# Laplacian Matrices

# Synchrony patterns in Laplacian networks

###### Abstract

A network of coupled dynamical systems is represented by a graph whose
vertices represent individual cells and whose edges represent couplings
between cells. Motivated by the impact of synchronization results of the
Kuramoto networks, we introduce the generalized class of Laplacian networks,
governed by mappings whose Jacobian at any point is a symmetric matrix with
row entries summing to zero. By recognizing this matrix with a weighted
Laplacian of the associated graph, we derive the optimal estimates of its
positive, null and negative eigenvalues directly from the graph topology.
Furthermore, we provide a characterization of the mappings that define
Laplacian networks. Lastly, we discuss stability of equilibria inside
synchrony subspaces for two types of Laplacian network on a ring with some
extra couplings.

Keywords. network, admissible mapping, graph Laplacian, singularity,
synchrony, stability.

AMS classification. 34C15, 37G40, 82B20, 34D06.

Tiago de Albuquerque Amorim111Email address<EMAIL_ADDRESS>
Mathematics Department, ICMC

University of São Paulo

13560-970 P.O. box 668, São Carlos, SP - Brazil

Miriam Manoel222Email address<EMAIL_ADDRESS>(corresponding author)

Mathematics Department, ICMC

University of São Paulo

13560-970 P.O. box 668, São Carlos, SP - Brazil

##  1 Introduction

About fifty years ago Kuramoto introduced in [15] a system of ordinary
differential equations that has since gained recognition as the simple (or
traditional) Kuramoto model. It has been proposed as a straightforward and
solvable framework to comprehend mutual synchronization within a cluster of
oscillators that are equally coupled to all other oscillators. Its
significance is further underscored by its applications across various
synchronization scenarios in chemical [7], biological [21], social systems and
in neuroscience phenomena [17]. Since then, numerous related articles have
emerged in the literature in distinct contexts of investigation, varying from
numerical analysis and stochastic methods [14] to criteria for the existence
and stability of (clusters of) synchrony [6]. In this paper we introduce a
more comprehensive framework for this particular type of system of coupled
identical individual systems, accounting for two distinct aspects:

  * •

the coupling may not necessarily be universally applied across all elements;

  * •

the Jacobian of the network vector field is a general Laplacian matrix.

As we shall see, the second requires that being bidirected is a necessary
condition for the associated graphs. We remark that the second aspect appears
naturally in gradient systems of coupled cell networks with an extra ${\bf
S}^{1}$-invariance condition on the vector field components; see [16], where
critical points are investigated for cells coupled in a ring. In fact, we
prove that vector fields with a Laplacian structure are gradient; see Theorem
4.10.

Recall from algebraic graph theory that eigenvalues of a weighted graph
Laplacian matrix hold substantial information of the graph. For instance, it
follows directly from the definition that zero is an eigenvalue, associated
with the eigenvector $(1,\ldots,1)$. This condition is already interesting
from the point of view of the local dynamics, as it implies that there are no
isolated singularities. Furthermore, in the case of non-negative weights, it
is well known that all the eigenvalues are non-negative and the algebraic
multiplicity of the zero eigenvalue is exactly the number of connected
components of the graph. This result has been extended in [5] to the case of
connected graphs allowing negative weights. Since it is expected that the
Jacobian matrix of the vector field has zero entries, recognizing this matrix
as the graph Laplacian naturally leads us to the inherent condition that the
graph may be disconnected. With the requirement to address this condition, we
have interpreted the absence of an edge connecting two vertices as the
existence of an edge with zero weight to extend the result to disconnected
graphs; see Theorem 4.1. The result is the optimal estimate for the number of
negative, null and positive eigenvalues from the number of connected
components of the subgraph determined by positive weights and the subgraph
determined by negative weights.

In the study of local dynamics, specially in bifurcation theory, the starting
point is the analysis of existence and nature of equilibria and periodic
orbits. An equilibrium is a critical point of the vector field expected to lye
on subspaces that are invariant under the dynamics. In equivariant dynamics,
these are fixed-point subspaces of subgroups of the symmetry group; in coupled
networks, these are synchrony subspaces determined from the associated graph.
Synchrony subspaces are robust polysynchronous subspaces; that is,
polysynchronous subspaces (determined by a set of equalities of the vector
components – the synchronized components) that are invariant under any vector
field admissible for the graph. For a comprehensive study of the formalism of
these two contexts, we refer to [10] and [12], respectively. We now raise the
issue regarding a connection between synchrony and symmetries of the
automorphism group of the network graph. Not all polysynchronous subspaces are
robust; not all robust polysynchronous (synchrony) subspaces are fixed-point
subspaces of symmetries. In fact, even for regular graphs, there are
synchronies that do not come from symmetries of the graph automorphisms, and
this has been named exotic synchrony by the authors in [4]. We prove that, for
regular networks, all polysynchrony subspaces are generically robust
polysynchronous (Proposition 3.3). Furthermore, we present two classes of
regular graphs for which synchrony and symmetry coincide. One is the class of
regular rings (any number of cells) and the other is the class of the so-
called $G_{n}$-graphs ($5\leq n\leq 9$ and $n=11$); see Propositions 3.4 and
3.6. $G_{n}$ is the graph of $n$ cells connected by edges with nearest and
next nearest neighbors. Subsection 5.2 presents the study of singularities of
a Kuramoto network with coupling determined by the graph $G_{6}$.

As the first step to extend the simple Kuramoto model, we characterize all
vector fields whose Jacobian at any point is a Laplacian matrix (Theorem 4.6).
We also deduce that these belong to a subset of gradient fields, namely those
whose potential function operates on the differences between variables. In
particular, it follows that the dynamics is completely understood from the
study of critical points, from the LaSalle’s principle of invariance. The
results on stability of critical points then follows from an investigation of
signs of the Laplacian matrix eigenvalues that can be deduced from Theorem
4.1. Yet based on the additive nature of Kuramoto systems, in Subsection 4.3
we impose an additive structure on the couplings. Under this condition, each
component of the mapping being odd is a necessary condition (Theorem 4.11);
for the Kuramoto network, this is the sine function. As an advantage, this
structure provides a direct way to associate an admissible graph to this
mapping, in a unique way, from the steps given in the formalism of [3]. This,
in turn, enables us to use graph topology tools to study synchronization.

The authors in [9] show that for the simple Kuramoto network the total
synchrony is an asymptotically stable family of critical points. Here we show
that this also holds for the generalized Laplacian additive networks (Theorem
5.3). As mentioned above, the authors in [16] describe the critical points on
synchrony subspaces for gradient networks of identical cells on a ring with
the additional condition of $S^{1}$-invariance in the coupling function. We
notice that these turn out to be a particular case of Laplacian additive
networks; for the details we refer to [16, Section 4]. Here we present the
complete list of critical points and their stability estimates for two
examples of Laplacian additive network. As a first example, we consider a
network with six identical cells with couplings determined by the regular
network graph $G_{6}$, with the coupling function to be the sine function as
in the traditional Kuramoto model (Subsection 5.2). In the second example we
consider a homogeneous network with six identical cells and two types of
couplings, namely a combination of the sine and the identity functions.

Let us summarize how this paper is organized. In Section 2 we briefly review
the network formalism introduced in [20]. In Section 3 we review the notions
of balanced relation and robust polysynchrony, also introduced in [20], and we
discuss about invariant subspaces of a regular graph network dynamics
regarding their connection with synchrony and symmetry. Section 4 is devoted
to our two main results on Laplacian networks, given by Theorems 4.1 and 4.6,
where we also discuss the case of additive structure on the couplings.
Finally, in Section 5 we deduce Lyapunov stability for totally synchronous
equilibria in additive Laplacian networks (Theorem 5.3) and for two particular
examples we study the existence and nature of critical points on the remaining
synchrony subspaces.

