A Virtual Space Embedding for the Analysis of Dynamical Networks
Giacomo Innocenti and Paolo Paoletti
Abstract— Networks of dynamical systems are widespread
both in nature and in technology applications, but the interplay
between the node dynamics, their interconnections and the
macroscopic behavior is still not completely understood. In
fact, the intrinsically large dimension of the phase space of
such networks represents the main obstacle for their analysis
and control. In this paper we propose a technique that is
based on embedding the network in a (potentially virtual)
spatial coordinate system to obtain a simplified description
of the overall behavior. We derive conditions under which
both the network dynamics and its stability properties can
be successfully analyzed using such approach. A network of
Fitzhugh-Nagumo systems is used as an illustrative example to
show the effectiveness of the proposed technique.
I. I NTRODUCTION
The analysis and control of networks composed by the
interconnection of nonlinear dynamical systems represent
intriguing fields of investigation due to their wide application
to the understanding of natural systems and to the control of
artificial ones. For example, in the last decades networks
of dynamical systems, such as cellular neural networks or
analog processor arrays, have been introduced in engineering
applications as a paradigm for analog computation, image
analysis and pattern generation [1], [2]. Similarly, dynamical
networks play crucial roles in biological systems for gener-
ating spatio-temporal patterns in nervous systems of various
organisms, including humans, see e.g. [3], [4]. Recently,
dynamical networks has also been exploited to understand
phenomena of opinion dynamics [5] and infectious diseases
spreading [6].
However, beside its relevance for a wide variety of sci-
entific fields, the interplay between the individual node
dynamics and interconnections, and the behavior exhibited
by the network itself is still not completely understood.
With this respect, the main obstacle is represented by the
intrinsically large dimension of the network phase space that
scales linearly with the number of nodes and that therefore
can easily reach the point where traditional analysis and
control techniques developed for low-dimensional systems
may not be easily applied anymore.
On the other hand, a dramatic reduction of the model
complexity can be achieved by assuming that the dynamics
of each node is actually a sampling of the dynamics of
a continuous system governed by an underlying partial
differential equation (PDE). The evolution of each node is
G. Innocenti is with the Dipartimento di Ingegneria
dell’Informazione, Universit` a di Firenze, I-50139 Firenze, Italy,
giacomo.innocenti@unifi.it
P. Paoletti is with the Centre for Engineering Dynamics, School
of Engineering, University of Liverpool, L69 3GH Liverpool, UK,
P.Paoletti@liverpool.ac.uk
required to be close (in some norm) to the one observed
at the point where such node is collocated, see [7] for an
example for the linear case and [8] for some preliminary
results for chains of nonlinear systems. We stress that the
continuous spatial coordinate system used to collocate the
nodes of the network needs not to represent a physical space.
It is also worth pointing out that, although it is known
that there exists phenomena, such as propagation failure,
that may occur on networks but not on PDEs [9], such
degenerate situations seem to play a negligible role for all the
applications mentioned before and this justifies the need for
further investigations on the relationships between networks
and behavior of PDEs.
In this paper we propose a technique for analyzing the be-
havior of networks where individual nodes are governed by a
nonlinear ordinary differential equation (ODE). In section II
such network is embedded in a continuous spatial coordinate
system to define a PDE whose dynamics interpolates the
behavior of each node. Conditions for the PDE to correctly
represent the network dynamics are stated in section III.
Here we focus mainly on propagating phenomena where
the PDE can be rewritten in a moving coordinate frame to
effectively obtain an ordinary differential equation whose
solution provides the profile of the propagating wave. A
procedure for determining the stability of such profile is
also discussed in the same section. We then introduce an
illustrative example in section IV to show how the proposed
approach can be successfully used to analyze the behavior
of a non trivial nonlinear dynamical network. Some final
remarks and future outlooks are discussed in section V.
Notation and preliminaries. Throughout the paper we
will denote the sets of real and complex numbers as, re-
spectively, R and C, whereas N
0
will represents the set of
natural number including 0, while Z will denote the integer
set. Given a function f (ξ ): R
w
→ R
n
we will refer to its
derivative as the linear operator
df (ξ)
dξ
=
∂f
1
∂ξ
1
...
∂f
1
∂ξw
.
.
.
∂fn
∂ξ
1
...
∂f
1
∂ξw
.
We will also make use of the multi–index notation, i.e.,
given the vectors h = [h
1
,...,h
w
]
T
∈ N
w
0
and x =
[x
1
,...,x
w
]
T
∈ R
w
we will refer to the following quantities
as follows
h!= h
1
! ...h
w
!, x
h
= x
h1
1
...x
hw
w
∂
h
x
=
∂
h1
∂x
h1
1
...
∂
hw
∂x
hw
w
=
∂
|h|
∂x
h1
1
...∂x
hw
w
.
52nd IEEE Conference on Decision and Control
December 10-13, 2013. Florence, Italy
978-1-4673-5717-3/13/$31.00 ©2013 IEEE 1331