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Contraction Theory and the Master Stability
Function: linking two approaches to study
synchronization of complex networks
Giovanni Russo and Mario di Bernardo
Abstract—In this paper contraction theory is applied to the
problem of synchronization of a network. Particularly, the associ-
ation between the contraction principle, the Lyapunov exponents
of a system and the Master Stability Function of the network are
pointed out. Novel, sufficient criteria warranting the fulfilment
of a synchronous state are derived. Numerical simulations are
used to validate the theoretical results.
I. INTRODUCTION
The study of coupled oscillators is a common problem in
a variety of different research areas such as mathematics,
biology, robotics, electronics and neuroscience (see e.g. [1],
[2], [3], [4], [5]). Indeed, many complex systems in nature
and technology can be described as networks (or graphs) of
interacting agents communicating over the network links. Ex-
amples include neural systems, the world-wide web, electrical
power grids [6]. A typical problem is to find conditions to
guarantee the synchronization of a network of identical non-
linear oscillators, so that all oscilators converge asymptotically
towards the same common evolution. Rigorous stability results
were obtained by constructing appropriate Lyapunov functions
(see for example [7]) or by means of local tools such as
the Master Stability Function (MSF) introduced in [8]. The
former approach requires the coupling matrix to be positive
definite and aims at proving global stability of the synchronous
evolution. The MSF, instead, uses Lyapunov exponents to
guarantee the local transversal stability of the synchronization
manifold, assuming that the coupling functions and vector
fields can be locally linearized and block-diagonalized.
In the above approaches, stability is considered with respect
to a particular solution or to some invariant set. However, in
synchronization problems we are typically interested in finding
conditions guaranteeing the evolution of all the trajectories, i.e.
solutions, of the nodes of a network of interest towards a stable
manifold which is unknown a priori. For this reason a possible
approach to the study of synchronization would be that of
studying the convergence properties of all solutions rather than
of one a priori solution or set. The first results on asymptotic
stability of all solutions of a nonlinear dynamical system are
due to the Russian mathematician Demidovich [9]. Several
G. Russo is with the Department of Systems and Computer Science,
University of Naples Federico II giovanni.russo2@unina.it
M. di Bernardo is with the Department of Systems and Computer Science,
University of Naples Federico II mario.dibernardo@unina.it
Copyright (c) 2008 IEEE. Personal use of this material is permitted.
However, permission to use this material for any other purposes must be
obtained from the IEEE by sending an email to pubs-permissions@ieee.org
decades after Demidovich’s publications the interest on the
stability properties of trajectories with respect to each other
revived: incremental stability [10] and contraction theory are
indeed related to these concepts. Contraction theory has been
established as an effective tool for analyzing the convergence
behavior of nonlinear systems in state-space form. It was
successfully applied to both nonlinear control and observer
problems, [11], [12] and, more recently, to synchronization
and consensus problems in complex networks [13], [14], [15].
A pressing open problem, only marginally addressed in the
existing literature [16], is to understand the relationship be-
tween the MSF approach, mostly developed (and used) within
the Physics community, and the contraction theory approach
proposed within the nonlinear control theoretic community.
The aim of this paper is to address this open problem. We
shall seek to establish a formal link between the MSF and the
contraction principle by giving a novel set of conditions under
which a given network (characterized by a general topology,
linear or nonlinear coupling and regular or chaotic dynamics
at the nodes) synchronizes.
II. PROBLEM STATEMENT
We consider graphs in which each node is a nonlinear
autonomous system; the coupling can be either a linear or non-
linear function of all or some of the system states. Specifically,
we assume that there are Nidentical nodes, with dynamics
given by
˙xi=f(xi),(1)
where xi∈Rmand f:Rm→Rm; then, the dynamics of
the whole network can be represented as:
˙x=F(x)−αL ⊗H(x),(2)
with x= (x1, ..., xN)T,F(x) = (f(x1), ..., f (xN))T,
H(x) = (h(x1), ..., h (xN))T,⊗indicating the direct prod-
uct, Lrepresenting the N×NLaplacian matrix and h:
Rm→Rma coupling function. The network is said to
be synchronized if all oscillators converge towards the same
synchronous state, characterized by the stability properties of
the synchronization manifold, defined as {x∈RmN :x1=
... =xN}.
III. PROVING SYNCHRONIZATION
In this section we briefly review the main features of
the two approaches of our concern used for the study of
synchronization: contraction theory and MSF (see [17] and
[8] for further details).