##  2 Coupled cell networks

In this section we briefly recall the set up of the network structure of a
dynamical system defined by an autonomous system of ordinary differential
equations

$\dot{x}\ =\ f(x),\ \ x\in M,$ (2.1)

where $M$ is a smooth manifold, the state space or phase space, and $f$ is a
smooth vector field. To simplify exposition, we shall mostly assume
$M={\mathbb{R}}^{n}$, so we consider $f$ to be a map
$f:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$, and write
$f(x_{1},\ldots,x_{n})=\bigl{(}f_{1}(x_{1},\ldots,x_{n}),\ldots,f_{n}(x_{1},\ldots,x_{n})\bigr{)},$
but we point out that in Subsection 5.2, for example, $M$ is the 6-torus
${\mathbb{T}}^{6}$.

The network structure appears when a component $f_{c}$, governing the dynamics
of an individual cell $c$, depends on a subset of variables, the input
variables of cell $c$, and is independent of the remaining variables. In this
setting, the phase space is a vector space of the form
$P=P_{1}\times\cdots\times P_{n}$, where $x_{c}\in P_{c}$ may be
multidimensional. Here we assume that the cells are identical, so $P_{c}$ is
the same for all $c$. We say that cell $c$ is interacting with cells
associated with its input set of variables. The network is then represented by
a graph, with the set of vertices representing the set of cells and the edges
representing the interactions.

More precisely, a coupled cell network $\mathcal{G}$ consists of a finite set
of cells $\mathcal{C}=\\{1,\ldots,n\\}$ and a finite set of arrows (or edges)
$\mathcal{E}\subset\mathcal{C}\times\mathcal{C}$, both equipped with
equivalence relations $\sim_{\mathcal{C}}$ and $\sim_{\mathcal{E}}$,
satisfying the compatibility condition

$\displaystyle(c_{1},d_{1})\sim_{\mathcal{E}}(c_{2},d_{2})\Rightarrow
c_{1}\sim_{\mathcal{C}}c_{2},\ d_{1}\sim_{\mathcal{C}}d_{2}.$

For simplicity we assume that there is no arrow of the form $(c,c)$. Distinct
interactions in a coupled cell network define distinct
$\sim_{\mathcal{E}}$-classes of edges, so to each class
$\xi\in\mathcal{E}/{\sim_{\mathcal{E}}}$ there corresponds an adjacency matrix
$A^{\xi},$

$\displaystyle(A^{\xi})_{ij}=\left\\{\begin{array}[]{c}1,\mbox{ if
}(j,i)\in\xi\\\ 0,\mbox{ if }(j,i)\not\in\xi\end{array}\right.$ (2.4)

If the model requires, we similarly associate a weighted graph to the network,
in which case

$(A^{\xi})_{ij}=w_{ij},$ (2.5)

where $w_{ij}$ denotes the weight of (the class of) the edge $(j,i)$. In the
present paper, the interest lies on bidirected graphs; that is, $(i,j)$ is an
edge if, and only if $(j,i)$ is an edge of the same type; see the remark after
Definition 4.4. Hence, the adjacency matrices are symmetric.

We now establish the algebraic set up of a coupled cell network. Let

$I(c)=\\{e\in\mathcal{E}:\ e=(d,c)\\}$

denote the input set of a cell $c$. Another equivalence relation among cells
is given by an identification of input sets: $c$ and $d$ are said input
equivalent if there exists an arrow-type preserving bijection

$\displaystyle\beta\colon I(c)\rightarrow I(d),$

that is, $\beta(e)\sim_{\mathcal{E}}e,$ for all $e\in I(c).$ Let $B(c,d)$
denote the set of all bijections as above.

We end with the two main definitions for the forthcoming sections.

######  Definition 2.1.

A network graph $\mathcal{G}$ is homogeneous if all cells are $I$-equivalent,
in which case $B(c,d)\neq\emptyset,$ for all $c,d\in\mathcal{C}$. A network
graph is regular if it is homogeneous with one type of edge.

If $I(c)=\\{i_{1},\ldots,i_{v}\\},$ let us denote
$x_{{I}(c)}=(x_{i_{1}},\ldots,x_{i_{\nu}})$. Then:

######  Definition 2.2.

A map $f\colon P\to P$ is $\mathcal{G}$-admissible if:

  * (a)

(Domain condition) For every $c\in\mathcal{C}$, the component $f_{c}$ depends
only on $x_{c}$ and on the variables associated with ${I}(c)$. That is, there
exists $\hat{f}_{c}\colon P_{c}\times P_{I(c)}\rightarrow P_{c}$ such that

$f_{c}(x)=\hat{f}_{c}(x_{c},x_{I(c)}).$

  * (b)

(Pullback condition) For every input equivalent pair $c,d\in\mathcal{C}$ and
every $\beta\in B(c,d)$,

$\hat{f}_{d}(x_{d},x_{{I}(d)})=\hat{f}_{c}(x_{d},x_{I_{\beta(c)}}).$

We denote the vector space of admissible maps by ${\cal M}({\mathcal{G}},P)$,
or just ${\cal M}({\mathcal{G}})$ when $P$ is clear from the context.

##  3 Invariant subspaces in regular networks

Subspaces that remain unchanged under the dynamics hold significance. This
section focuses on synchrony patterns, as they constitute subspaces of
dynamics invariance in the network context. We initiate by briefly revisiting
the fundamental concepts.

In broad terms, synchrony is a natural concept in network dynamics related to
a partition of the cells into subsets (often called clusters) such that all
cells in the same cluster are synchronous. If synchrony occurs in the network
system (2.1), then two or more cells of a solution $x(t)$ behave identically,
that is, if $c$ and $d$ are any two of these cells, then

$x_{c}(t)=x_{d}(t),\,\forall t\in D,$

where $D$ is a domain in ${\mathbb{R}}$ for which $x(t)$ exists. The authors
in [11, 20] characterize occurrence of synchrony in a network dynamics from
the graph architecture. This is given as follows.

Given a network graph $\mathcal{G}$, for an edge $e\in{\cal E}$, let
$\mathcal{T}(e)$ denote the vertex given by the tail of $e$. Let $\bowtie$ be
an equivalence relation on $\mathcal{C}$. Then

  * •

$\bowtie$ is said to be balanced if $c\bowtie d$ implies that there exists
$\beta\in B(c,d)$ such that
$\mathcal{T}(e)\bowtie\mathcal{T}(\beta(e)),\,\forall e\in I(c)$;

  * •

$\bowtie$ is robustly polysynchronous if for any choice of total phase space
$P$ we have $f(\Delta_{\bowtie})\subseteq\Delta_{\bowtie}$, for all
$f\in\mathcal{M}(\mathcal{G})$, where

$\Delta_{\bowtie}=\\{x\in P\,|\,x_{c}=x_{d}\Leftrightarrow c\bowtie d\\}.$

We notice that the first of the notions above is associated with the graph
$\mathcal{G}$, whereas the second is related to the vector space
$\mathcal{M}(\mathcal{G})$, but it turns out that both notions are equivalent,
as established in [11, Theorem 4.3]. Moreover, the authors prove that these
are also equivalent to

$\displaystyle
A_{\mathcal{G}}^{\xi}(\Delta_{\bowtie})\subseteq\Delta_{\bowtie},\,\,\,\forall\xi\in\mathcal{E}/{\sim_{\mathcal{E}}},\
\ {\rm when}\ P={\mathbb{R}}^{n},$ (3.1)

that is, synchrony subspaces of ${\mathbb{R}}^{n}$ are those left invariant
under all adjacency matrices of $\mathcal{G}$.

