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# On network dynamical systems with a nilpotent singularity

Hildeberto Jardón-Kojakhmetov  Dynamical Systems, Geometry & Mathematical
Physics, University Of Groningen, Nijenborg 9, 9747 AG, Groningen,The
Netherlands  Christian Kuehn  School of Computation Information and
Technology, Department of Mathematics, Technical University of Munich,
Boltzmannstraße, 85748, Garching bei München, Germany

(2023; …)

###### Abstract

Network dynamics is nowadays of extreme relevance to model and analyze complex
systems. From a dynamical systems perspective, understanding the local
behavior near equilibria is of utmost importance. In particular, equilibria
with _at least_ one zero eigenvalue play a crucial role in bifurcation
analysis. In this paper, we want to shed some light on nilpotent equilibria of
network dynamical systems. As a main result, we show that the blow-up
technique, which has proven to be extremely useful in understanding degenerate
singularities in low-dimensional ordinary differential equations, is also
suitable in the framework of network dynamical systems. Most importantly, we
show that the blow-up technique preserves the network structure. The further
usefulness of the blow-up technique, especially with regard to the
desingularization of a nilpotent point, is showcased through several examples
including linear diffusive systems, systems with nilpotent internal dynamics,
and an adaptive network of Kuramoto oscillators.

###### keywords:

Nilpotent singularities; Network dynamical systems; Blow-up method; Geometric
desingularization.

##  1 Introduction

We consider a networked system of $\displaystyle N$ _scalar_ nodes
$\displaystyle x_{i}=x_{i}(t)\in\mathbb{R}$:

$\begin{split}\dot{x}_{i}=f_{i}(x_{i},\mu_{i})+\sum_{j=1}^{N}w_{ij}h_{ij}(x_{i},x_{j},\lambda_{ij}),\qquad
i=1,\ldots,N,\end{split}$ (1.1)

where $\displaystyle f_{i}$ and $\displaystyle h_{ij}$ are smooth functions
with some “internal” parameters $\displaystyle\mu_{i}\in\mathbb{R}^{u_{i}}$,
“interaction parameters” $\displaystyle\lambda_{ij}\in\mathbb{R}^{l_{ij}}$,
and “weights” $\displaystyle w_{ij}\in\mathbb{R}$. We refer to $\displaystyle
f_{i}$ as _the internal dynamics_ and to $\displaystyle h_{ij}$ as _the
interaction_. The topology of the network is encoded in $\displaystyle
w_{ij}$, which are the coefficients of a weighted adjacency matrix. In compact
form, we can write (1.1) as

$\dot{x}=F(x,\mu)+H(x,w,\lambda),$ (1.2)

where $\displaystyle F$ includes all the internal dynamics and $\displaystyle
H$ all the interactions; and with $\displaystyle x\in\mathbb{R}^{N}$,
$\displaystyle\mu\in\mathbb{R}^{u_{1}+\cdots+u_{N}}=\mathbb{R}^{u}$,
$\displaystyle w\in\mathbb{R}^{N^{2}}$ and
$\displaystyle\lambda\in\mathbb{R}^{l}$ with $\displaystyle
l=\sum_{i,j=1}^{N}l_{ij}$. We shall say that a system _has a network
structure_ [26, 30, 34] if it can be (re-)written in the form (1.1), or
equivalently (1.2). The main idea is that one can clearly distinguish the
internal, or uncoupled, dynamics from the network interaction. In the rest of
this paper, we shall concentrate on studying the dynamics near a nilpotent
equilibrium point of (1.1). We recall that given a differential equation
$\displaystyle\dot{x}=f(x)$, $\displaystyle x\in\mathbb{R}^{n}$, an
equilibrium point $\displaystyle x^{*}$ is said to be nilpotent if the
eigenvalues of the Jacobian $\displaystyle\textnormal{D}_{x}f(x^{*})$ are all
zero.

The main conclusion drawn from the analysis presented in this paper is that
_the blow-up transformation preserves the network structure_. We point out
that, a priori, there is no guarantee for a coordinate change to preserve any
sort of structure. This can be assessed by simply attempting a linear
transformation on e.g. (1.2). The blow-up (see section 2.1) is a singular
coordinate transformation that has been extensively used to desingularize the
local dynamics of (mostly) low dimensional differential equations. Some
further important observations are the following (for the details see section
2):

  * $\displaystyle\bullet$

A node-directional blow-up induces dynamics on the “blown-up” edges. However,
these dynamics occur at higher orders. That is, up to leading order terms, the
blown-up network is still static, see section 2.3.

  * $\displaystyle\bullet$

A parameter-directional blow-up is, essentially, a rescaling such that a
distinguished parameter is set to $\displaystyle 1$ and the rest are kept
small, see section 2.4.

  * $\displaystyle\bullet$

The blow-up seems to be especially useful for slowly adaptive networks, where,
for example, an edge-directional blow-up leads to a static network with a
distinguished edge fixed to $\displaystyle 1$. Moreover, the adaptation rule
that defines the dynamics of such distinguished edge is visible, globally, in
the whole network, see section 2.5.

Before going into the technical details, let us motivate our interest with
some examples.

###  1.1 Motivating examples

In this section, we motivate considering networked systems with nilpotent
equilibria through some examples.

Nilpotent internal dynamics:

Consider a weakly coupled network of the form (1.1) given by

$\begin{split}\dot{x}_{i}=f_{i}(x_{i},\mu_{i})+\varepsilon\sum_{j=1}^{N}w_{ij}h_{ij}(x_{i},x_{j},\lambda_{ij}),\qquad
i=1,\ldots,N,\end{split}$ (1.3)

where $\displaystyle 0<\varepsilon\ll 1$, and such that there is a fixed set
of parameters $\displaystyle\mu_{i}^{*}$, and a point $\displaystyle
x^{*}=(x_{1}^{*},\ldots,x_{N}^{*})$, such that $\displaystyle
f_{i}(x_{i}^{*},\mu_{i}^{*})=\frac{\partial f_{i}}{\partial
x_{i}}(x_{i}^{*},\mu_{i}^{*})=0$. In this case, the point $\displaystyle
x^{*}$ is nilpotent for $\displaystyle\varepsilon=0$ and an approach to
investigate the dynamics for $\displaystyle\varepsilon$ small could be via the
blow-up technique, as we shall present in this paper. See a concrete example
in section 3.2.

######  Remark 1.

Even though we focus on networks of scalar nodes, dynamic networks with
nilpotent internal dynamics frequently appear once the nodes are considered
multidimensional. For example, one may consider coupled neuronal models [32]
where each neuron modeled, e.g. via the Hodgkin-Huxley model or its
simplifications contains nilpotent points in its internal dynamics. In a more
general setting, nilpotent Hopf bifurcations have been considered for coupled
cell networks [8], see also [13, 12, 24]. In such a case, an adapted version
of [14] together with the blow-up analysis considered here could be applicable
as well.

A class of nonlinear consensus protocols:

Consider the general consensus protocol

$\dot{x}=-L\left(Ax+\sum_{i=2}^{p}F_{i}(x)\right),$ (1.4)

where $\displaystyle x\in\mathbb{R}^{N}$, $\displaystyle p\geq 2$,
$\displaystyle L$ is a Laplacian matrix, $\displaystyle A$ is a non-singular
matrix and $\displaystyle F_{i}$ is a homogeneous polynomial vector field of
degree $\displaystyle i$. The classical consensus protocol is obtained when
$\displaystyle A=I$ and $\displaystyle F_{i}=0$ for all $\displaystyle i\geq
2$. On the other hand, if $\displaystyle A=0$ we immediately have that the
origin is nilpotent. Furthermore, even if one simply considers
$\displaystyle\dot{x}=-LF_{2}(x)$, any $\displaystyle x^{*}$ so that
$\displaystyle F(x^{*})\in\ker L$ would correspond to a nilpotent point. Since
the kernel of $\displaystyle L$ is non-trivial, nilpotent points become
relevant in this situation. See [5] for a particular case study, and [16] for
a similar, but adaptive, example.

Canceling linear dynamics:

Consider (1.1) in the particular matrix form

$\dot{x}=Ax+Bx+\cdots,$ (1.5)

where $\displaystyle A$ is diagonal, accounting for the (leading part of the)
internal dynamics, and $\displaystyle B$ is a matrix describing the (leading
part of the) interaction; the dots stand for higher-order terms111Throughout
this document, “higher-order terms” means terms of order
$\displaystyle\mathcal{O}(|x|^{k})$ as $\displaystyle|x|\rightarrow 0$ for
some $\displaystyle k>1$ to be specified when appropriate.. One can then
realize that, upon variation of parameters, some components of these linear
parts may “cancel” in the sense of having a nilpotent form. Clearly, this does
not require $\displaystyle A=-B$ in general. For example, consider the
homogeneous case where $\displaystyle A=aI_{N}$ with $\displaystyle
a\in\mathbb{R}$. So, it suffices to notice that if $\displaystyle T$ is the
(complex) matrix that brings $\displaystyle B$ into its Jordan canonical form,
then the transformation $\displaystyle x\mapsto T^{-1}x$ implies
$\displaystyle T(aI_{N}+B)T^{-1}=aI_{N}+J_{B}$, where $\displaystyle J_{B}$ is
the Jordan canonical form of $\displaystyle B$, and is upper-triangular. In
turn, $\displaystyle aI_{N}+J_{B}$ can be nilpotent, or at least have some
zero eigenvalues, for $\displaystyle B\neq-A$. For an example see section 3.1,
and [27] for a related scenario.