A. Contraction theory
A nonlinear dynamical system is called contracting if initial
conditions or temporary perturbations decay exponentially
fast. We consider general deterministic systems of the form
(1). All quantities are assumed to be real and sufficiently
smooth. Recall that a virtual displacement is an infinitesimal
displacement at fixed time. Formally it defines a linear tangent
differential form [18]. Considering two neighboring trajecto-
ries of the flow, the virtual displacement δx between them and
the virtual velocity δ˙x, we have:
δ˙x=∂f
∂x (x)δx. (3)
From (3) we get
d
dt δxTδx= 2δxT∂f
∂x δx ≤2λmaxδxTδx, (4)
where λmax (x)is the largest eigenvalue of the symmetric part
of the Jacobian ∂f
∂x . Hence, if λmax (x)in (4) is uniformly
strictly negative, any infinitesimal length converges exponen-
tially to zero. By path integration, this immediately implies
that the length of any finite path in phase space converges
exponentially to zero, i.e. that distances shrink in phase space.
The contraction principle, derived in [17] and [13] can be
stated as follows.
Theorem 1: Let x(t)and ˜x(t)be two generic trajectories
of (1). Say Mt:Bε(x(t)) and let C⊆Rmbe a contracting
region in phase space, defined as: C:= {x∈Rm:
1
2∂f
∂x +∂fT
∂x ≤ −βI,β > 0,∀t∈R+}.If ˜x(t)is such
that: (i) ˜x(t0)∈M0;(ii) Mt⊂C∀t∈R+, then: (I)
˜x(t)∈Mt,∀t∈R+; (II) δxTδx ≤kδxT
0δx0e−βt ,k≥1,
β≥0,∀t∈R+. And viceversa.
Thus, all the solutions of (1) starting from any initial condition
inside the contraction region will remain in Cand converge
exponentially to a single trajectory. Note that: (i) in Theorem
1, Mtis an invariant, time-varying, set; (ii) the dynamics
along a chaotic attractor are not contracting as they are charac-
terised by local exponential divergence of nearby trajectories.
To apply contraction theory to synchronization of networks
the construction of a virtual system is needed [12]. Such a
system depends on the state variables of the oscillators and
on some virtual state variables. The substitution of the i-th
node state variables into the virtual state variables returns the
dynamics of the i-th node of the network. The proof of the
contracting property with respect to these virtual state variables
immediately implies synchronization.
Example 1: Consider two coupled nonlinear systems:
˙z=f(z) + h(w)−h(z),(5)
˙w=f(w) + h(z)−h(w),(6)
where his some output function. A suitable virtual system
can be chosen as
˙x=f(x)−2h(x) + h(z) + h(w) := ϕ(x, z, w).(7)
Trajectories of the nodes are particular solutions (x-solutions)
of the virtual system, e.g. ϕ(z, z, w) = f(z) + h(w)−h(z)
and ϕ(w, z, w) = f(w) + h(z)−h(w). If (7) is contracting
with respect to the xstate variable the two particular x-
solutions zand wwill converge to each other. Synchronization
is then attained. To prove synchronization of (5) and (6), it
will suffice to show that (7) is contracting with respect to the
xstate variable. Differentiation of (7) with respect to xyields
δ˙x=∂f (x)
∂x −2∂ h (x)
∂x δx. (8)
Contractivity is then ensured by making the Jacobian matrix
uniformly negative definite (see [14], [19]).
B. The Master Stability Function approach
Let xs(t)be a trajectory of (1) with initial conditions xs0. In
[20] the Lyapunov exponents for xs(t)are defined as follows.
Definition 1: The Lyapunov exponents of the flow φ(xs0)
are defined to be the Lyapunov exponents of the associated
stroboscopic time-T map.
Thus, the Lyapunov exponents of a flow are defined in terms
of the Lyapunov exponents of a map, i.e. the time-T map.
Namely, let g:Rm→Rmbe a smooth map and say g(n)
the n-th iterate of g. Define Jn=∂ g(n)
∂xnand let Σbe the m-
dimensional sphere of unitary radius with JnΣrepresenting
the deformation of the sphere after niterations of the map.
Also, let rn
kbe the length of the k-th longest orthogonal axis
of the ellipsoid JnΣfor an orbit with initial point xs0∈Σ,
for k= 1, ..., m .