######  Definition 3.1.

When $\bowtie$ is balanced, $\Delta_{\bowtie}$ is called a synchrony subspace.

The practical way to express the resulting $\bowtie$-classes of a synchrony
pattern is coloring vertices: cells in the same class have the same color and
‘receive from edges’ vertices of the same color.

######  Remark 3.2.

Based on (3.1), the authors in [1] develop, for a given network graph, an
algorithm whose output is the set of synchrony subspaces. This is done through
polydiagonal invariant subspaces from the eigenvectors of the Jordan
decomposition of the adjacency matrices. As they show, it suffices to
implement the algorithm for the case of regular graph, because all the lattice
of synchrony subspaces can be associated with a regular network through the
notion they introduce of minimal synchrony subspaces that are sum-irreducible.
Following this approach, we have recently implemented this algorithm with a
code using the software Mathematica. The code has been useful to compute the
synchrony patterns presented in [3] as well as all the synchrony patterns in
the present work.

###  3.1 Invariant subspaces from synchronies

In this subsection we relate synchrony subspaces of a network graph
$\mathcal{G}$ to invariant subspaces under the dynamics of (2.1) for
$f\in{\cal M}(\mathcal{G})$. As we have pointed out in [3], even for regular
graphs a coupled dynamical system may admit invariant polydiagonal subspaces
that are not patterns of synchrony. However, the next result shows that
generically those are in fact synchrony patterns.

######  Proposition 3.3.

For any regular network graph $\mathcal{G}$ on any phase space $P$,
polydiagonal subspaces of $P$ that are invariant under an admissible map are
generically synchrony subspaces.

###### Proof.

Based on (3.1) we assume, without loss of generality, that each cell phase
space is one dimensional, so $P={\mathbb{R}}^{n}$, where $n$ is the number of
cells. Since $\mathcal{G}$ is regular, the Jacobian of an admissible map $f$
at any point of total synchrony $\nu=(t,\ldots,t)\in\mathbb{R}^{n}$ is

$\displaystyle Jf(\nu)=\alpha(\nu)I+\beta(\nu)A,$ (3.2)

where $A$ is the adjacency matrix of $\mathcal{G}$. The result now follows
observing that $\nu\in\Theta$ for any polydiagonal $\Theta$ and taking the
generic condition as $b(\nu)\neq 0$ for some $\nu$. ∎

In Subsection 5.2 we investigate existence and nature of singularities inside
synchrony subspaces. For the ODE (5.3), it is straightforward that $b(\nu)=1$,
for any $\nu\in{\mathbb{T}}^{n}$, so the singularities we find are in fact all
the possible equilibria inside invariant subspaces, by Proposition 3.3.

###  3.2 Synchrony and symmetry in regular networks

Existence of symmetries and synchronies force existence of invariant
subspaces, because of their fixed-point sets and robustly polysynchronous
subspaces, respectively. These are related notions and we now turn to the
issue of synchronies with respect to symmetries of a network graph.

A symmetry of $\mathcal{G}$, or an automorphism of $\mathcal{G}$, is a
permutation $\gamma_{\mathcal{C}}\colon\mathcal{C}\rightarrow\mathcal{C}$ such
that

$(c,d)\sim_{\mathcal{E}}(\gamma_{\mathcal{C}}(c),\gamma_{\mathcal{C}}(d)),$

which also induces a permutation on $\mathcal{E}$. The group ${\rm
Aut}(\mathcal{G})$ of all such automorphisms is the symmetry group of
$\mathcal{G}$.

It is well known that symmetries of a network graph determine synchronies, for
every subgroup $\Sigma<{\rm Aut}(\mathcal{G})$ defines a balanced equivalence
relation on $\mathcal{G}$ ([4, Proposition 3.3]):

$c\bowtie_{\Sigma}d\Leftrightarrow\exists\gamma\in\Sigma:\gamma(c)=d.$

This is the same as saying that the synchrony subspace
$\Delta_{\bowtie_{\Sigma}}$ is the fixed-point subspace of $\Sigma$,

${\rm Fix}(\Sigma)=\\{x\in P\ :\ \sigma x=x,\ \forall\sigma\in\Sigma\\}.$

As it is also well known, the converse does not hold in general; that is, not
all synchrony subspaces are fixed-point subspaces. Such synchrony has been
named an exotic synchrony after [4]. a balanced relation $\bowtie$ is called
exotic if there is no subgroup $\Sigma<{\rm Aut}(\mathcal{G})$ for which
$\bowtie=\bowtie_{\Sigma}$.

Exotic patterns can appear in homogeneous graphs and even in the most simple
cases of regular graphs. The first example in the literature was presented in
[13], the regular graph $G_{12}$ of twelve vertices with nearest and next
nearest neighbors. Exotic synchronies also appear in [3] in the classification
of admissible homogeneous graphs of a class of vector fields on
${\mathbb{R}}^{6}$ (see [3, Table 2]). Figure 3.1 illustrates an exotic
pattern of the graph $\mathcal{G}$ which is number 6 in [3, Table 2]).
Disregard the colors in Fig. 3.1 for the picture of $\mathcal{G}$. We have
that ${\rm Aut}(\mathcal{G})$ is the group ${\mathbb{Z}}_{2}=\ <(12)(36)(45)>$
and the synchrony pattern is $\\{x_{1}=x_{4},x_{2}=x_{5},x_{3}=x_{6}\\}$,
which is clearly exotic.

Figure 3.1: A homogeneous graph with six identical cells and two types of
edges. The coloring shows an exotic synchrony.

There are however classes of network graphs for which all synchronies are
determined by the symmetries of ${\rm Aut}(\mathcal{G})$. For example, the
regular network graph ${\rm G}_{6}$ of six cells with nearest and next nearest
neighbor coupling of Subsection 5.2, which falls in the set of graphs given in
Proposition 3.6. The next proposition states that this is also the case for
the regular ring of $n$ cells with nearest neighbor coupling, for which ${\rm
Aut}(\mathcal{G})={\mathbb{D}}_{n}$. Although the result is not surprising,
the proof is not straightforward and, to our knowledge, it has not been found
in the literature, so it is included here.

######  Proposition 3.4.

There are no exotic pattern of synchrony for the regular ring of $n$ cells,
with nearest neighbor coupling, for $n\geq 3$.

###### Proof.

If $\bowtie$ is a balanced equivalence relation, we prove that $\bowtie\ =\
\bowtie_{\Sigma}$ for some subgroup $\Sigma<\mathbb{D}_{n}$.

The following is a necessary condition for the equivalence relation $\bowtie$
to be balanced for this network graph:

$\displaystyle c\bowtie d\quad\Rightarrow\left\\{\begin{array}[]{c}(c-1\bowtie
d-1,\,\,c+1\bowtie d+1)\\\ \mbox{ or }\\\ (c-1\bowtie d+1,\,\,c+1\bowtie
d-1).\end{array}\right.$ (3.6)

First, suppose that there are three consecutive $\bowtie$-equivalent cells. By
(3.6), all cells must be $\bowtie$-equivalent, and so $\bowtie\ =\
\bowtie_{\mathbb{D}_{n}}$.

We now suppose $c\bowtie d$ such that

$\displaystyle c-1\bowtie d-1,\,\,c+1\bowtie d+1.$ (3.7)

We need to identify $\Delta_{\bowtie}$ as a fixed-point subspace of a subgroup
$\Sigma<{\mathbb{D}_{n}}$, that is, find a rotation multiple of ${2\pi}/{n}$
or a line reflection generating $\Sigma$.