(Slowly) adaptive networks:

As in the classical (low-dimensional) setting, nilpotent singularities are
extremely relevant for slow-fast systems [21, 36]. Consider the (slowly)
adaptive dynamic network [3, 31]

$\begin{split}\dot{x}_{i}&=f_{i}(x_{i},\mu_{i})+\sum_{j=1}^{N}w_{ij}h_{ij}(x_{i},x_{j},\lambda_{ij})\\\
\dot{w}_{ij}&=\varepsilon g_{ij}(x,w_{ij}).\end{split}$ (1.6)

If one were to attempt the analysis of (1.6) with techniques derived from
Geometric Singular Perturbation Theory (GSPT), the nilpotent points of (1.6)
restricted to $\displaystyle\varepsilon=0$ define a class of singularities
nearby which the solutions may exhibit a quite intricate behavior for
$\displaystyle 0<\varepsilon\ll 1$. As a particular example consider a family
of adaptive Kuramoto oscillators

$\begin{split}\phi_{i}^{\prime}&=\Omega-\frac{1}{N}\sum_{j=1}^{N}w_{ij}\sin(\phi_{i}-\phi_{j}+\alpha)\\\
w_{ij}^{\prime}&=-\varepsilon\left(\sin(\phi_{i}-\phi_{j}+\beta)+w_{ij}\right),\end{split}$
(1.7)

with parameters $\displaystyle\alpha,\beta$ close to zero. A phase-locked
equilibrium is given by

$\begin{split}\phi_{i}^{*}&=\psi_{i}\\\
w_{ij}^{*}&=-\sin(\psi_{i}-\psi_{j}+\beta)\\\
\Omega&=-\frac{1}{N}\sum_{j=1}^{N}\sin(\psi_{i}-\psi_{j}+\beta)\sin(\psi_{i}-\psi_{j}+\alpha),\end{split}$
(1.8)

for some $\displaystyle\psi_{i}\in[0,2\pi)$, $\displaystyle i=1,\ldots,N$. As
an example, let us take a 4-node network and
$\displaystyle\psi=(\psi_{1},\psi_{2},\psi_{3},\psi_{4})=\left(0,\frac{\pi}{2},\pi,\frac{3\pi}{2}\right)$.
For this choice of equilibrium phases, the phase-locked equilibrium is
nilpotent for $\displaystyle\alpha=\beta=0$ and
$\displaystyle\Omega=-\frac{1}{2}$. Indeed we have that the relevant part of
the linearization, at the aforementioned phase-locked equilibrium, is given by

$\begin{split}J&=\frac{1}{4}\begin{bmatrix}w_{13}&0&-w_{13}&0\\\
0&w_{24}&0&-w_{24}\\\ -w_{31}&0&w_{31}&0\\\
0&-w_{42}&0&w_{42}\end{bmatrix}.\end{split}$ (1.9)

So, further noting that $\displaystyle
w_{13}^{*}=w_{31}^{*}=w_{24}^{*}=w_{42}^{*}=0$ we have that $\displaystyle
J=0$, implying that the phase-locked equilibrium is nilpotent. This example is
further detailed in section 3.3.

######  Remark 2.

Adaptivity is, of course, not necessary. See for example [33] where nilpotent
equilibria of the classical Kuramoto model are considered.

As motivated in the previous examples, we are interested in network dynamical
systems (1.1), equivalently (1.2), with a nilpotent singularity. Nilpotent
points are highly important because they hint at the possibility of
complicated bifurcations occurring in their vicinity. From now on, we shall
assume that there is a set of parameters $\displaystyle\mu^{*}$,
$\displaystyle\lambda^{*}$ and edge weights $\displaystyle w^{*}$ such that
$\displaystyle x^{*}$ is an _isolated_ nilpotent equilibrium point of (1.2).

######  Remark 3.

Assuming $\displaystyle x^{*}$ being isolated is not too strict, at least for
our arguments regarding the blow-up transformation preserving the network
structure, because one can also blow-up higher-dimensional sets of equilibria
in a similar way. We present a brief description of the blow-up transformation
and several relevant references in section 2.1.

Without loss of generality, we further assume that the nilpotent point is at
$\displaystyle x^{*}=0$ for
$\displaystyle(\mu^{*},\lambda^{*},\omega^{*})=(0,0,0)$. However, we emphasize
that, as presented in the examples, a nilpotent point does not necessarily
have to be related to a disconnected network. Let $\displaystyle
a_{i}(\mu)=\frac{\partial f_{i}}{\partial x_{i}}(0,\mu)$, $\displaystyle
D(\mu)=\operatorname{diag}\left\\{a_{i}(\mu)\right\\}_{i=1}^{N}$, and
$\displaystyle A(\omega,\lambda)=\frac{\partial H}{\partial
x}(0,\omega,\lambda)$. So, we can write (1.2) as

$\dot{x}=(D(\mu)+A(\omega,\lambda))x+\tilde{F}(x,\mu)+\tilde{H}(x,w,\lambda),$
(1.10)

where, from our assumptions, $\displaystyle D(0)+A(0,0)$ is a nilpotent matrix
(and non of which needs to be the zero matrix) and $\displaystyle\tilde{F}$
and $\displaystyle\tilde{H}$ stand, respectively, for the higher-order terms
(by this we mean monomials of degree higher than one) of the internal dynamics
and of the interaction.

It is an important question whether one may indeed expect the leading term
$\displaystyle D(\mu)+A(\omega,\lambda)$ in (1.10) to be nilpotent for
parameters
$\displaystyle(\mu,\omega,\lambda)=(\mu^{*},\omega^{*},\lambda^{*})$. Roughly
speaking, since $\displaystyle n$ eigenvalues are required to be zero, we
expect that this can be achieved, generically, for $\displaystyle n$-parameter
families of dynamic networks (1.10). More formally, since $\displaystyle
GL(n)$ has dimension $\displaystyle n^{2}$ and since there are at most
$\displaystyle n(n-1)$ “different” nilpotent matrices, one expects that such
matrices appear generically whenever $\displaystyle D(\mu)+A(\omega,\lambda)$
has at least $\displaystyle n$ independent parameters. In other words, if
$\displaystyle D(\mu)+A(\omega,\lambda)$ has at least $\displaystyle n$
independent parameters, then there is at least one particular choice
$\displaystyle(\mu^{*},\omega^{*},\lambda^{*})$ such that $\displaystyle
D(\mu^{*})+L(\omega^{*},\lambda^{*})$ is nilpotent. Moreover, since the only
nilpotent symmetric matrix is the zero matrix, we shall consider that the
interaction is _directed_.

######  Remark 4.

We emphasize that we do not put $\displaystyle D(0)+A(0,0)$ in normal form
since, generally speaking, a similarity transformation destroys the network
structure of the system.

##  2 Blowing up preserves the network structure

Our aim in this section is to show that the (singular) coordinate
transformation known as blow-up, when applied to a nilpotent singularity of a
network dynamical system, preserves the network structure. Due to the
importance of the blow-up, although the literature contains already plenty of
information on it [1, 17, 21], we prefer to provide a brief recollection of
what is known as a blow-up in the way it is used in this paper. More
importantly, in this section, we provide certain terminology and fix some
notations that are later used in the examples of section 3.

###  2.1 The blow-up

Let $\displaystyle X:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a smooth vector
field222What follows also holds for smooth vector fields on smooth manifolds
by taking a chart in a neighborhood of the nilpotent point. such that
$\displaystyle X(0)=0$ and $\displaystyle 0\in\mathbb{R}^{n}$ is a nilpotent
point. Let $\displaystyle\phi:\mathbb{S}^{n-1}\times I\to\mathbb{R}^{n}$ be a
weighted polar transformation given by

$(\bar{x}_{1},\ldots,\bar{x}_{n},r)\mapsto(r^{\alpha_{1}}\bar{x}_{1},\ldots,r^{\alpha_{n}}\bar{x}_{n})=(x_{1},\ldots,x_{n}),$
(2.1)

where $\displaystyle\alpha_{i}\in\mathbb{N}$ for all $\displaystyle
i=1,\ldots,n$, $\displaystyle\sum_{i=1}^{n}\bar{x}_{i}^{2}=1$ (i.e.
$\displaystyle(\bar{x}_{1},\ldots,\bar{x}_{n})\in\mathbb{S}^{n-1}$) and
$\displaystyle r\in I\subset\mathbb{R}$ where $\displaystyle I$ is an interval
containing the origin. We notice that $\displaystyle\phi^{-1}$ maps the origin
$\displaystyle 0\in\mathbb{R}^{n}$ to the sphere
$\displaystyle\mathbb{S}^{n-1}\times\left\\{0\right\\}$ and we usually say
that “the origin is blown-up to a sphere”. On the other hand,
$\displaystyle\phi$ is a diffeomorphism whenever $\displaystyle r>0$. Due to
the weights in the transformation, one refers to the transformation as
quasihomogeneous (and homogeneous when all weights are the same). These
weights often facilitate the desingularization (see a description below), but
their appropriate choice can be quite complicated. If the diagram of figure 1
commutes, then we call $\displaystyle\bar{X}$ the blown-up vector field.

$\displaystyle\mathbb{S}^{n-1}\times
I$$\displaystyle\mathbb{R}^{n}$$\displaystyle T\mathbb{S}^{n-1}\times
I$$\displaystyle\mathbb{R}^{n}$$\displaystyle\phi$$\displaystyle\textnormal{D}\phi$$\displaystyle\bar{X}$$\displaystyle
X$ Figure 1: Commutative diagram defining a blow-up transformation.

Since $\displaystyle X(0)=0$, it follows that $\displaystyle\bar{X}|_{r=0}=0$.
However, if the weights are appropriately chosen, one can often
_desingularize_ $\displaystyle\bar{X}$ by dividing by some power of
$\displaystyle r$. In other words, one can define the _desingularized_ vector
field $\displaystyle\tilde{X}=\frac{1}{r^{k}}\bar{X}$ such that
$\displaystyle\tilde{X}(0)\neq 0$ and $\displaystyle\tilde{X}$ has semi-
hyperbolic singularities only (and no more nilpotent ones). Recalling that
$\displaystyle\phi$ is a diffeomorphism for $\displaystyle r>0$, the main
advantage now is that for $\displaystyle r>0$ small (i.e. near
$\displaystyle\mathbb{S}^{n-1}\times\left\\{0\right\\}$) one can recover the
dynamics of $\displaystyle\bar{X}$ from those of $\displaystyle\tilde{X}$, and
in turn these describe the dynamics of $\displaystyle X$ near the origin. In
practice, however, working with spherical coordinates can become cumbersome
very quickly. Hence, one prefers to work with _directional_ blow-ups, which
are defined by replacing $\displaystyle\mathbb{S}^{n-1}$ by its charts.
Moreover, blowing up using a sphere is not necessary, as one can blow-up, for
example, to a hyperbolic manifold [22].

Although the origins of the blow-up transformation are in algebraic geometry,
the blow-up transformation has proven extremely useful in dynamical systems to
study nilpotent singularities. It has been, for example, used to study
degenerate bifurcation problems and their unfoldings [35, 6], see also the
survey [1]; or, of particular relevance for this paper, to study some non-
hyperbolic points of slow-fast systems [7, 20], see also the survey [17].
Indeed, in the aforementioned context, one can consider nowadays that the
blow-up transformation is a standard tool of Geometric Singular Perturbation
Theory [21, 36]. Over the past decades, the blow-up transformation has been
mostly used for low-dimensional problems. In fact, it is known that analytic
vector fields up to dimension three can be desingularized after a finite
number of blow-ups [28, 29]. In this paper, we will see the usefulness of the
blow-up transformation on systems that are usually considered large-
dimensional, dynamic networks. As dynamic networks are, usually, of large
dimension, a relevant direction in this regard is the blow-up in the context
of PDEs [15, 19, 10, 9], which so far has been considered without networks
structure and, in some cases, can be regarded as mean-field limits of dynamic
networks [23, 11].

###  2.2 Structure preservation

This section is dedicated to showing that a general directional blow-up
preserves the network structure of a given dynamical system.

