of a local dynamical system displays the system in a ``standard"" or ``simple"" form. Normal forms are an important tool in the study of the dynamics and bifurcations of maps and vector fields near equilibria; cf. [14], [22]. The goal of this section is to prove Theorem 9.6 below. This theorem states that it can be arranged that the normal form of a dynamical system possesses the same quiver symmetry as the original system. For simplicity, we do not consider parameter-dependent vector fields in this section (but it is straightforward to prove the same result for systems with parameters as well).
We start by recalling one of the classical results of normal form theory in Theorem 9.1. To this end, let us consider a smooth ODE
dx
dt = F (x) = F
0(x) + F1(x) + F2(x) + . . .
on a finite dimensional vector space E. That is, F \in C\infty (E) is a smooth vector field on E, F (0) = 0, and Fk \in Pk(E) where
Pk(E) := \{ Fk : E \rightarrow E | Fk is a homogeneous polynomial of degree k + 1\} . The idea is that we now try to make local coordinate transformations
x \mapsto \rightarrow y = \Phi (x) = x + \scrO (| | x| | 2)
that simplify (in one way or another) the higher order terms Fk(k = 1, 2, . . .) of F . There are various ways to define such coordinate transformations, and there are various ways to define what it means to ``simplify"" a local ODE. Theorem 9.1 states one of the many well-known results.
Theorem 9.1 (normal form theorem). Let E be a finite dimensional real vector space and let F \in C\infty (E) be a smooth vector field with F (0) = 0 and Taylor expansion
F = F0+ F1+ F2+ . . . , where Fk \in Pk(E) .
Then, for every 1 \leq r < \infty , there exists an analytic diffeomorphism \Phi , sending an open neighborhood of 0 in E to an open neighborhood of 0 in E, so that the coordinate transformation x \mapsto \rightarrow y = \Phi (x) = x + \scrO (| | x| | 2) transforms the ODE
dx
dt = F (x)
into an ODE of the form
dy
dt = F (y) with
F = F0+ F1+ F2+ . . . with Fk\in Pk(E) ,
while at the same time it holds that
etLS \circ Fk= Fk\circ etLS for all 1 \leq k \leq r and t \in \BbbR . (9.1)
Here, LS = (F0)S denotes the semisimple part of F
0 and etL
S
its (linear) time-t flow.
To clarify the statement in this theorem, we will now make a number of definitions and observations. A sketch of the proof of Theorem9.1will be given afterward. First of all, for any two smooth vector fields F, G \in C\infty (E) on E one may define the Lie bracket [F, G] \in C\infty (E) as the vector field
[F, G](x) := d dt \bigm| \bigm| \bigm| \bigm| t=0 \bigl( etF\bigr)
\ast G(x) = DF (x) \cdot G(x) - DG(x) \cdot F (x) .
(9.2)
Here, etF denotes the time-t flow of F (which is defined near each x \in E for some positive time) and (etF)\ast G(x) := DetF \cdot G(e - tF(x)) is the pushforward of the vector field G by the
time-t flow of F . We say that F and G commute if [F, G] = 0, which is equivalent to their flows etF and etG commuting, and equivalent to F being equivariant under the flow etG, and equivalent to G being equivariant under the flow etF. In particular, (9.1) is equivalent to
[LS, Fk] = 0 for all 1 \leq k \leq r .
We shall also define, for F \in C\infty (E), the linear operator
adF : C\infty (E) \rightarrow C\infty (E) by adF(G) := [F, G] .
It follows from (9.2) that if Fk\in Pk(E) and Gl\in Pl(E), then [Fk, Gl] \in Pk+l(E). In other
words, adFk : Pl(E) \rightarrow Pk+l(E).
We will also use the following result, which we state here without proof.
Proposition 9.2. Let L : E \rightarrow E be a linear map on a finite dimensional vector space E. Recall that there are a unique semisimple linear map LS : E \rightarrow E and a unique nilpotent linear map LN : E \rightarrow E so that
L = LS+ LN and [LS, LN] := LSLN - LNLS = 0 . LS is called the semisimple part of L and LN the nilpotent part of L.
The semisimple and nilpotent parts of the restriction adL : Pk(E) \rightarrow Pk(E) of adL to
Pk(E) are then given by
(adL)S = adLS and (adL)N = adLN.
Corollary 9.3. It holds that i) Pk(E) = im ad
LS\oplus ker adLS;
ii) im adLS\cap Pk(E) \subset im adL\cap Pk(E); iii) ker adL\cap Pk(E) \subset ker adLS\cap Pk(E);
iv) adL: im adLS \cap Pk(E) \rightarrow im adLS\cap Pk(E) is an isomorphism.
Proof. For any linear map M on a finite dimensional real vector space V it holds that V = im MS \oplus ker MS, im MS \subset im M and ker M \subset ker MS. So these identities hold in
particular for M = adL and V = Pk(E).
To prove point iv), note that M (MS(x)) = MS(M (x)) so M sends im MS into itself.
Because ker M \subset ker MS, we have that ker M \cap im MS = \{ 0\} .
Proof of Theorem 9.1. We sketch the well-known construction of the normal form by means of ``Lie transformations,"" providing only those details that are necessary to prove Theorem9.6 below, and leaving out any analytical estimates.
First of all, recall that for any smooth vector field G \in C\infty (E) satisfying G(0) = 0, the time-t flow etG defines a diffeomorphism of an open neighborhood of 0 in E to another open neighborhood of 0 in E. Thus we can consider, for any other smooth vector field F \in C\infty (E), the curve t \mapsto \rightarrow (etG)\ast F of transformed vector fields. This curve satisfies the initial condition
\bigl( e0G\bigr)
\ast F = F
together with the linear differential equation d dt\bigl( e tG\bigr) \ast F = d dh \bigm| \bigm| \bigm| \bigm| h=0 \bigl( ehG\bigr)
\ast \bigl( \bigl( e tG\bigr)
\ast F\bigr) = [G, \bigl( e tG\bigr) \ast F ] = adG\bigl( \bigl( etG \bigr) \ast F\bigr) . (9.3)
The second equality holds by definition of the Lie bracket. We conclude that \bigl( eG\bigr)
\ast F = e
\mathrm{a}\mathrm{d}G(F ) = F + [G, F ] +1
2[G, [G, F ]] + . . . .
The diffeomorphism \Phi in the statement of the theorem is now constructed as the composition of a sequence of time-1 flows eGk (1 \leq k \leq r) with Gk \in Pk(E). We first take G1 \in P1(E),
so that F is transformed by eG1 into
\bigl( eG1\bigr) \ast F = e \mathrm{a}\mathrm{d}G1(F ) = F0+ F1,1+ F2,1+ . . . in which F1,1 = F1+ [G1, F0] \in P1(E) , F2,1 = F2+ [G1, F1] +21[G1, [G1, F0]] \in P2(E) , F3,1 = F3+ . . . \in P3(E) , etc.
From now on we shall often use the notation L := F0.