We can assume that $d-c>0$ is minimal with respect to (3.7). There is a
sequence of $\bowtie$-classes with length $d-c$ that repeats inside the ring.
In fact, by (3.6),

$c\bowtie d,\ c+1\bowtie d+1\Rightarrow c+2\bowtie
d+2\Rightarrow\ldots\Rightarrow c+(d-c)=d\bowtie d+(d-c).$

Hence, $d-c$ divides $n$ and then a rotation of ${2(d-c)\pi}/{n}$ is a
generator of $\Sigma$.

Finally, suppose that there are two cells $c^{\prime}\bowtie d^{\prime}$ such
that $c<c^{\prime}<d^{\prime}<d$ and by minimality of $d-c>0$ we have that
$c^{\prime}-1\bowtie d^{\prime}+1,\,\,c^{\prime}+1\bowtie d^{\prime}-1$. The
line reflection that takes $c^{\prime}$ into $d^{\prime}$ also takes
$c^{\prime}-1$ into $d^{\prime}+1$, $c^{\prime}+1$ into $d^{\prime}-1$ and so
on. Then, this reflection is a generator of $\Sigma$. ∎

######  Remark 3.5.

A study of a gradient network of identical oscillators on a ring is carried
out in [16]. We notice that in the description given in [16, Table 1] a
symmetry in one type of critical points in that table is missing. In fact,
inside their list of critical points with trivial isotropy, there is a
subclass with nontrivial isotropy, namely ${\mathbb{Z}}_{2}<{\mathbb{D}}_{n}$,
which is supported by Proposition 3.4 above. Nonetheless, we point out that
this does not affect the list of all possible critical points presented in
that table, which is in fact complete.

The following result gives another set of regular network graphs for which all
synchronies are symmetries:

######  Proposition 3.6.

There are no exotic pattern of synchrony for the network graph $G_{n}$, for
$5\leq n\leq 9$ and $n=11$.

###### Proof.

This follows by an extensive use of the Mathematica code; see Remark 3.2. ∎

The list of exotic patterns of $G_{10}$ is given in [4, Section 7], which we
have confirmed with our code. The network graph $G_{12}$ presents an exotic
pattern, as already mentioned above. Due to the computational complexity, we
have not investigated existence of exotic patterns in $G_{n}$ for $n\geq 13$
yet. Some attempts suggest that there may be a general procedure to answer the
question for bigger $n$ and we intend to go in this direction soon.

##  4 Laplacian networks

This section is devoted to the main object of this paper, namely the class of
networks whose admissible maps are attached to the associated graph not only
through its architecture but also through the Laplacian matrix of the graph.
As mentioned in the introduction, these generalize the traditional Kuramoto
model.

In Subsection 4.1 we recall the graph Laplacian matrix and present a result
from algebraic graph theory for Laplacian matrices. Subsection 4.2 is
dedicated to present the general algebraic expression of an admissible map of
a Laplacian network. Finally, the extra admissible additive structure behind
the Kuramoto networks is dealt with in Subsection 4.3.

###  4.1 Graph Laplacian matrix

For a bidirected graph $G$ with $n$ vertices, recall that the Laplacian matrix
is

$L=D-A,$

where $D$ is the diagonal matrix of the valencies of the vertices and $A$ is
the adjacency matrix. The $ii$-entry of $D$ is the degree of the vertex $i$ if
$G$ is unweighted, and ${\sum_{j=1}^{n}w_{ij}}$ if $G$ is a weighted graph.
$A$ is, as usual, the sum of the adjacency matrices (2.4) (for unweighted edge
classes) or (2.5) (for weighted edge classes).

It is direct from the definition that $0$ is an eigenvalue of $L$ with
eigenvector $(1,\ldots,1)$. For weighted connected graphs with possibly
negative weights, the authors in [5, Theorem 2.10] give the best possible
bounds on the numbers of positive, negative and zero eigenvalues. The next
result generalizes their result for disconnected graph, which is used in the
next sections.

Let $G_{+}$ (resp. $G_{-}$) denote the subgraph with the same vertex set as
$G$ together with the edges of positive (resp. negative) weights. Let
$n_{0},n_{-}$ and $n_{+}$ denote the numbers of zero, negative and positive
eigenvalues of the Laplacian $L$, and let $c(G)$ denote the number of
connected components of $G$.

######  Theorem 4.1.

If $G$ is a (possibly disconnected) graph, then

$\displaystyle c(G_{+})-c(G)\,\,\leq$ $\displaystyle n_{+}(L)$
$\displaystyle\leq\,\,n-c(G_{-})$ $\displaystyle c(G_{-})-c(G)\,\,\leq$
$\displaystyle n_{-}(L)$ $\displaystyle\leq\,\,n-c(G_{+})$ (4.1)
$\displaystyle c(G)\,\,\leq$ $\displaystyle n_{0}(L)$
$\displaystyle\leq\,\,n+2c(G)-c(G_{+})-c(G_{-}).$

###### Proof.

If $G$ has $k$ connected components, for $i=1,\ldots k$, let $G_{i}$ denote
the connected components with $n_{i}$ vertices, so $n_{1}+\ldots+n_{k}=n$.
Reorder the rows and columns of $L$ if necessary to assume that $L$ is a block
diagonal matrix

$L=diag(L_{1},\ldots,L_{k}).$

By the estimates in [5], for each $i=1,\ldots,k$ we have
$c(\mathcal{G}_{i+})-1\,\,\leq n_{+}(L_{i})\leq\,\,n_{i}-c(G_{i-})$, so

$\displaystyle c(G_{1+})+\ldots+c(G_{k+})-k\,\,$ $\displaystyle\leq$
$\displaystyle n_{+}(L_{1})+\ldots+n_{+}(L_{k})\leq$ $\displaystyle\leq$
$\displaystyle\,\,n_{1}+\ldots+n_{k}-(c(G_{1-})+\ldots+c(G_{k-})),$

so the first estimate follows. The other inequalities follow similarly. ∎

If the graph $G$ is connected, then $c(G)=1$ and (4.1) are the bounds given in
[5].

We notice that this result involves only topological information about the
graph, namely the connectivity of the graph and the sign information on the
edge weights. In particular, it follows that, for a graph with $n$ vertices,
the difference between the upper and lower bounds in (4.1) is an integer that
can vary between 0 and $n-1$.

###  4.2 Laplacian mappings

In this subsection we use the idea of inserting weights on an unweighted graph
$G$ to adapt the algebraic graph theory for weighted graphs to the associated
Laplacian network graph $\mathcal{G}$. Let us explain: for a given unweighted
graph $G$, we associate an admissible coupled system (2.1). For our case of
interest, the Jacobian $Jf(x)$ of the governing vector field is, at any point
$x$, a Laplacian matrix. The idea is to look at it as a weighted Laplacian of
$G$, considering $G$ as a weighted graph. Under this approach, we use Theorem
4.1.

We start with three definitions:

######  Definition 4.2.

A matrix $L=(l_{ij})$ of order $n$ is a Laplacian matrix if it is symmetric
with $l_{ii}=-\sum_{j\neq i}l_{ij}$, for $i=1,\ldots,n$.

Our interest here is to deduce an algebraic expression of a map $f:P\to P$ on
a real vector space $P$, so for simplicity we assume $P={\mathbb{R}}^{n}$.

######  Definition 4.3.

A map $f:{\mathbb{R}^{n}}\to{\mathbb{R}}^{n}$ of class $C^{1}$ is a Laplacian
map if its Jacobian $Jf(x)$ at any $x\in\mathbb{R}^{n}$ is a linear map whose
matrix is a Laplacian matrix.