######  Remark 5.

As described in the previous section, usually one expects that after
desingularization via blow-up one gains a certain degree of hyperbolicity. The
number of blow-up transformations to completely desingularize a nilpotent
singularity is, however, not known a priori. This means that, after a blow-up,
some of the new singularities may be semi-hyperbolic or (even still)
nilpotent. This is nevertheless already good progress. In the former case,
further analysis can be restricted to a lower dimensional subset, the
corresponding center manifold, where the blow-up can be applied again. In the
latter, the process of blowing-up can also be repeated in a system that is,
usually, less degenerate than the starting one. Nevertheless, one expects that
after a finite number of blow-ups, one can fully describe the dynamics of a
system near a nilpotent singularity.

We emphasize that in this section, we only focus on structure preservation via
blowing up. The question of (full) desingularization is a nontrivial one and
is highly dependent on the particular problem at hand, and the choice of
weights in the quasihomogeneous blow-up (a nontrivial task in itself). We
discuss some cases where one can achieve desingularization later in the
examples of section 3.

Using the notation introduced above, let us consider a vector field of the
form

$X:\left\\{\begin{aligned}
\dot{x}&=F(x,\sigma)+H(x,\sigma)=(D_{\sigma}+A_{\sigma})x+\tilde{F}(x,\sigma)+\tilde{H}(x,\sigma)\\\
\dot{\sigma}&=0,\end{aligned}\right.$ (2.2)

where $\displaystyle F$ stands for the internal dynamics, $\displaystyle H$
the interaction, and $\displaystyle\sigma\in\mathbb{R}^{p}$ includes all
possible parameters, hence $\displaystyle D_{0}+A_{0}$ is nilpotent. _Our
objective is to show that a local vector field obtained by a directional blow-
up has the same identifiable structure of internal dynamics plus interaction_.

######  Remark 6.

The blow-up technique is local. Hence, without further mentioning it, we
assume that the vector fields we are considering are polynomial.

Let us define a quasihomogeneous blow-up via the map
$\displaystyle\Xi:\mathbb{R}_{\geq
0}\times\mathbb{S}^{N+p-1}\to\mathbb{R}^{N+p}$ given by:

$\Xi:(r,\bar{x}_{1},\ldots,\bar{x}_{n},\bar{\sigma}_{1},\ldots,\bar{\sigma}_{p})\mapsto\underbrace{(r^{\alpha_{1}}\bar{x}_{1},\dots,r^{\alpha_{n}}\bar{x}_{n},r^{\beta_{1}}\bar{\sigma}_{1},\ldots,r^{\beta_{p}}\bar{\sigma}_{p})}_{=(x_{1},\ldots,x_{N},\sigma_{1},\ldots,\sigma_{p})},$
(2.3)

and where

$\sum_{i=1}^{N}\bar{x}_{i}^{2}+\sum_{j=1}^{p}\bar{\sigma}_{j}^{2}=1.$ (2.4)

We can define a _directional_ blow-up by fixing one of the blown-up variables
in $\displaystyle\mathbb{S}^{N+p-1}$ to $\displaystyle\pm 1$ and let the rest
be coordinates in $\displaystyle\mathbb{R}^{n-1}$ (this is, of course,
reminiscent of a stereographic projection). The sign is, for now, rather
inessential, so let us consider charts

$\begin{split}K_{i}&=\left\\{\bar{x}_{i}=1\right\\},\qquad i=1,\ldots,N\\\
Q_{j}&=\left\\{\bar{\sigma}_{j}=1\right\\},\qquad j=1,\ldots,p.\end{split}$
(2.5)

Let us denote a corresponding directional blow-up by

$\begin{split}\Phi_{i}&=\Xi|_{\bar{x}_{i}=1},\\\
\Psi_{j}&=\Xi|_{\bar{\sigma}_{j}=1}.\end{split}$ (2.6)

The blown-up vector field in the chart $\displaystyle K_{i}$ (resp.
$\displaystyle Q_{j}$) is then obtained via the application of the
corresponding change of coordinates $\displaystyle\Phi_{i}$ (resp.
$\displaystyle\Psi_{j}$). For convenience, let

$\begin{split}\bar{X}_{i}&=(\textnormal{D}\Phi_{i})^{-1}X\circ\Phi_{i},\qquad
i=1,\ldots,N\\\
\bar{Y}_{j}&=(\textnormal{D}\Psi_{j})^{-1}X\circ\Psi_{j},\qquad
j=1,\ldots,p.\end{split}$ (2.7)

######  Remark 7.

  * $\displaystyle\bullet$

The quasi-homogeneous blow-up requires a careful choice of weights so that the
local vector fields are well defined. In this section, however, we are only
interested in the form the blown-up vector fields take and hence assume that
(2.7) are all well defined, especially at $\displaystyle r=0$ (possibly after
a division by some power of $\displaystyle r$).

  * $\displaystyle\bullet$

$\displaystyle\Phi_{i}$ corresponds to a blow-up of the node $\displaystyle
x_{i}$. Hence, within the context of networks, we refer to it as a “node blow-
up”. On the other hand, the need to blow-up in the direction of a parameter
arises when such a parameter is responsible for some singular behavior, for
example when it induces a nilpotent singularity. So, if for example,
$\displaystyle\sigma_{k}$ corresponds to some “weight parameter”
$\displaystyle w_{ij}$, we shall refer to $\displaystyle\Psi_{k}$ as an “edge
blow-up”. Of course, everything that we will present below can be extended to
network dynamical systems given by

$\begin{split}\dot{x}_{i}&=f_{i}+\sum_{j=1}^{N}w_{ij}h_{ij},\\\
\dot{w}_{ij}&=g_{ij},\end{split}$ (2.8)

where the “edge-directional blow-up” would be more apparent. This
consideration is, however, inessential for the purposes of the paper, which is
to show the preservation of the network structure after the blow-up. See more
details in section 2.5.

Since the compositions $\displaystyle X\circ\Phi_{i}$ and $\displaystyle
X\circ\Psi_{j}$ do not change the network structure, we only need to look at
the effect of pre-multiplication by the inverse of the derivative of the
directional blow-up.

###  2.3 Node directional blow-up

We first focus on the node blow-up, that is on the
$\displaystyle\bar{X}_{i}$’s. For convenience and simplicity of the
exposition, for each $\displaystyle i=1,\ldots,N$ let us reorder the
components of $\displaystyle X$ according to the coordinates

$\displaystyle(x_{i},x_{1},\ldots,x_{i-1},x_{i+1},\ldots,x_{N},\sigma_{1},\ldots,\sigma_{p}).$

We call such a vector filed $\displaystyle X_{i}$, but notice that we are
simply swapping the position of the $\displaystyle i$-th component333If (2.2)
is given by $\displaystyle X=\sum_{i=1}^{N}f_{i}\frac{\partial}{\partial
x_{i}}+\sum_{j=1}^{p}0\frac{\partial}{\partial\sigma_{j}}$, then
$\displaystyle X_{i}=f_{i}\frac{\partial}{\partial x_{i}}+\sum_{k=1,k\neq
i}^{N}f_{k}\frac{\partial}{\partial
x_{k}}+\sum_{j=1}^{p}0\frac{\partial}{\partial\sigma_{j}}.$. In this way the
corresponding directional blow-up $\displaystyle\Phi_{i}$ is given in
coordinate form by

$\begin{pmatrix}r^{\alpha_{i}}\\\ r^{\alpha_{1}}\bar{x}_{1}\\\ \vdots\\\
r^{\alpha_{i-1}}\bar{x}_{i-1}\\\ r^{\alpha_{i+1}}\bar{x}_{i+1}\\\ \vdots\\\
r^{\alpha_{N}}\bar{x}_{N}\\\ r^{\beta_{1}}\bar{\sigma}_{1}\\\ \vdots\\\
r^{\beta_{p}}\bar{\sigma}_{p}\end{pmatrix}=\begin{pmatrix}x_{i}\\\ x_{1}\\\
\vdots\\\ x_{i-1}\\\ x_{i+1}\\\ \vdots\\\ x_{N}\\\ \sigma_{1}\\\ \vdots\\\
\sigma_{p}\end{pmatrix},$ (2.9)

and therefore

$\textnormal{D}\Phi_{i}=\begin{bmatrix}\rho_{i}&0_{1\times(N-1)}&0_{1\times
p}\\\ \chi_{i}&R_{i}&0_{(N-1)\times p}\\\
\zeta&0_{p\times(N-1)}&B\end{bmatrix}$ (2.10)

where

$\begin{split}\rho_{i}&=\alpha_{i}r^{\alpha_{i}-1},\\\
\chi_{i}&=(\alpha_{1}r^{\alpha_{1}-1}\bar{x}_{1},\cdots,\alpha_{i-1}r^{\alpha_{i-1}-1}\bar{x}_{i-1},\alpha_{i+1}r^{\alpha_{i+1}-1}\bar{x}_{i+1},\cdots,\alpha_{N}r^{\alpha_{N}-1}\bar{x}_{N})^{\top},\\\
R_{i}&=\operatorname{diag}\left\\{r^{\alpha_{1}},\cdots,r^{\alpha_{i-1}},r^{\alpha_{i+1}},\cdots,r^{\alpha_{N}}\right\\},\\\
\zeta&=(\beta_{1}r^{\beta_{1}-1},\cdots,\beta_{p}r^{\beta_{p}-1}\bar{\sigma}_{p})^{\top},\\\
B&=\operatorname{diag}\left\\{r^{\beta_{1}},\cdots,r^{\beta_{p}}\right\\}.\end{split}$
(2.11)

Since $\displaystyle\textnormal{D}\Phi_{i}$ is lower triangular, its inverse
is also lower triangular. In fact, we have:

$(\textnormal{D}\Phi)^{-1}=\frac{1}{\rho_{i}}\begin{bmatrix}1&0_{1\times(N-1)}&0_{1\times
p}\\\ -R_{i}^{-1}\chi_{i}&\rho_{i}R_{i}^{-1}&0_{(N-1)\times p}\\\
-B^{-1}\zeta&0_{p\times(N-1)}&\rho_{i}B^{-1}.\end{bmatrix}$ (2.12)

######  Remark 8.

As expected, $\displaystyle\textnormal{D}\Phi$ is invertible for
$\displaystyle r\neq 0$.

Thus, the local vector field
$\displaystyle\bar{X}_{i}=(\textnormal{D}\Phi_{i})^{-1}X_{i}\circ\Phi_{i}$ can
be written in local coordinates444We recycle the coordinate notation in each
chart to avoid introducing obfuscating new terminology. So, for example, the
local coordinates in the chart $\displaystyle K_{i}$ are
$\displaystyle(r,\bar{x}_{1},\ldots,\bar{x}_{i-1},\bar{x}_{i+1},\ldots,\bar{x}_{N},\bar{\sigma}_{1},\ldots,\bar{\sigma}_{p})$,
and similarly the local coordinates in the chart $\displaystyle Q_{j}$ are
$\displaystyle(r,\bar{x}_{1},\ldots,\bar{x}_{N},\bar{\sigma}_{1},\ldots,\bar{\sigma}_{j-1},\bar{\sigma}_{j+1},\ldots,\bar{\sigma}_{p})$.
The distinction between these local coordinates is only necessary when
transitioning from one chart to another. Furthermore, along the text we use
the common notation $\displaystyle
r^{\alpha}\bar{x}=(r^{\alpha_{1}}\bar{x}_{1},\ldots,r^{\alpha_{N}}\bar{x}_{N})$,
which within the chart $\displaystyle K_{i}$ should be understood as
$\displaystyle
r^{\alpha}\bar{x}=(r^{\alpha_{i}},r^{\alpha_{1}}\bar{x}_{1},\ldots,r^{\alpha_{i-1}}\bar{x}_{i-1},r^{\alpha_{i+1}}\bar{x}_{i+1},r^{\alpha_{N}}\bar{x}_{N})$,
and similarly for $\displaystyle r^{\beta}\bar{\sigma}$. as

$\bar{X}_{i}:\begin{cases}r^{\prime}&=\frac{r^{1-\alpha_{i}}}{\alpha_{i}}\left(\bar{f}_{i}+\bar{h}_{i}\right),\\\
\bar{x}_{k}^{\prime}&=\frac{1}{r^{\alpha_{k}}}\left(\bar{f}_{k}+\bar{h}_{k}-\frac{\alpha_{k}}{\alpha_{i}}r^{\alpha_{k}-\alpha_{i}}\bar{x}_{k}(\bar{f}_{i}+\bar{h}_{i})\right),\\\
\bar{\sigma}_{j}^{\prime}&=-\frac{\beta_{j}}{\alpha_{i}}r^{-\alpha_{i}}\bar{\sigma}_{j}(\bar{f}_{i}+\bar{h}_{i}),\end{cases}$
(2.13)

where $\displaystyle k=1,\ldots,i-1,i+1,\ldots,N$, $\displaystyle
j=1,\ldots,p$ and where $\displaystyle\bar{f}_{\ell}$ and
$\displaystyle\bar{h}_{\ell}$ are the $\displaystyle\ell$-th component of
$\displaystyle F\circ\Phi_{i}$ and $\displaystyle H\circ\Phi_{i}$
respectively. Notice already that the equations for $\displaystyle\bar{x}_{k}$
keep, in some sense to be detailed below, the network structure.

Let $\displaystyle[M]_{ij}$ denote the $\displaystyle i,j$ entry of the matrix
$\displaystyle M$. Recall from (1.10) that $\displaystyle
F+H=(D_{\sigma}+A_{\sigma})x+\cdots$. This implies that, up to leading order
terms, we have:

$\begin{split}\bar{f}_{i}+\bar{h}_{i}&=(a_{i}(r^{\beta}\bar{\sigma})+[A_{r^{\beta}\bar{\sigma}}]_{ii})r^{\alpha_{i}}+\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{N}[A_{r^{\beta}\bar{\sigma}}]_{ij}r^{\alpha_{j}}\bar{x}_{j}+\cdots\\\
\bar{f}_{k}+\bar{h}_{k}&=[A_{r^{\beta}\bar{\sigma}}]_{ki}r^{\alpha_{i}}+(a_{k}(r^{\beta}\bar{\sigma})+[A_{r^{\beta}\bar{\sigma}}]_{kk})r^{\alpha_{k}}\bar{x}_{k}+\sum_{\begin{subarray}{c}j=1\\\
j\neq i\\\ j\neq
k\end{subarray}}^{N}[A_{r^{\beta}\bar{\sigma}}]_{kj}r^{\alpha_{j}}\bar{x}_{j}+\cdots,\end{split}$
(2.14)

which hints to the choice of the blow-up weights
$\displaystyle\alpha_{i}=\alpha$ for all $\displaystyle i=1,\dots,N$ so that
every local vector field $\displaystyle\bar{X}_{i}$ is well defined for
$\displaystyle r=0$. Under such a choice we have, up to leading order terms,
the blown-up vector field $\displaystyle\bar{X}_{i}$ given by:

$\begin{split}r^{\prime}&=\frac{r}{\alpha}\left((a_{i}(r^{\beta}\bar{\sigma})+[A_{r^{\beta}\bar{\sigma}}]_{ii})+\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{N}[A_{r^{\beta}\bar{\sigma}}]_{ij}\bar{x}_{j}+\cdots\right)\\\
\bar{x}_{k}^{\prime}&=\left([A_{r^{\beta}\bar{\sigma}}]_{ki}+(a_{k}(r^{\beta}\bar{\sigma})+[A_{r^{\beta}\bar{\sigma}}]_{kk})\bar{x}_{k}+\sum_{\begin{subarray}{c}j=1\\\
j\neq i\\\ j\neq
k\end{subarray}}^{N}[A_{r^{\beta}\bar{\sigma}}]_{kj}\bar{x}_{j}+\cdots\right)\\\
&\qquad-\bar{x}_{k}\left((a_{i}(r^{\beta}\bar{\sigma})+[A_{r^{\beta}\bar{\sigma}}]_{ii})+\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{N}[A_{r^{\beta}\bar{\sigma}}]_{ij}\bar{x}_{j}+\cdots\right)\\\
\bar{\sigma}_{j}^{\prime}&=-\frac{\beta_{j}}{\alpha}\bar{\sigma}_{j}\left((a_{i}(r^{\beta}\bar{\sigma})+[A_{r^{\beta}\bar{\sigma}}]_{ii})+\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{N}[A_{r^{\beta}\bar{\sigma}}]_{ij}\bar{x}_{j}+\cdots\right).\end{split}$
(2.15)

For $\displaystyle r=0$ we have that $\displaystyle r^{\prime}=0$ and

$\begin{split}\bar{x}_{k}^{\prime}&=[A_{0}]_{ki}+(a_{k}(0)+[A_{0}]_{kk}-a_{i}(0)-[A_{0}]_{ii})\bar{x}_{k}+\sum_{\begin{subarray}{c}j=1\\\
j\neq i\\\ j\neq
k\end{subarray}}^{N}[A_{0}]_{kj}\bar{x}_{j}+\mathcal{O}(|\bar{x}|^{2})\\\
\bar{\sigma}_{j}^{\prime}&=-\frac{\beta_{j}}{\alpha}\left(a_{i}(0)+[A_{0}]_{ii}\right)\bar{\sigma}_{j}+\mathcal{O}(\bar{\sigma}_{j}\bar{x}),\end{split}$
(2.16)

where $\displaystyle k=1,\ldots,i-1,i+1,\ldots,N$. We can see that the blown-
up system (2.15) has “dynamic” parameters $\displaystyle\bar{\sigma}$.
However, for $\displaystyle r=0$ (i.e. (2.16)) the network is static.
Moreover, we see that the equilibrium point of $\displaystyle\bar{x}^{\prime}$
may not be at the origin any more.

Interpretation:

one can give several network interpretations to the equation for
$\displaystyle\bar{x}^{\prime}$ in (2.16). For this, it is convenient to
rewrite the first part of (2.16) as

$\bar{x}_{k}^{\prime}=[A_{0}]_{ki}+(a_{k}(0)-a_{i}(0)-[A_{0}]_{ii})\bar{x}_{k}+\sum_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{N}[A_{0}]_{kj}\bar{x}_{j}+\cdots,$ (2.17)

where we can notice a network with $\displaystyle N-1$ nodes
$\displaystyle(\bar{x}_{1},\ldots,\bar{x}_{i-1},\bar{x}_{i+1},\ldots,\bar{x}_{N})$.

  * $\displaystyle\bullet$

As a first interpretation, one may say that the internal dynamics are now
modified (“shifted” by $\displaystyle[A_{0}]_{ki}$ and “scaled” by
$\displaystyle[A_{0}]_{ii}$) by the influence of node $\displaystyle i$ (which
has been blown-up and “does not exists anymore”). Regarding the interaction,
that remains the same, except that the node $\displaystyle i$ does not appear
anymore.

  * $\displaystyle\bullet$

Another interpretation could be given when we re-write (2.16) as:

$\bar{x}_{k}^{\prime}=[A_{0}]_{ki}+a_{k}(0)\bar{x}_{k}+\sum_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{N}[\tilde{A}_{0}]_{kj}\bar{x}_{j}+\cdots,$ (2.18)

where $\displaystyle[\tilde{A}_{0}]_{kj}=-a_{i}(0)-[L_{0}]_{ii}+[A_{0}]_{kk}$
if $\displaystyle j=k$ and $\displaystyle[\tilde{A}_{0}]_{kj}=[A_{0}]_{kj}$
otherwise. This, instead can be interpreted as the “self-interaction” having
been modified by the (nonexistent) node $\displaystyle i$, while the term
$\displaystyle[A_{0}]_{ki}$ can be regarded as a constant input, which is
nonzero if and only if there is a connection from node $\displaystyle i$ to
node $\displaystyle k$.

Finally, if $\displaystyle r=0$ is hyperbolic in the $\displaystyle
r$-direction (which would be the case under desingularization), then we expect
that (2.15) is a regular perturbation of (2.16) for $\displaystyle r>0$
sufficiently small. This means that the dynamics of (2.2) in a small
neighborhood of the nilpotent origin can be determined from to those of
(2.16), see a relevant example in section 3.

###  2.4 Parameter directional blow-up

We now turn our attention to the blow-up in the parameter direction (the
$\displaystyle\bar{Y}_{i}$’s in (2.7)). For convenience, let us recall that we
are considering the (extended) vector field

$X:\left\\{\begin{aligned}
\dot{x}&=F(x,\sigma)+H(x,\sigma)=(D_{\sigma}+L_{\sigma})x+\tilde{F}(x,\sigma)+\tilde{H}(x,\sigma)\\\
\dot{\sigma}&=0,\end{aligned}\right.$ (2.19)

where $\displaystyle L_{0}+D_{0}$ is nilpotent. If one would rewrite (2.19) as
$\displaystyle X=\sum_{i=1}^{N}f_{i}\frac{\partial}{\partial
x_{i}}+\sum_{j=1}^{p}0\frac{\partial}{\partial\sigma_{j}}$, it is rather
convenient to consider the vector field

$Y_{j}=0\frac{\partial}{\partial\sigma_{j}}+\sum_{i=1}^{N}f_{i}\frac{\partial}{\partial
x_{i}}+\sum_{k=1,\,k\neq j}^{p}0\frac{\partial}{\partial\sigma_{k}},$ (2.20)

with the directional blow-up $\displaystyle\Psi_{j}$ given by local
coordinates:

$\begin{pmatrix}r^{\beta_{j}}\\\ r^{\alpha}\bar{x}_{1}\\\ \vdots\\\
r^{\alpha}\bar{x}_{N}\\\ r^{\beta_{1}}\bar{\sigma}_{1}\\\ \vdots\\\
r^{\beta_{j-1}}\bar{\sigma}_{j-1}\\\ r^{\beta_{j+1}}\bar{\sigma}_{j+1}\\\
\vdots\\\
r^{\beta_{p}}\bar{\sigma}_{p}\end{pmatrix}=\begin{pmatrix}\sigma_{j}\\\
x_{1}\\\ \vdots\\\ x_{N}\\\ \sigma_{1}\\\ \vdots\\\ \sigma_{j-1}\\\
\sigma_{j+1}\\\ \vdots\\\ \sigma_{p}\end{pmatrix}.$ (2.21)

Recall that from the previous section we have already chosen the blow-up
weights $\displaystyle\alpha_{i}=\alpha$ for all $\displaystyle i=1,\ldots,N$.
Therefore

$\textnormal{D}\Psi_{j}=\begin{bmatrix}\rho_{j}&0_{1\times
N}&0_{1\times(p-1)}\\\ \chi&R&0_{N\times(p-1)}\\\ \zeta_{j}&0_{(p-1)\times
N}&B_{j}\end{bmatrix},$ (2.22)

where

$\begin{split}\rho_{j}&=\beta_{j}\bar{\sigma}_{j}^{\beta_{j}-1}\\\
\chi&=(\alpha r^{\alpha-1}\bar{x}_{1},\ldots,\alpha
r^{\alpha-1}\bar{x}_{N})^{\top}\\\
R&=\operatorname{diag}\left\\{r^{\alpha},\ldots,r^{\alpha}\right\\}\\\
\zeta_{j}&=(\beta_{1}r^{\beta_{1}-1}\bar{\sigma}_{1},\ldots,\beta_{j-1}r^{\beta_{j-1}-1}\bar{\sigma}_{j-1},\beta_{j+1}r^{\beta_{j+1}-1}\bar{\sigma}_{j+1},\ldots,\beta_{p}r^{\beta_{p}-1}\bar{\sigma}_{p})^{\top}\\\
B_{j}&=\operatorname{diag}\left\\{r^{\beta_{1}},\ldots,r^{\beta_{j-1}},r^{\beta_{j+1}},\ldots,r^{\beta_{p}}\right\\}.\end{split}$
(2.23)

Following similar steps as in the previous section, we find that the
corresponding local vector field (in the chart $\displaystyle Q_{j}$) reads as

$\bar{Y}_{j}=\begin{cases}r^{\prime}&=0\\\
\bar{x}_{i}^{\prime}&=\frac{1}{r^{\alpha}}(\bar{f}_{i}+\bar{h}_{i})\\\
\bar{\sigma}_{k}^{\prime}&=0,\end{cases}$ (2.24)

where $\displaystyle k=1,\ldots,j-1,j+1,\ldots,p$, and
$\displaystyle\bar{f}_{\ell}$ and $\displaystyle\bar{h}_{\ell}$ are the
$\displaystyle\ell$-th component of $\displaystyle F\circ\Psi_{j}$ and
$\displaystyle H\circ\Psi_{j}$ respectively. Again, it is evident that the
blown-up vector field $\displaystyle\bar{Y}_{j}$ still has a network
structure. More specifically we have:

$\begin{split}\bar{f}_{i}&=a_{i}(r^{\beta}\bar{\sigma})r^{\alpha}\bar{x}_{i}+\mathcal{O}(r^{2\alpha})\\\
\bar{h}_{i}&=\sum_{k=1}^{N}[A_{r^{\beta}\bar{\sigma}}]r^{\alpha}\bar{x}_{k}+\mathcal{O}(r^{2\alpha}).\end{split}$
(2.25)

We recall that in the chart $\displaystyle Q_{j}$ we use the short-hand
notation

$\displaystyle
r^{\beta}\bar{\sigma}=(r^{\beta_{1}}\bar{\sigma}_{1},\ldots,r^{\beta_{j-1}}\bar{\sigma}_{j-1},r^{\beta_{j}},r^{\beta_{j+1}}\bar{\sigma}_{j+1},\ldots,r^{\beta_{p}}\bar{\sigma}_{p}).$

Interpretation:

in this case the blow-up is essentially a rescaling, where one wants to focus
on the influence of the particular parameter $\displaystyle\sigma_{j}$. After
successful desingularization (which depends on the vector fields and the
choice of blow-up weights), the blown-up vector field
$\displaystyle\bar{Y}_{j}$ is, qualitatively speaking, $\displaystyle Y_{j}$
with the parameter $\displaystyle\bar{\sigma}_{j}=1$ (or
$\displaystyle\bar{\sigma}_{j}=-1$ depending on the direction of the blow-up).
See the example in section 3.3.

From the above insight, since a parameter $\displaystyle\sigma_{i}\in\sigma$
could be an edge weight, and due to its relevance in applied sciences, we now
discuss the edge-directional blow-up in the context of adaptive networks.

###  2.5 Edge-directional blow-up for adaptive networks

Let us now consider a slowly adaptive network

$\begin{split}\dot{x}_{i}&=f_{i}(x_{i})+\sum_{j=1}^{N}w_{ij}h_{ij}(x_{i},x_{j})\\\
\dot{w}_{ij}&=\varepsilon g_{ij}(x,w_{ij}).\end{split}$ (2.26)

######  Remark 9.

Frequently, the adaptation rule $\displaystyle g_{ij}$ in applications depends
only on the nodes $\displaystyle(i,j)$, see e.g. [3]; but for our purposes
such specification is not relevant.

Consider an edge-directional blow-up for the edge $\displaystyle(k,l)$ (for
simplicity we only consider the positive direction)

$x_{i}=r\bar{x}_{i},\,w_{ij}=r\bar{w}_{ij},\,\varepsilon=r\bar{\varepsilon},\,w_{kl}=r,\qquad
ij\neq kl.$ (2.27)

######  Remark 10.

In this section, we only focus on the preservation of network structure and
its interpretation. If more details are known from the particular functions in
(2.26), then a more appropriate choice of blow-up (2.27) can be made.

The corresponding blown-up system reads as:

$\begin{split}\dot{r}&=r\bar{\varepsilon}\bar{g}_{kl}(r,\bar{x})\\\
\dot{\bar{x}}_{i}&=\frac{\bar{f}_{i}(r,\bar{x}_{i})}{r}+\sum_{j=1}^{N}\bar{w}_{ij}\bar{h}_{ij}(r,\bar{x}_{i},\bar{x}_{j})-\bar{\varepsilon}\bar{x}_{i}\bar{g}_{kl}(r,\bar{x}),\qquad
i\neq k\\\
\dot{\bar{x}}_{k}&=\frac{\bar{f}_{k}(r,\bar{x}_{k})}{r}+\sum_{j=1,j\neq
l}^{N}\bar{w}_{kj}\bar{h}_{kj}(r,\bar{x}_{k},\bar{x}_{j})+\bar{h}_{kl}(r,\bar{x}_{k},\bar{x}_{l})-\bar{\varepsilon}\bar{x}_{k}\bar{g}_{kl}(r,\bar{x}),\\\
\dot{\bar{w}}_{ij}&=\bar{\varepsilon}(\bar{g}_{ij}(r,\bar{x},\bar{w}_{ij})-\bar{w}_{ij}\bar{g}_{kl}(r,\bar{x})),\quad
ij\neq kl\\\
\dot{\bar{\varepsilon}}&=-\bar{\varepsilon}^{2}\bar{g}_{kl}(r,\bar{x})\end{split}$
(2.28)

The dynamics restricted to the invariant subset
$\displaystyle\left\\{\bar{\varepsilon}=0\right\\}$ are often useful. One
accordingly has that (2.28) reduces to

$\begin{split}\dot{r}&=0\\\
\dot{\bar{x}}_{i}&=\frac{\bar{f}_{i}(r,\bar{x}_{i})}{r}+\sum_{j=1}^{N}\bar{w}_{ij}\bar{h}_{ij}(r,\bar{x}_{i},\bar{x}_{j}),\qquad
i\neq k\\\
\dot{\bar{x}}_{k}&=\frac{\bar{f}_{k}(r,\bar{x}_{k})}{r}+\sum_{j=1,j\neq
l}^{N}\bar{w}_{kj}\bar{h}_{kj}(r,\bar{x}_{k},\bar{x}_{j})+\bar{h}_{kl}(r,\bar{x}_{k},\bar{x}_{l}),\\\
\dot{\bar{w}}_{ij}&=0,\quad ij\neq kl\\\ \end{split}$ (2.29)

which is the model of a static network, with weights
$\displaystyle\bar{w}_{ij}$ and fixed weight $\displaystyle\bar{w}_{kl}=1$. If
$\displaystyle\bar{\varepsilon}$ remains small, then (2.28) can be interpreted
as a small perturbation of the static model (2.29). In this case, we notice
that the weights $\displaystyle\bar{w}_{ij}$, $\displaystyle ij\neq kl$, would
evolve slowly, while $\displaystyle\bar{w}_{kl}$ is still fixed to
$\displaystyle 1$. Moreover, we notice that the adaptation effects of
$\displaystyle w_{kl}$ are “globally transported” to the dynamics of the
nodes. This may have important applications to elucidate the influence of
certain edges on the rest of the network. If (2.29) has been desingularized,
then the perturbation (2.28) would be regular.

##  3 Examples

In this section, we present several examples that highlight the use of the
blow-up for network dynamical systems. In particular, we focus on network
structure preservation and the possibility of desingularization after blow-up.

###  3.1 Linearly Diffusive systems

Our first example concerns a set of diffusively coupled nodes with linear
internal dynamics:

$\dot{x}_{i}=a_{i}x_{i}+\sum_{j=1}^{N}w_{ij}(x_{j}-x_{i}),$ (3.1)

with parameters $\displaystyle a_{i}\in\mathbb{R}$, $\displaystyle a_{i}\neq
0$, and $\displaystyle w_{ij}\in\mathbb{R}$ such that the origin is nilpotent.
For the moment, we focus on describing the flow near the origin _with the
parameters fixed_.

To start, let us consider the case where $\displaystyle N=2$, that is:

$\begin{split}\dot{x}_{1}&=a_{1}x_{1}+w_{12}(x_{2}-x_{1})\\\
\dot{x}_{2}&=a_{2}x_{2}+w_{21}(x_{1}-x_{2}).\end{split}$ (3.2)

For this, the nilpotency condition is given by the two simultaneous equations

$\begin{split}a_{1}-w_{12}+a_{2}-w_{21}&=0\\\
a_{1}a_{2}-a_{1}w_{21}-a_{2}w_{12}&=0.\end{split}$ (3.3)

Notice that for these equations to have a nontrivial solution it is necessary
that $\displaystyle w_{12}\neq w_{21}$ and $\displaystyle a_{1}\neq a_{2}$
which is fulfilled, generically, by having (at least) two independent
parameters. Assuming that $\displaystyle a_{1},a_{2}$ are independent, we have
that $\displaystyle w_{ij}=w_{ij}^{*}:=\frac{a_{i}^{2}}{a_{i}-a_{j}}$, makes
the origin nilpotent.