The set of Laplacian maps shall be denoted by $LS(\mathbb{R}^{n})$.

It follows from the two definitions above that being bidirected is a necessary
assumption in this context. We have:

######  Definition 4.4.

For a bidirected graph, an associated network $\mathcal{G}$ is a Laplacian
network if the admissible map is a Laplacian map.

For the main result, we need:

######  Lemma 4.5.

If $h:{\mathbb{R}}^{n}\to{\mathbb{R}}$ is such that

$\displaystyle\sum_{i=1}^{n}\frac{\partial h}{\partial x_{i}}(x)=0.$ (4.3)

then

$h(x_{1},\ldots,x_{n})=\alpha(x_{1}-x_{n},\ldots,x_{n-1}-x_{n}),$ (4.4)

for some $\alpha:\mathbb{R}^{n-1}\to\mathbb{R}$.

###### Proof.

Consider the change of coordinates $t_{i}=x_{i}-x_{n},$ $i=1\ldots,n-1$,
$t_{n}=x_{n}$. We then have

$h(x_{1},\ldots,x_{n})=h(t_{1}+t_{n},\ldots,t_{n-1}+t_{n},t_{n})=\tilde{\alpha}(t_{1},\ldots,t_{n}).$

But

$\frac{\partial\tilde{\alpha}}{\partial t_{n}}=\sum_{i}\frac{\partial
h}{\partial x_{i}}=0.$

Hence, $\tilde{\alpha}(t_{1},\ldots,t_{n})=\alpha(t_{1},\ldots,t_{n-1}),$ and
the result follows. ∎

We now present the characterization of the general form of a Laplacian
mapping:

######  Theorem 4.6.

$f=(f_{1},\ldots,f_{n})\in LS(\mathbb{R}^{n})$ if, and only if,

$\displaystyle f_{i}(x_{1},\ldots,x_{n})$ $\displaystyle=$
$\displaystyle\dfrac{\partial g}{\partial
t_{i}}(x_{1}-x_{n},\ldots,x_{n-1}-x_{n}),\quad i=1,\ldots,n-1,$ $\displaystyle
f_{n}$ $\displaystyle=$ $\displaystyle-f_{1}-\ldots-f_{n-1}+k.$

for some $g\in C^{2}(\mathbb{R}^{n-1})$ and some constant $k\in\mathbb{R}$.

###### Proof.

From Lemma 4.5, we have

$\displaystyle f_{i}(x_{1},\ldots,x_{n})$ $\displaystyle=$
$\displaystyle\alpha_{i}(x_{1}-x_{n},\ldots,x_{n-1}-x_{n}),\quad
i=1,\ldots,n.$

The Jacobian $Jf(x)$ is symmetric, so

$\displaystyle\frac{\partial\alpha_{i}}{\partial
t_{j}}=\frac{\partial\alpha_{j}}{\partial t_{i}},\quad i,j=1,\ldots,n-1,i\neq
j,$ (4.5) $\displaystyle\frac{\partial\alpha_{n}}{\partial
t_{i}}=-\frac{\partial\alpha_{i}}{\partial
t_{1}}-\ldots-\frac{\partial\alpha_{i}}{\partial t_{n-1}}\quad
i=1,\ldots,n-1,$ (4.6)

which imply that

$\displaystyle\frac{\partial\alpha_{n}}{\partial
t_{i}}=-\frac{\partial\alpha_{1}}{\partial
t_{i}}-\ldots-\frac{\partial\alpha_{n-1}}{\partial
t_{i}}=\frac{\partial}{\partial t_{i}}(-\alpha_{1}-\ldots-\alpha_{n-1}),\quad
i=1,\ldots,n-1.$ (4.7)

Hence, $\alpha_{n}=-\sum_{i=1}^{n-1}\alpha_{i}+k$, for some constant $k.$

Finally, given any function $g\colon\mathbb{R}^{n-1}\rightarrow\mathbb{R}$ of
class $C^{2}$, we can take $\alpha_{i}=\dfrac{\partial g}{\partial t_{i}}$,
$i=1,\ldots,n-1$. Conversely, any map
$\alpha=(\alpha_{1},\ldots,\alpha_{n-1})$ of class $C^{1}$ satisfying (4.5) is
a gradient mapping. In fact, just set

$g(t_{1},\ldots,t_{n-1})=\sum_{i=1}^{n-1}\int_{0}^{t_{i}}\alpha_{i}(0,\ldots,0,s,t_{i+1},\ldots,t_{n-1})ds.$

∎

The example below is a simple illustration of Theorem 4.6.

######  Example 4.7.

For $n=3$, consider $g(y,z)=\dfrac{y^{2}z^{2}}{2}$. The following are the
coordinate functions of a Laplacian mapping
$f\colon\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$:

$\displaystyle f_{1}(x_{1},x_{2},x_{3})$ $\displaystyle=$
$\displaystyle(x_{1}-x_{3})(x_{2}-x_{3})^{2}$ $\displaystyle
f_{2}(x_{1},x_{2},x_{3})$ $\displaystyle=$
$\displaystyle(x_{1}-x_{3})^{2}(x_{2}-x_{3})$ $\displaystyle
f_{3}(x_{1},x_{2},x_{3})$ $\displaystyle=$
$\displaystyle-(x_{1}-x_{3})(x_{2}-x_{3})^{2}-(x_{1}-x_{3})^{2}(x_{2}-x_{3}).$

The next two corollaries follow straightforwardly:

######  Corollary 4.8.

The following linear isomorphism holds:

$LS(\mathbb{R}^{n})\
\simeq(C^{2}(\mathbb{R}^{n-1})/\mathbb{R})\oplus\mathbb{R}.$

######  Corollary 4.9.

If $f=(f_{1},\ldots,f_{n})\in LS(\mathbb{R}^{n})$, then $\sum_{i=1}^{n}f_{i}$
is constant.

We end with the characterization of the Laplacian maps, which is also direct
from Theorem 4.6:

######  Theorem 4.10.

$f=(f_{1},\ldots,f_{n})\in LS(\mathbb{R}^{n})$ if, and only if, $f$ is a
gradient map $f=\nabla\bar{g}$, where
$\bar{g}(x_{1},\ldots,x_{n})=g(x_{1}-x_{n},\ldots,x_{n-1}-x_{n})+kx_{n}$.

###  4.3 Laplacian networks with additive structure

Here we consider Laplacian admissible maps $f\in LS(\mathbb{R}^{n})$ with the
extra condition that each component $f_{c}$ of cell $c$ has an additive input
structure, namely

$f_{c}(x)=\sum_{d\neq c}\phi_{cd}(x_{c},x_{d}).$ (4.8)

It is worth mentioning that this structure provides the simplest way to
associate an admissible graph following the formalism in [3]: for such a map,
we consider the following coupling rule:

$\displaystyle(a,b)\sim_{\mathcal{E}}(c,d)\ \Leftrightarrow\
\phi_{ba}=\phi_{dc}.$ (4.9)

The question of realization of admissible graphs for a given map has been
answered in [3], and the notion of the optimized admissible graph for this map
is given. In this setting, we point out that, in the present context, the
condition (4.9) realizes a unique optimized admissible graph for the map as in
(4.8); see [3, Remark 3.1 b].