Consider the node-directional blow-up555It can be easily checked that these
two charts are sufficient as in the charts $\displaystyle
K_{2}^{\pm}=\left\\{\bar{x}_{2}=\pm 1\right\\}$ one obtains analogous local
vector fields.

$x_{1}=\pm r,\;x_{2}=r\bar{x}_{2},$ (3.4)

where the $\displaystyle\pm$ sign accounts for the charts $\displaystyle
K_{1}^{\pm}:\left\\{\bar{x}_{1}=\pm 1\right\\}$, and which leads to the local
vector field

$\begin{split}r^{\prime}&=r\left(a_{1}\pm w_{12}(\bar{x}_{2}\mp 1)\right)\\\
\bar{x}_{2}^{\prime}&=\pm w_{21}+\bar{x}_{2}(-a_{1}+a_{2}+w_{12}-w_{21}\mp
w_{12}\bar{x}_{2}),\end{split}$ (3.5)

whereby substituting $\displaystyle
w_{ij}=w_{ij}^{*}=\frac{a_{i}^{2}}{a_{i}-a_{j}}$ we get

$\begin{split}r^{\prime}&=-\frac{a_{1}r}{a_{1}-a_{2}}(a_{2}\mp
a_{1}\bar{x}_{2})\\\ \bar{x}_{2}^{\prime}&=\mp\frac{(a_{2}\mp
a_{1}\bar{x}_{2})^{2}}{a_{1}-a_{2}}.\end{split}$ (3.6)

The equilibrium point of (3.6), namely
$\displaystyle(r,x_{2})=(0,\pm\frac{a_{2}}{a_{1}})$, is still nilpotent, and
in fact the linearization matrix at the equilibrium point is the zero matrix.
However, as a slight advantage, (3.6) is in triangular form and can be
desingularized666The desingularized vector is found by dividing (3.6) by
$\displaystyle(a_{2}\mp a_{1}\bar{x}_{2})$, and provides the phase-portrait of
(3.6) away from $\displaystyle(a_{2}\mp a_{1}\bar{x}_{2})=0$ after reversing
the direction of the flow in the region $\displaystyle(a_{2}\mp
a_{1}\bar{x}_{2})<0$. to find the classification shown in figure 2.

$\displaystyle\frac{a_{1}}{a_{1}-a_{2}}>0$$\displaystyle\frac{a_{1}}{a_{1}-a_{2}}<0$
$\displaystyle x_{1}$$\displaystyle x_{1}$$\displaystyle x_{2}$$\displaystyle
x_{2}$ Figure 2: Classification of the flow of the blown-up system (3.6); left
for $\displaystyle\frac{a_{1}}{a_{1}-a_{2}}>0$ and right for
$\displaystyle\frac{a_{1}}{a_{1}-a_{2}}<0$. We show in the first row the
global phase-portrait which is obtained by gluing together the flow on the
charts $\displaystyle K_{1}^{\pm}$. The dotted line corresponds to the line of
equilibria $\displaystyle x_{2}=\frac{a_{2}}{a_{1}}x_{1}$. The shown line is a
representative of the case $\displaystyle a_{1}>0$, $\displaystyle a_{2}>0$.
In the second row, we show a representative simulation of the original system
(3.2) with $\displaystyle w_{ij}=w_{ij}^{*}=\frac{a_{i}^{2}}{a_{i}-a_{j}}$.
For these simulations we use $\displaystyle a_{1}=2$, $\displaystyle a_{2}=1$
on the left; $\displaystyle a_{1}=1$, $\displaystyle a_{2}=2$ on the right.

We further emphasize that the equilibrium of (3.6) is as degenerate as it can
get, in the sense that shifting the equilibrium point to the origin one gets
(re-using the variables)

$\begin{split}r^{\prime}&=\frac{a_{1}^{2}}{a_{1}-a_{2}}r\bar{x}_{2}\\\
\bar{x}_{2}^{\prime}&=-\frac{a_{1}^{2}}{a_{1}-a_{2}}\bar{x}_{2}^{2},\end{split}$
(3.7)

which has linearization matrix $\displaystyle\begin{bmatrix}0&0\\\
0&0\end{bmatrix}$, instead of the less degenerate form
$\displaystyle\begin{bmatrix}0&1\\\ 0&0\end{bmatrix}$, which is mostly
considered in the literature. Hence, for example, the techniques of [25]
cannot be applied to achieve the stability of the origin via the introduction
of higher-order terms. Of course, stabilization by modifying the weights, or
equivalently by introducing new linear terms, is quite straightforward. If one
were to consider weights

$w_{ij}=w_{ij}^{*}+\delta_{ij},$ (3.8)

in (2.16), one finds equilibria $\displaystyle(r^{*},\bar{x}_{2}^{*})$ given
by $\displaystyle r^{*}=0$ and

$\bar{x}_{2}^{*}=\frac{\pm(a_{2}-a_{1})\sqrt{4a_{1}\delta_{21}+4a_{2}\delta_{12}+(\delta_{12}+\delta_{21})^{2}}+2a_{1}a_{2}+a_{1}\delta_{12}-a_{1}\delta_{21}-a_{2}\delta_{12}+a_{2}\delta_{21}}{2\left(a_{1}^{2}+a_{1}\delta_{12}-a_{2}\delta_{12}\right)}$
(3.9)

with corresponding Jacobian

$J=\begin{bmatrix}\frac{1}{2}\left(\pm\sqrt{4a_{1}\delta_{21}+4a_{2}\delta_{12}+(\delta_{12}+\delta_{21})^{2}}-\delta_{12}-\delta_{21}\right)&0\\\
0&\mp\sqrt{4a_{1}\delta_{21}+4a_{2}\delta_{12}+(\delta_{12}+\delta_{21})^{2}}\\\
\end{bmatrix}$ (3.10)

from which one can conclude, for example, that if
$\displaystyle-\delta_{12}-\delta_{21}<0$ and $\displaystyle\delta$
$\displaystyle 4a_{1}\delta_{21}+4a_{2}\delta_{12}<0$ then there is one stable
node and one saddle, and the flow on both $\displaystyle r$-eigenspaces is
contracting. This then leads to a stable origin for (3.2), see figure 3 for an
example.

Figure 3: Phase-portrait of (3.2) with $\displaystyle
w_{ij}=w_{ij}^{*}+\delta_{ij}$ with $\displaystyle a_{1}=-2$, $\displaystyle
a_{2}=1$, $\displaystyle\delta_{ij}=\frac{1}{5}$.

It would be interesting to investigate, however, if higher-order interactions
[4] can stabilize the origin. This is, in general, a very complicated problem
and we limit ourselves to providing a proof of concept. Suppose one introduces
higher-order terms as follows:

$\begin{split}\dot{x}_{1}&=a_{1}x_{1}+w_{12}(x_{2}-x_{1})-wx_{2}^{3}+w(x_{1}+x_{2})^{3}\\\
\dot{x}_{2}&=a_{2}x_{2}+w_{21}(x_{1}-x_{2})+wx_{2}^{3},\end{split}$ (3.11)

where $\displaystyle w\in\mathbb{R}$. Upon a time-rescaling, one can assume
that all parameters are small. To simplify the analysis, but keeping the
problem still interesting, let $\displaystyle a_{1}=\alpha$ and $\displaystyle
a_{2}=-\alpha$ with $\displaystyle\alpha>0$, hence the uncoupled dynamics are
unstable. We emphasize that the origin is still nilpotent for (3.11) with
$\displaystyle w_{ij}=w_{ij}^{*}$. Consider the polar blow-up

$(x,y)=(r\cos(\theta),r\sin(\theta))$ (3.12)

together with the rescaling of parameters $\displaystyle\alpha=r^{3}A,w=rW$.
In these new coordinates, and after division by $\displaystyle r^{3}$, we
obtain the vector field

$\begin{split}r^{\prime}&=rf_{1}(\theta)\\\
\theta^{\prime}&=f_{2}(\theta)\end{split}$ (3.13)

where, having substituted $\displaystyle w_{ij}=w_{ij}^{*}$, we have

$\begin{split}f_{1}(\theta)&=\frac{1}{8}(4A\cos(2\theta)+W(6\sin(2\theta)+3\sin(4\theta)-\cos(4\theta)+9))\\\
f_{2}(\theta)&=\frac{1}{8}(-2(2A+3W)\sin(2\theta)-4A+W\sin(4\theta)+3W\cos(4\theta)-3W).\end{split}$
(3.14)

It follows that if we let777One can obtain the equilibria and their
corresponding Jacobians numerically for any value of $\displaystyle A>0$ and
$\displaystyle W<0$ leading to a similar qualitative behavior. It just happens
that for this choice, the equilibria are independent of the parameters.
$\displaystyle W=-A$, then (3.13) has eight hyperbolic equilibria for
$\displaystyle r=0$ given by

$\\{\theta_{i}\\}\approx\\{0.431808,1.26918,2.35619,2.54775,3.5734,4.41077,5.49779,5.68935\\},\;i=1,\ldots,8,$
(3.15)

where in particular $\displaystyle(r,\theta)=(0,\theta_{k})$, $\displaystyle
k=2,4,6,8$, are stable nodes and the rest are saddles with the $\displaystyle
r$-direction stable. This leads to the diagram shown in figure 4.

######  Remark 11.

Notice that thanks to the higher-order terms, the equilibria of the blown-up
system are hyperbolic. That is, one can indeed desingularize a nilpotent
equilibrium of a network dynamical system via the blow-up.

Due to hyperbolicity, a qualitatively similar result, as the one stated above,
holds for $\displaystyle W\approx-A$. Moreover, since the result does not
depend on the rescaling of the parameters, just on their signs, we conclude
that the origin of (3.11) is stable for $\displaystyle a_{1}=\alpha$,
$\displaystyle a_{2}=-\alpha$ with $\displaystyle\alpha>0$ and $\displaystyle
w<0$, see a simulation in figure 4.

(a)  (b)(c) Figure 4: (a) Global phase-portrait of (3.13), where all
equilibria are hyperbolic. Panels (b) and (c) show a simulation of (3.11) with
$\displaystyle a_{1}=1$, $\displaystyle a_{2}=-1$, $\displaystyle
w_{ij}=w_{ij}^{*}$, $\displaystyle w=-1/10$ and $\displaystyle w=-2$
respectively.

The main conclusions of the previous analysis are summarized as follows.

######  Proposition 1.

Consider the 2-node network

$\begin{split}\dot{x}_{1}&=a_{1}x_{1}+w_{12}(x_{2}-x_{1})-wx_{2}^{3}+w(x_{1}+x_{2})^{3}\\\
\dot{x}_{2}&=a_{2}x_{2}+w_{21}(x_{1}-x_{2})+wx_{2}^{3},\end{split}$ (3.16)

with parameters satisfying (3.3) so that the origin is nilpotent. If
$\displaystyle w=0$ then the origin is unstable. On the other hand, if
$\displaystyle a_{1}=-a_{2}=\alpha$ with $\displaystyle\alpha>0$ and
$\displaystyle w<0$, then the origin is locally asymptotically stable.

The general case (3.1) quickly becomes analytically intractable. For example,
it is quite difficult to find closed expressions for the parameters such that
the origin is nilpotent. Nevertheless, we conjecture that the dynamics shall
be reminiscent of what is shown in figure 2, and that as for the case
$\displaystyle N=2$, higher-order interactions may stabilize nilpotent points.
A special case that is simple to analyze is considered in the following
proposition.

######  Proposition 2.

Consider

$\dot{x}_{i}=a_{i}x_{i}+\sum_{j=1}^{N}w_{ij}(x_{j}-x_{i})+h_{i}(x),$ (3.17)

with $\displaystyle h_{i}(0)=0$, $\displaystyle h_{i}\in\mathcal{O}(|x|^{2})$,
and with parameters $\displaystyle a_{i}\in\mathbb{R}$, $\displaystyle
a_{i}\neq 0$, and $\displaystyle w_{ij}\in\mathbb{R}$ such that the origin is
nilpotent. Suppose that there exists a smooth function $\displaystyle
V:\mathbb{R}^{n}\to\mathbb{R}$ such that $\displaystyle
h(x)=(h_{1}(x),\ldots,h_{n}(x))=-\nabla V(x)$ and $\displaystyle x=0$ is a
(degenerate, since $\displaystyle h_{i}\in\mathcal{O}(|x|^{2})$) local minimum
of $\displaystyle V$. Then the origin is locally asymptotically stable.

###### Proof.

System (3.1) can be rewritten as the gradient system
$\displaystyle\dot{x}=-\nabla W$ where $\displaystyle W=(W_{1},\ldots,W_{n})$
with $\displaystyle W_{i}=-(a_{i}-w_{ii})x_{i}^{2}-\sum_{i=1,\,i\neq
j}^{n}w_{ij}x_{i}x_{j}+V$. The statement is now straightforward from noticing
that the nilpotent assumption guarantees that the origin is a (degenerate)
local minimum of $\displaystyle W$. ∎

######  Remark 12.

Proposition 2 concerns network dynamical systems for which the leading part is
nilpotent and the higher-order terms are of gradient form. The proposition
roughly tells us that near the nilpotent point, the system behaves as a
gradient system, hence the higher-order terms, if chosen appropriately,
stabilize the nilpotent origin. Naturally, many network dynamical systems do
not admit a gradient structure. In particular, one can check that (3.11) is
not of such a gradient form.

###  3.2 An example of nilpotent internal dynamics

Consider the following system of weakly and diffusely coupled scalar
nodes888This model can, alternatively, be studied with the tools developed in
[18]. Here we present an analysis fitting the blow-up method.

$\begin{split}\dot{x}_{i}&=a_{i}x_{i}^{3}(1-x_{i})+\varepsilon\sum_{j=1}^{N}w_{ij}(x_{j}-x_{i}),\end{split}$
(3.18)

where $\displaystyle a_{i}\neq 0$, $\displaystyle w_{ij}\in\mathbb{R}$, and
for simplicity we assume that $\displaystyle x_{i}(0)\in(0,1)$, and that
$\displaystyle w_{ii}=0$ for all $\displaystyle i=1,\ldots,N$ (there are no
self-couplings). Since for $\displaystyle\varepsilon=0$ the origin is
nilpotent, we are going to use the blow-up technique to elucidate the
stability of the origin for $\displaystyle\varepsilon$ small.

A node directional blow-up given by

$(r,r\bar{x}_{1},\ldots,r\bar{x}_{i-1},r\bar{x}_{i+1},\ldots,r\bar{x}_{N},r^{2}\bar{\varepsilon})=(x_{i},x_{1},\ldots,x_{i-1},x_{i+1},\ldots,x_{N},\varepsilon)$
(3.19)

leads, after desingularization, to

$\begin{split}r^{\prime}&=r\left(a_{i}+\bar{\varepsilon}\sum_{j=1}^{N}w_{ij}(\bar{x}_{j}-1)-a_{i}r\right)\\\
\bar{x}_{k}^{\prime}&=a_{k}\bar{x}_{k}^{3}+\bar{\varepsilon}\sum_{j\neq
i}^{N}w_{kj}(\bar{x}_{j}-\bar{x}_{k})+\bar{\varepsilon}w_{ki}(1-\bar{x}_{k})-\bar{x}_{k}\left(a_{i}+\bar{\varepsilon}\sum_{j=1}^{N}w_{ij}(\bar{x}_{j}-1)-a_{i}r\right)-ra_{k}\bar{x}_{k}^{4}\\\
\bar{\varepsilon}^{\prime}&=-2\bar{\varepsilon}\left(a_{i}+\bar{\varepsilon}\sum_{j=1}^{N}w_{ij}(\bar{x}_{j}-1)-a_{i}r\right),\end{split}$
(3.20)

$\displaystyle k=1,\ldots,N$ with $\displaystyle k\neq i$. In this model
$\displaystyle x_{i}$ is replaced by $\displaystyle r$, hence one is
interested in the stability of $\displaystyle r=0$, which we recall
corresponds to the origin via the blow-up transformation.

As is usual in the blow-up analysis, one is interested in the dynamics
restricted to the invariant subspaces
$\displaystyle\left\\{r=\bar{\varepsilon}=0\right\\}$,
$\displaystyle\left\\{\bar{\varepsilon}=0\right\\}$ and
$\displaystyle\left\\{r=0\right\\}$ (although in this case, it suffices to
take the last two subspaces). For $\displaystyle\bar{\varepsilon}=0$ we have

$\begin{split}r^{\prime}&=a_{i}r(1-r)\\\
\bar{x}_{k}^{\prime}&=-a_{i}\bar{x}_{k}+a_{k}\bar{x}_{k}^{3}+r(a_{i}\bar{x}_{k}-a_{k}\bar{x}_{k}^{4})\\\
\bar{\varepsilon}^{\prime}&=0,\end{split}$ (3.21)

where it is straightforward to see, from the leading order terms, that if
$\displaystyle a_{l}<0$ for all $\displaystyle l=1,\ldots,N$ we have that,
locally, $\displaystyle(r,\bar{x}_{k})=(0,0)$ is a _hyperbolic saddle_ while
$\displaystyle(r,\bar{x}_{k})=(0,\pm\sqrt{a_{i}/a_{k}})$ are _hyperbolic_
sinks, with Jacobians

$J_{(0,0)}=\begin{bmatrix}a_{i}&0\\\
0&-a_{i}\end{bmatrix}\qquad\textnormal{and}\qquad
J_{\left(0,\pm\sqrt{a_{i}/a_{k}}\right)}=\begin{bmatrix}a_{i}&0\\\
*&2a_{i}\end{bmatrix}$ (3.22)

respectively. It will be useful for our arguments below to notice that these
stability properties depend only on the $\displaystyle a_{i}$ coefficient.

Since the above equilibria are hyperbolic after the blow-up, one can conclude
that for $\displaystyle\bar{\varepsilon}>0$ small, the local stability
properties of the equilibria are preserved.

In the $\displaystyle\bar{\varepsilon}$-direction (the rescalling chart) we
obtain the desingularized system

$\begin{split}\dot{\bar{x}}_{i}&=a_{i}\bar{x}_{i}^{3}(1-\bar{x}_{i})+\sum_{j=1}^{N}w_{ij}(\bar{x}_{j}-\bar{x}_{i}),\end{split}$
(3.23)

The leading order term is now the interaction. Therefore, the origin is not
nilpotent anymore, but semi-hyperbolic. It follows that if all the weights
$\displaystyle w_{ij}$ are nonnegative, and the digraph is strongly connected,
then the subspace
$\displaystyle\textnormal{span}\left\\{1,\ldots,1\right\\}\in\mathbb{R}^{N}$
is locally attracting. The dynamics on such a space are thus given by the
scalar equation

$\bar{x}=a_{i}\bar{x}^{3}(1-\bar{x}),$ (3.24)

for which the origin is locally asymptotically stable provided that
$\displaystyle a_{i}<0$.

From our previous analysis, we have proven the following:

######  Proposition 3.

If $\displaystyle a_{i}<0$, $\displaystyle w_{ij}\leq 0$ for all
$\displaystyle i,j=1,\ldots,N$ and the underlying digraph of (3.18) is
strongly connected, then the (nilpotent) origin of (3.18) is locally
asymptotically stable for $\displaystyle\varepsilon>0$ sufficiently small.

###  3.3 A slowly adaptive network

Let us consider an adaptive network of $\displaystyle N$ homogeneous Kuramoto
oscillators given by [2]

$\begin{split}\dot{\phi}_{i}&=\omega-\frac{1}{N}\sum_{j=1}^{N}\kappa_{ij}\sin(\phi_{i}-\phi_{j}+\alpha)\\\
\dot{\kappa}_{ij}&=-\varepsilon(\sin(\phi_{i}-\phi_{j}+\beta)+\kappa_{ij}).\end{split}$
(3.25)

Without loss of generality, one may consider $\displaystyle\omega=0$ by
changing to the co-rotating frame $\displaystyle\phi_{i}\mapsto\phi_{i}+\omega
t$. Thus, from now on we consider

$\begin{split}\dot{\phi}_{i}&=-\frac{1}{N}\sum_{j=1}^{N}\kappa_{ij}\sin(\phi_{i}-\phi_{j}+\alpha)\\\
\dot{\kappa}_{ij}&=-\varepsilon(\sin(\phi_{i}-\phi_{j}+\beta)+\kappa_{ij}).\end{split}$
(3.26)

A one-cluster solution999Even though the one-cluster solutions for this system
have already been described in [2], we use it as a simple example to validate
the application of the blow-up technique. is defined as
$\displaystyle\phi_{i}(t)=s(t)+a_{i}$ for some constant $\displaystyle
a_{i}\in[0,2\pi)$, $\displaystyle i=1\ldots,N$. Depending on the phases
$\displaystyle a_{i}$ the aforementioned one-cluster solutions receive
different names, see [2, Definition 2.3]. Upon substitution of the one-cluster
solution in (3.26) one finds that $\displaystyle\dot{s}=$constant, meaning
that one-cluster solutions are, in fact, of the form
$\displaystyle\phi_{i}(t)=\Omega t+a_{i}$, with some
$\displaystyle\Omega\in\mathbb{R}$. It is well-known that the one-cluster
solutions with
$\displaystyle\Omega=\Omega^{*}:=\frac{1}{N}\sum_{j}\sin(a_{i}-a_{j}+\alpha)\sin(a_{i}-a_{j}+\beta)$,
together with $\displaystyle k_{ij}(t)=-\sin(a_{i}-a_{j}+\beta)$ form a set of
relative equilibria of (3.26). In other words, consider the co-rotating
coordinates $\displaystyle\psi_{i}=\phi_{i}-(\Omega^{*}t+a_{i})$, and the
shifted weights
$\displaystyle\sigma_{ij}=\kappa_{ij}+\sin(a_{i}-a_{j}+\beta)$. Then (3.26) is
re-written as

$\begin{split}\dot{\psi}_{i}&=-\Omega^{*}+\frac{1}{N}\sum_{j=1}^{N}\sin(a_{i}-a_{j}+\beta)\sin(\psi_{i}+a_{i}-\psi_{j}-a_{j}+\alpha)\\\
&\quad-\frac{1}{N}\sum_{j=1}^{N}\sigma_{ij}\sin(\psi_{i}+a_{i}-\psi_{j}-a_{j}+\alpha)\\\
\dot{\sigma}_{ij}&=-\varepsilon(\sin(\psi_{i}+a_{i}-\psi_{j}-a_{j}+\beta)-\sin(a_{i}-a_{j}+\beta)+\sigma_{ij})\end{split}$
(3.27)

It follows, indeed, that $\displaystyle(\psi_{i}^{*},\sigma_{ij}^{*})=(0,0)$
is an equilibrium of (3.27). It turns out that this equilibrium is non-
hyperbolic from a slow-fast perspective. Moreover, for certain parameters,
one-cluster solutions can be nilpotent.