We notice that, by Lemma 4.5,

$\displaystyle
f_{n}(x_{1},\ldots,x_{n})=\alpha_{n}(x_{1}-x_{n},\ldots,x_{n-1}-x_{n})$ (4.10)

and from (4.8),

$\displaystyle\frac{\partial^{2}\alpha_{n}}{\partial t_{i}\partial
t_{j}}=\frac{\partial^{2}f_{n}}{\partial x_{i}\partial x_{j}}=0,\quad i\neq
j<n.$ (4.11)

Hence, $\alpha_{n}$ ‘splits the variables’ $x_{i}$ and $x_{j}$, $i\neq j<n$;
more precisely,

$f_{n}(x_{1},\ldots,x_{n})=\sum_{d<n}\phi_{nd}(x_{n}-x_{d}).$

By the same reasoning, the same condition holds for other components, and so

$f_{c}(x)=\sum_{d\neq c}\phi_{cd}(x_{c}-x_{d}),\quad\forall c=1,\ldots,n.$

Since the graph is bidirected, we have $(a,b)\sim_{\mathcal{E}}(b,a)$, and so
$\phi_{ba}=\phi_{ab}$. Furthermore, by the Laplacian condition,

$\phi_{ab}^{\prime}(x_{a}-x_{b})=-\dfrac{\partial f_{a}}{\partial
x_{b}}(x)=-\dfrac{\partial f_{b}}{\partial
x_{a}}(x)=\phi_{ba}^{\prime}(x_{b}-x_{a}).$

Therefore,

$\phi_{ab}^{\prime}(x_{a}-x_{b})=\phi_{ab}^{\prime}(-(x_{a}-x_{b})),$

that is, all coupling functions must be odd. We have just proved:

######  Theorem 4.11.

Let $\mathcal{G}$ be a bidirected graph network. If $f$ is a
$\mathcal{G}$-admissible map, then $f\in LS(\mathbb{R}^{n})$ has an additive
structure if, and only if, each component of $f$ is of the form

$\displaystyle f_{c}(x)=k_{[c]}+\sum_{d\in I(c)}\phi_{[(c,d)]}(x_{d}-x_{c}),$
(4.12)

where $\phi_{[(c,d)]}$ is an odd function that depends only on the
$[(c,d)]$-class of the edge $(c,d)$ and $k_{[c]}$ is a constant that depends
only on the $I$-class $[c]$ of $c$.

The two theorems above imply the following:

######  Corollary 4.12.

Let $\mathcal{G}$ be a bidirected graph network. Let $f$ be a
$\mathcal{G}$-admissible map. Then $f\in LS(\mathbb{R}^{n})$ has an additive
structure if, and only if, $f$ is a gradient mapping $f=-\nabla{g}$, where

${g}(x)=\sum_{(c,d)\in\mathcal{E}}\psi_{[(c,d)]}(x_{d}-x_{c})+\sum_{c\in\mathcal{C}}k_{[c]}x_{c},$

$\psi_{[(c,d)]}$ is an even function that depends only on the $[(c,d)]$-class
of the edge $(c,d)$ and $k_{[c]}$ is a constant determined by the $I$-class of
$c$.

The example below illustrates the theorem above. This is a nonhomogeneous
network graph with six cells, with two $I$-classes $\\{1,2,4,5\\}$ and
$\\{3,6\\}$, given in Fig. 4.1.

Figure 4.1:  A nonhomogeneous bidirected graph with six identical cells and
two types of edges.

######  Example 4.13.

For a bidirected graph with six identical cells and two types of edges as
given in Fig. 4.1, a Laplacian network $\mathcal{G}$ on ${\mathbb{R}}^{6}$
with additive structure is given by

$\dot{x}=f(x),$

where $f=(f_{1},\ldots,f_{6})\colon\mathbb{R}^{6}\rightarrow\mathbb{R}^{6},$

$\displaystyle f_{1}(x)$ $\displaystyle=$
$\displaystyle\kappa+\theta(x_{1}-x_{2})+\theta(x_{1}-x_{5})+\phi(x_{1}-x_{6})$
$\displaystyle f_{2}(x)$ $\displaystyle=$
$\displaystyle\kappa+\theta(x_{2}-x_{1})+\theta(x_{2}-x_{4})+\phi(x_{2}-x_{3})$
$\displaystyle f_{3}(x)$ $\displaystyle=$
$\displaystyle\ell+\phi(x_{3}-x_{2})+\phi(x_{3}-x_{4})$ $\displaystyle
f_{4}(x)$ $\displaystyle=$
$\displaystyle\kappa+\theta(x_{4}-x_{2})+\theta(x_{4}-x_{3})+\phi(x_{4}-x_{5})$
$\displaystyle f_{5}(x)$ $\displaystyle=$
$\displaystyle\kappa+\theta(x_{5}-x_{1})+\theta(x_{5}-x_{4})+\phi(x_{5}-x_{6})$
$\displaystyle f_{6}(x)$ $\displaystyle=$
$\displaystyle\ell+\phi(x_{6}-x_{1})+\phi(x_{6}-x_{5}),$

for $\theta,\phi$ any odd functions of class $C^{1}$ and $\kappa,\ell$
constant.

##  5 Critical points on synchrony subspaces of additive Laplacian networks

The starting point in the study of a dynamical behaviour of a system, or
bifurcations with variations of possible external parameters, is the analysis
of existence and stability of equilibrium points. Here we proceed in this
direction.

In Subsection 5.1 we prove that, for any homogeneous additive Laplacian
network, Lyapunov stability holds generically for totally synchronyous
critical points. In the last two subsections we choose two examples to search
for critical points with the remaining possible synchrony.

The following result ensures that the investigation of trajectory stability in
Laplacian networks is based on the eigenvalues of the Jacobian evaluated at
equilibrium points.

######  Lemma 5.1.

Let $f\in LS(\mathbb{R}^{n})$. The $\alpha$-limit and $\omega$-limit of any
trajectory are equilibrium points.

###### Proof.

From Theorem 4.10, $f\in LS(\mathbb{R}^{n})$ is gradient, so the result
follows from LaSalle’s invariance principle (see for example [19, Section
8.3]). ∎

In Subsection 5.2 we consider the homogeneous Kuramoto network of six cells
with coupling $G_{6}$ (identical edges connecting nearest and next nearest
neighbors). We list the critical points inside the remaining synchrony
subspaces. For each case, we give the estimate for the degree of linear
instability through the range of the number of positive eigenvalues. We end
with Subsection 5.3 doing the same analysis for a homogeneous coupled network
of six cells with two types of couplings.

###  5.1 Total synchrony subspace

Let $\mathcal{G}$ be a connected homogeneous graph with $n$ cells. Let $f$ be
an admissible vector field as in (4.12) . We search for critical points, so
$k=0$.

Suppose that there exists $\varepsilon>0$ such that

$\displaystyle\alpha\phi_{[(c,d)]}(\alpha)>0,\forall\alpha\in(-\varepsilon,\varepsilon),\alpha\neq
0.$ (5.1)

Notice that this holds if, in particular, $\phi^{\prime}_{[(c,d)]}(0)>0$, for
all $(c,d)\in\mathcal{E}$, which can be the case if the primitive of
$\phi_{[(c,d)]}$ has a local minimum at zero.

Let $\Omega$ be the biggest open hipercilinder around of the total synchrony
subspace $\Delta$ such that $x\in\Omega$ implies that
$|x_{d}-x_{c}|<\varepsilon$, for all $(c,d)\in\mathcal{E}$. We then have the
following:

######  Lemma 5.2.

  * (i)

$\Omega$ is a flow invariant set.

  * (ii)

for $x\in\Omega$, we have $f(x)=0$ if, and only if, $x\in\Delta$.

Before proving this lemma, we state the following theorem, which is an
immediate consequence of this Lemma, since $f$ is gradient (see Corollary
4.12).

######  Theorem 5.3.

Let $\mathcal{G}$ be a connected homogeneous graph. Let $f$ be an admissible
vector field as in (4.12) such that (5.1) holds. The total synchrony subspace
is asymptotically stable on $\Omega$ in the sense of Lyapunov.