######  Proposition 4.

Consider the layer equation of (3.27). Then the equilibrium point
$\displaystyle(\psi_{i}^{*},\sigma_{ij}^{*})=(0,0)$ is non-hyperbolic. In
particular, the antipodal solution, which corresponds to $\displaystyle
a_{i}\in\left\\{0,\pi\right\\}$, is nilpotent for $\displaystyle\beta=0$.

######  Remark 13.

These are not the only nilpotent solutions. In fact, for
$\displaystyle\alpha=\beta=0$ all one-cluster solutions in [2, Corollary 4.3]
are nilpotent. In this example, however, we only concentrate on the antipodal
case.

###### Proof.

Since we are looking at the layer equation of (3.27), we only need to consider
the Jacobian $\displaystyle J=[J_{ij}]_{i,j=1,\ldots,N}$ where $\displaystyle
J_{ij}=\frac{\partial\dot{\psi}_{i}}{\partial\psi_{j}}\Big{|}_{\psi_{i}=\psi_{j}=\sigma_{ij}=0}$.
So, we have

$\begin{split}J_{ii}&=\frac{1}{N}\sum_{j=1}^{N}\sin(a_{i}-a_{j}+\beta)\cos(a_{i}-a_{j}+\alpha)\\\
J_{ij}&=\frac{1}{N}\sin(a_{i}-a_{j}+\beta)\cos(a_{i}-a_{j}+\alpha).\end{split}$
(3.28)

Since $\displaystyle J_{ii}+\sum_{j\neq i}^{N}J_{ij}=0$, it follows that
$\displaystyle\ker J$ is non-trivial (this is reminiscent, of course, of
Laplacian matrices), showing that the equilibrium point
$\displaystyle(\psi_{i}^{*},\sigma_{ij}^{*})=(0,0)$ is non-hyperbolic.
Finally, if $\displaystyle\beta=0$ and $\displaystyle
a_{i}\in\left\\{0,\pi\right\\}$ we have that $\displaystyle J=0$, i.e., the
equilibrium point $\displaystyle(\psi_{i}^{*},\sigma_{ij}^{*})=(0,0)$ is
nilpotent. ∎

The question one would now like to answer is: what is the stability of the
anti-podal solutions for $\displaystyle\varepsilon,\,\beta$ small? To answer
this question we are going to use the blow-up technique. We remark that the
blow-up technique is a local tool, hence, it is in principle applied to
polynomial vector fields. Here we shall simply expand (3.27) for
$\displaystyle(\psi_{i},\sigma_{ij})$ close to the origin. Let $\displaystyle
a_{i}\in\left\\{0,\pi\right\\}$. Up to a constant shift above, we can assume
without loss of generality that $\displaystyle a_{i}=0$ for all $\displaystyle
i=1,\ldots,N$. Then (3.27) reads as

$\begin{split}\dot{\psi}_{i}&=-\Omega^{*}+\frac{1}{N}\sum_{j=1}^{N}\sin\beta\sin(\psi_{i}-\psi_{j}+\alpha)-\frac{1}{N}\sum_{j=1}^{N}\sigma_{ij}\sin(\psi_{i}-\psi_{j}+\alpha)\\\
\dot{\sigma}_{ij}&=-\varepsilon(\sin(\psi_{i}-\psi_{j}+\beta)-\sin\beta+\sigma_{ij}).\end{split}$
(3.29)

Expanding $\displaystyle\sin(\psi_{i}-\psi_{j}+\rho)$ for
$\displaystyle\psi_{i}\approx\psi_{j}$, i.e.
($\displaystyle\sin(\psi_{i}-\psi_{j}+\rho)=\sin\rho+\cos\rho(\psi_{i}-\psi_{j})+\cdots$),
with $\displaystyle\rho=\alpha,\beta$, and accounting for the fact that in
this setup we have $\displaystyle-\Omega^{*}+\sin\alpha\sin\beta=0$, we get

$\begin{split}\dot{\psi}_{i}&=\frac{1}{N}\sum_{j=1}^{N}\left[\sin\beta\cos\alpha(\psi_{i}-\psi_{j})-\sigma_{ij}(\sin\alpha+\cos\alpha(\psi_{i}-\psi_{j})\right]+\cdots\\\
\dot{\sigma}_{ij}&=-\varepsilon\left(\cos\beta(\psi_{i}-\psi_{j})+\sigma_{ij}+\cdots\right),\end{split}$
(3.30)

where the $\displaystyle\cdots$ indicate higher-order terms. Naturally, the
origin $\displaystyle(\psi_{i},\sigma_{ij})=(0,0)$ is (still) nilpotent for
$\displaystyle\varepsilon=\beta=0$, but now it is straightforward to notice
that $\displaystyle\alpha=\pm\frac{\pi}{2}$ also induces that the origin is
nilpotent (this case is not further discussed in this example). For the rest
of this example, we only consider the leading part of (3.30).

Let us propose the $\displaystyle\beta$-directional blow-up (in the rescaling
chart) given by101010Notice that this is equivalent to first going to the
rescaling chart and then blowing up in the $\displaystyle\pm\beta$-direction.

$\psi_{i}=ru_{i},\quad\sigma_{ij}=rs_{ij},\quad\alpha=rA,\quad\beta=\pm
r,\quad\varepsilon=r.$ (3.31)

This leads to the desingularized vector field (where we take only the leading
order terms of the Taylor expansion of the trigonometric functions)

$\begin{split}\dot{r}&=0\\\
\dot{u}_{i}&=\frac{1}{N}\sum_{j=1}^{N}\pm(u_{i}-u_{j})-s_{ij}(A+(u_{i}-u_{j}))\\\
\dot{s}_{ij}&=-((u_{i}-u_{j})+s_{ij}).\end{split}$ (3.32)

Now, the following conclusions on the local stability of the line of
equilibria $\displaystyle u_{i}=u_{j},\,s_{ij}=0$, which we denote by
$\displaystyle\gamma$, are straightforward111111It suffices to consider the
leading part $\displaystyle\dot{u}_{i}=\frac{\pm
1+A}{N}\sum_{i=1}^{N}(u_{i}-u_{j}).$: a) if $\displaystyle\beta>0$,
corresponding to the plus sign in (3.32), then $\displaystyle\gamma$ cannot be
stable; b) if $\displaystyle\beta<0$, corresponding to the negative sign in
(3.32), then $\displaystyle\gamma$ is locally stable for $\displaystyle A<1$
and unstable for $\displaystyle A>1$. From this analysis, and recalling the
blow-up map (3.31), the following statement is proven (compare with [2, figure
4.2 (b)]):

######  Proposition 5.

The antipodal solutions of (3.26) are locally asymptotically stable for
$\displaystyle\beta<-\alpha$ and $\displaystyle\varepsilon>0$ sufficiently
small.

##  4 Conclusions and Discussion

We have studied the problem of structure preservation under the blow-up
transformation of network dynamical systems, including adaptive ones. This
transformation is a useful tool to rigorously analyze the dynamics of
differential equations close to nilpotent equilibria. The essential philosophy
of the method is to transform a singular problem into a regular one.

The main conclusion from our analysis is that, indeed, the blow-up
transformation preserves the network structure. This means that the local
vector fields obtained after a directional blow-up can still be interpreted as
a network dynamical system. The precise interpretation turns out to depend on
the direction of blow-up and the model at hand. Moreover, via a series of
examples, we have shown that (as in the classical low-dimensional setting) the
blow-up helps to desingularize nilpotent singularities.

Besides structure preservation, we have noticed that the blow-up induces
parameters to become dynamic. Nevertheless, such dynamics seem to occur at
higher-orders, which could yield another natural way to uncover higher-
interactions in networks. In the particular case of edge-directional blow-up,
including the case of adaptive networks, the main message is that singular
networks with static topology are to be regularized as network dynamical
systems with co-evolving edges; hence, this provides a route to potentially
uncover adaptive network dynamics.

Several open problems stem from the analysis presented here, from which we
highlight the following:

  1. 1.

The problem of desingularization has been addressed here only in the examples.
It would be interesting to relate the network topology and the nature of
internal dynamics, interaction, and possible adaptation, to better describe
the desingularization procedure via blow-up. In particular, one may want to
study classes of networked systems where the corresponding vector field is
quasi-homogeneous.

  2. 2.

For high-dimensional problems, it may become unrealistic to blow-up in all
possible directions. This raises new directions for improving and better
adapting the blow-up method for networks: a) one may want to blow-up in
several directions simultaneously, b) one may want to devise techniques that
distinguish between regular and singular directions, or c) one may want to
develop symbolic computational tools that automatically provide the local
vector fields in the relevant charts where the dynamics have been
desingularized.

  3. 3.

Choosing the blow-up weights is, even for low dimensions, quite challenging.
In the context of networks, it is not unfeasible to expect that the topology
of the network induces, or forces, some patterns in the blow-up weights. For
example, the analysis carried out in section 2 suggests that, due to the
network structure, all nodes are to be blown up with the same weight, provided
that the linear part is not identically zero. In this regard, it is known that
the quasi-homogeneous blow-up for planar systems can be related to a Newton
polyhedron, see [1]. In higher dimensions, the quasihomogeneous blow-up should
naturally be related to a polytope. It would be interesting to see if there is
a relation between the blow-up polytope and the topology of the network to be
desingularized.

  4. 4.

A way to describe the dynamics of large networks is via mean field limits,
which essentially provides a low-dimensional averaged description (e.g. via a
PDE, but in certain cases even a low-dimensional system of ODEs) of the
network. It would be interesting to investigate if a desingularization of a
mean-field limit corresponds, in any way, to the desingularization of the
corresponding large, but finite, network.

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