######  Proof of Lemma 5.2.

(i) From Corollary 4.9, $f(x)\cdot(1,\ldots,1)=0$ for any $x$. Also, $f$ is
constant along the lines parallel to $\Delta$. So it suffices to show that
$\Omega\cap\pi$ is flow invariant, where $\pi$ is the hiperplane
$x\cdot(1,\ldots,1)=0$. To see this, as $\Omega\cap\pi$ is a hipersphere, we
just need to show that $f(x)\cdot x<0$ for all
$x\in\Omega\cap\pi\backslash\\{0\\}$. In fact,

$\displaystyle f(x)\cdot x$ $\displaystyle=$
$\displaystyle\sum_{c\in\mathcal{C}}\sum_{d\in
I(c)}x_{c}\phi_{[(c,d)]}(x_{d}-x_{c})$ (5.2) $\displaystyle=$
$\displaystyle\sum_{(c,d)\in\mathcal{E}}(x_{c}-x_{d})\phi_{[(c,d)]}(x_{d}-x_{c})\leq
0,$

where at least one portion is strictly less than zero. As $\mathcal{G}$ is
connect, (5.2) also shows (ii). ∎

We finally mention that if all the functions $\phi_{[(c,d)]}$ are periodic
with a comum period, we can restrict the analysis of critical points on the
$n$-torus.

###  5.2 Kuramoto network with $G_{6}$ coupling

We consider the Kuramoto network system with six cells and coupling $G_{6}$
(Fig. 5.1):

$\displaystyle\dot{x}_{i}=\sum_{j=-2}^{2}\sin({x_{i-j}-x_{i}}),\qquad
i=1,\ldots,6.$ (5.3)

Figure 5.1: $G_{6}$

We search for the critical points inside synchrony subspaces. It follows from
Proposition 3.6 that these are all fixed-point subspaces of subgroups of the
automorphism group $Aut(G_{6})$, which is the octahedral group. We consider
the representation given by permutations,

$Aut(G_{6})=\langle\mathbb{D}_{6},(14),(25)\rangle$

(see [4, Lemma 2.1]).

The computation has been carried out starting with the smallest subgroups
$\Sigma$ of $Aut(G_{6})$ up to conjugacy, for these will have the largest
fixed-point subspaces; and we then continue refining from there. More
precisely, we consider the diagram (5.12) with the lattice of fixed-point
subspaces, where the arrows represent the refinement by inclusion. The total
synchrony has been omitted in this diagram and, apart from it, this is the
complete lattice up to conjugacy. Based on this diagram, we start by
considering the subgroups $\Sigma$ such that Fix($\Sigma$) correspond to
patterns 1 and 3.

(5.12)

Without loss of generality, for all cases we assume that a critical point
$x=(x_{1},\ldots,x_{6})$ satisfies $x_{1}=0$.

For $\Sigma=\langle(14)\rangle$, we search for critical points of the form
$(0,a,b,0,c,d)$ for $a,b,c,d\in{\mathbb{R}}$ mod $2\pi$:

$\displaystyle 2\sin(-a)+\sin(b-a)+\sin(d-a)=0$ $\displaystyle
2\sin(-b)+\sin(a-b)+\sin(c-b)=0$ $\displaystyle
2\sin(-c)+\sin(b-c)+\sin(d-c)=0$ $\displaystyle
2\sin(-d)+\sin(a-d)+\sin(c-d)=0,$

which can be rewritten as

$\displaystyle 2\left[\begin{array}[]{c}\sin a\\\ \sin
c\end{array}\right]=(\sin b+\sin d)\left[\begin{array}[]{c}\cos a\\\ \cos
c\end{array}\right]-(\cos b+\cos d)\left[\begin{array}[]{c}\sin a\\\ \sin
c\end{array}\right]$ (5.19) $\displaystyle 2\left[\begin{array}[]{c}\sin b\\\
\sin d\end{array}\right]=(\sin a+\sin c)\left[\begin{array}[]{c}\cos b\\\ \cos
d\end{array}\right]-(\cos a+\cos c)\left[\begin{array}[]{c}\sin b\\\ \sin
d\end{array}\right]$ (5.26)

Case 1: If $(\sin a,\sin c),(\cos a,\cos c)$ are linearly independent, then
(5.19) implies that $\cos b+\cos d=-2$, and so $b=d=\pi$. Using that in
(5.26), we get $\sin a=-\sin c$, and so $c=-a$ or $a=c+\pi$. These give the
following two families of critical points:

$(0,a,\pi,0,-a,\pi),\ \ (0,a,\pi,0,a+\pi,\pi).$

If $(\sin b,\sin d),(\cos b,\cos d)$ are linearly independent, then
analogously we get the other two families

$(0,\pi,b,0,\pi,-b),\ \ (0,\pi,b,0,\pi,b+\pi).$

Case 2: If both pairs of the previous case are linearly dependent, then each
pair form a matrix with zero determinant, and so $\sin(c-a)=0$ and
$\sin(d-b)=0$. Hence, $c=a$ or $c=a+\pi$ and $b=d$ or $b=d+\pi$.

  * •

If $c=a$ and $d=b$, then $\sin a=\sin(b-a)$ and $\sin b=\sin(a-b)$. So the
critical points are

$(0,\pi,0,0,\pi,0),\ \
(0,\dfrac{4\pi}{3},\dfrac{2\pi}{3},0,\dfrac{4\pi}{3},\dfrac{2\pi}{3},0).$

  * •

If $c=a$ and $d=b+\pi$, then the critical points are

$(0,0,\pi,0,0,0),\ \ (0,\pi,b,0,\pi,b+\pi).$

For $\Sigma=\langle(12)(45)\rangle$, we search for critical points of the form
$(0,0,a,b,b,c)$, for $a,b,c\in{\mathbb{R}}$ mod $2\pi$:

$\displaystyle 2\sin(-a)+2\sin(b-a)=0$
$\displaystyle\sin(-b)+\sin(a-b)+\sin(c-b)=0$ $\displaystyle
2\sin(-c)+2\sin(b-c)=0.$

These give the following two families of critical points:

$(0,0,a,\pi,\pi,-a),\ \ (0,0,a,\pi,\pi,a+\pi).$

In [3] we have listed the eight patterns of synchrony of $G_{6}$ that are
distinct from the total synchrony pattern; see [3, Table1]. Here we follow
that order to present in Table 1 the compilation of all (families of)
equilibrium points inside the corresponding synchrony pattern.

We then apply Theorem 4.1 to the Jacobian matrix $Jf(x)$, at every critical
point $x$, thought of as a weighted Laplacian of the graph. We refer to the
beginning of Subsection 4.2 for the explanation.

For the critical point $x=(0,0,{\pi}/{2},\pi,\pi,{3\pi}/{2})$, the Jacobian is

$Jf(x)\ =\ \left(\begin{array}[]{rrrrrr}0&+1&0&0&-1&0\\\ +1&0&0&-1&0&0\\\
0&0&0&0&0&0\\\ 0&-1&0&0&+1&0\\\ -1&0&0&+1&0&0\\\
0&0&0&0&0&0\end{array}\right).$

Thought of as a graph Laplacian, the graph is as given in Fig. 5.2. We then
have $c(G)=3,$ $c(G_{+})=4,$ $c(G_{-})=4,$ which gives $1\leq n_{+}\leq 2.$
This is one of the possible cases of pattern number 3 of Table 1.

Figure 5.2: The disconnected graph obtained from $G_{6}$ by assigning $Jf(x)$
as the weighted Laplacian, for $x=(0,0,{\pi}/{2},\pi,\pi,{3\pi}/{2})$.

It follows that for all cases in Table 1 we have $n_{+}\geq 1$. Therefore, all
equilibria are unstable, and asymptotic stability occurs only for the total
synchrony subspace in the sense of Lyapunov as given in the previous
subsection.

$\sharp$ |  Pattern |  Critical point | $c(G_{+})$ | $c(G_{-})$ | $c(G)$ | $n_{+}({\cal L})$
---|---|---|---|---|---|---
|  of synchrony |  representative |  |  |  |
1 |  |  - | - | - | - | -
2 |  |  $(0,0,\pi,0,0,0)$ | 2 | 2 | 1 | $1\leq n_{+}\leq 4$
3 |  |  $(0,0,\alpha,\pi,\pi,-\alpha)$ $(0,0,\alpha,\pi,\pi,\alpha+\pi)$ $\alpha\neq 0,\pi,\pm\frac{\pi}{2}$ | 2 | 1 | 1 | $1\leq n_{+}\leq 5$
|  $(0,0,\pm\frac{\pi}{2},\pi,\pi,\mp\frac{\pi}{2})$ $(0,0,\pm\frac{\pi}{2},\pi,\pi,\pm\frac{\pi}{2}+\pi)$ | 4 | 4 | 3 | $1\leq n_{+}\leq 2$
4 |  |  $(0,\pi,\alpha,0,\pi,-\alpha)$ $(0,\pi,\alpha,0,\pi,\alpha+\pi)$ $\alpha\neq 0,\pi,\pm\frac{\pi}{2}$ | 2 | 1 | 1 | $1\leq n_{+}\leq 5$
|  $(0,\pi,\pm\frac{\pi}{2},0,\pi,\mp\frac{\pi}{2})$ $(0,\pi,\pm\frac{\pi}{2},0,\pi,\pm\frac{\pi}{2}+\pi)$ | 6 | 3 | 3 | $3\leq n_{+}\leq 3$
|  $(0,\pi,\pi,0,\pi,0)$ | 2 | 2 | 1 | $1\leq n_{+}\leq 4$
5 |  |  $(0,\pi,0,0,\pi,0)$ | 3 | 1 | 1 | $2\leq n_{+}\leq 5$
6 |  |  $(0,0,0,\pi,\pi,\pi)$ | 2 | 1 | 1 | $1\leq n_{+}\leq 5$
7 |  |  $(0,0,0,\pi,\pi,0)$ | 2 | 1 | 1 | $1\leq n_{+}\leq 5$
8 |  |  $(0,\frac{4\pi}{3},\frac{2\pi}{3},0,\frac{4\pi}{3},\frac{2\pi}{3})$ | 6 | 1 | 1 | $5\leq n_{+}\leq 5$
Table 1: The first column shows the synchrony patterns extracted from [3,
Table 1]; the corresponding class of critical points is given in the second
column by a representative up to the symmetries of $Aut(G_{6})$; the last
column gives the number $n_{+}$ of positive eigenvalues of the Jacobian at the
corresponding critical point; the remaining three columns are the data used to
apply Theorem 4.1.

###  5.3 A homogeneous network $\tilde{G}_{6}$ with two types of couplings

The aim of this example is to illustrate that there are stable equilibria from
a ‘modification’ of $G_{6}$. We consider a network system with six cells and
coupling defined from the graph $\tilde{G}_{6}$ given in Fig. 5.3. The
symmetry group of this graph is

$Aut(\tilde{G}_{6})\ =\
{\mathbb{Z}}_{2}^{1}\times{\mathbb{Z}}_{2}^{2}\times{\mathbb{Z}}_{3},$

where ${\mathbb{Z}}_{2}^{1}=\langle(15)(24)\rangle$ and
${\mathbb{Z}}_{2}^{2}=\langle(12)(36)(45)\rangle$, which are not conjugate,
and ${\mathbb{Z}}_{3}=\langle(156)(234)\rangle.$

We choose the coupling functions $\phi(\theta)=\sin\theta$, which is
represented by the continuous arrow type in Fig. 5.3, and
$\psi(\theta)=\theta$, represented by the dashed arrow type, so the network
system is

$\begin{array}[]{ll}\dot{x}_{1}=&\sin(x_{2}-x_{1})+\sin(x_{3}-x_{1})+x_{5}+x_{6}-2x_{1}\\\
\dot{x}_{2}=&\sin(x_{1}-x_{2})+\sin(x_{6}-x_{2})+x_{3}+x_{4}-2x_{2}\\\
\dot{x}_{3}=&\sin(x_{1}-x_{3})+\sin(x_{5}-x_{3})+x_{2}+x_{4}-2x_{3}\\\
\dot{x}_{4}=&\sin(x_{5}-x_{4})+\sin(x_{6}-x_{4})+x_{2}+x_{3}-2x_{4}\\\
\dot{x}_{5}=&\sin(x_{3}-x_{5})+\sin(x_{4}-x_{5})+x_{1}+x_{6}-2x_{5}\\\
\dot{x}_{6}=&\sin(x_{2}-x_{6})+\sin(x_{4}-x_{6})+x_{1}+x_{5}-2x_{6}\\\
\end{array}$

Figure 5.3: A homogeneous $\tilde{G}_{6}$

By direct investigation from the output of our code (Remark 3.2), there are no
exotic synchrony patterns in this network graph.

The analysis follows analogously as in the previous subsection, so we shall
just present a summary with the results. The diagram in (5.33) is the lattice
of non-trivial synchrony subspace up to conjugacy.

(5.33)

For $\Sigma=\langle(15)(24)\rangle$, we search for critical points of the form
$(0,a,b,a,0,c)$, for $a,b,c,\in{\mathbb{R}}$:

$\displaystyle\sin(-a)+\sin(c-a)+(b-a)=0$ $\displaystyle 2(a-b)+2\sin(-b)=0$
$\displaystyle-2c+\sin(a-c)=0,$

which imply that

$\sin b+b=a=c+\arcsin c,$

Since $t\mapsto\sin(t)+t$ is injective, then $c=\sin b$. Together with the
first equality, this gives $\sin a=-2\sin b,$ and so $\sin(b+\sin b)+2\sin
b=0$. Hence, $b=k\pi,k\in\mathbb{Z},$ and also $a=b$. Therefore, the critical
points are

$(0,k\pi,k\pi,k\pi,0,0),\quad\quad k\in\mathbb{Z}.$

For $\Sigma=\langle(14)(25)(36)\rangle$, we search for critical points of the
form $(0,a,b,0,a,b)$, $a,b\in{\mathbb{R}}$:

$\displaystyle\sin a+\sin b+a+b=0$ $\displaystyle\sin(b-a)+\sin(-a)-2b+a=0,$

which imply that $\sin a+a=\sin(-b)-b$, and then $a=-b$. Together with the
second equality, this gives $\sin(2a)+2a=\sin(-a)-a$, and then $a=b=0$. This
gives the total synchrony, which has already been considered.

For $\Sigma=\langle(12)(36)(45)\rangle$, we search for critical points of the
form $(a,a,0,b,b,0)$, $a,b\in{\mathbb{R}}$:

$\displaystyle b=2a+\sin a$ $\displaystyle a=2b+\sin b,$

which also imply $a=b=0$. It follows from the computation of $n_{+}$ and
$n_{-}$ by Theorem 4.1 that $(0,k\pi,k\pi,k\pi,0,0)$ is a one paramenter
family of stable equilibria for $k$ even and unstable if $k$ is odd.

Acknowledgments. TAA acknowledges financial support by FAPESP grant number
2019/2130-0. MM acknowledges financial support by FAPESP grant 2019/21181-0.

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