Definition 2: The k-th Lyapunov number, Lk, of xs0is
defined as
Lk= lim
n→∞ (rn
k)1/n ,(9)
if this limit exists. The k-th Lyapunov exponent of xs0is hk=
ln Lk. Notice that by definition L1≥L2≥... ≥Lmand
h1≥h2≥... ≥hm.
Thus, the Lyapunov exponents measure the rates of divergence
of nearby points along morthogonal directions determined
by the dynamics of the flow. The MSF approach makes use
of the following assumptions, as shown in [21] and [22]: (i)
the coupled oscillators (nodes) and the coupling functions are
all identical; (ii) the synchronization manifold is an invariant
manifold; (iii) the coupling functions are well approximated
near the synchronous state by a linear operator. The main idea
in [8] is to derive the variational equation from equation (2)
describing small variations, ξk, of the trajectories of (2) from
the synchronous evolution, say xs(t). This equation is then
block diagonalized to give:
˙
ξk=∂f (xs)
∂x −αλk
∂H (xs)
∂x ξk.(10)
For k= 0, we have the variational equation along the
synchronization manifold; all other ks correspond to transverse
eigenmodes to such manifold. Hence, it is shown that local
stability of the synchronous evolution can be captured by the
computation of the maximum Lyapunov exponent as a function
of α, i.e. the MSF.
IV. A COMPARISON BETWEEN CONTRACTION THEORY
AND MASTER STABILITY FUNCTION
We start by looking at the representative example of two
coupled Rossler oscillators of the form:
˙x1=−(y1+z1) + εx(α(x2−x1))
˙y1=x1+ay1+εy(α(y2−y1))
˙z1=b+z1(x1−c) + εz(α(z2−z1))
,(11)
˙x2=−(y2+z2) + εx(α(x1−x2))
˙y2=x2+ay2+εy(α(y1−y2))
˙z2=b+z2(x2−c) + εz(α(z1−z2))
.(12)
We study the cases when coupling is active only on the x-
variable (εy=εz= 0), or only on the y-variable (εx=εz=
0), or on all variables (εx6= 0,εy6= 0,εz6= 0). A shown in
Section III-A, a possible virtual system is:
˙x=−(y+z) + εx(−2αx +αx1+αx2)
˙y=x+ay +εy(−2αy +αy1+αy2)
˙z=b+z(x−c) + εz(−2αz +αz1+αz2)
.(13)
We assume a= 0.2,b= 0.2,c= 2.5, so that each Rossler
system has a chaotic attractor. The virtual velocities of (13)
are expressed as:
"δ˙x
δ˙y
δ˙z#="−2αεx
−1−1
1 0.2−2αεy0
z0x−2.5−2αεz#" δx
δy
δz #.
(14)
The symmetric part of the Jacobian in (14), in the case of
coupling on the x-variable (εx= 1), is:
Js=
−2α0 (z−1) /2
0 0.2 0
(z−1) /2 0 x−2.5
.(15)
Analytical computation of the eigenvalues of (15) reveals that
an eigenvalue is always positive, implying that (13) is not
contracting. However, it is well known that the two systems
can be synchronized for small coupling strengths: indeed in
[23], [24], [22] it is shown that the MSF is negative for small
α. The symmetric part of the Jacobian in (14) in the case of
coupling on the y-variable (εy= 1) is:
Js=
0 0 (z−1) /2
0 0.2−2α0
(z−1) /2 0 x−2.5
.(16)
The analytical expressions of the eigenvalues of Js, in (16),
reveals that an eigenvalue is positive definite and indepen-
dent on α, implying that (13) is not contracting. However,
we know that chaotic Rossler systems can be synchronized
for a sufficiently large coupling strength, i.e. their MSF is
negative for large coupling strengths. The symmetric part of
the Jacobian in (14) in the case of coupling on all the state
variables (εx=εy=εz= 1) is:
Js=
−2α0 (z−1) /2
0 0.2−2α0
(z−1) /2 0 x−2α−2.5
.(17)
The eigenvalues of (17) are all dependent on the coupling
strength: particularly, the increase of the coupling strength
causes the decrease of the eigenvalues. Thus, for large enough
coupling strengths, the system is contracting and the two
Rossler systems synchronize. This result is confirmed by
studying the MSF, which is negative for large α. The example
indicates that if a virtual system is contracting, the MSF is
negative, but that the viceversa, as expected, is not true.
Remark 1: The MSF provides local conditions for synchro-
nization that need to be checked numerically, the construction
of a virtual system and the ensuing analysis provide a stronger
stability result which is global and can be proven analytically.
Remark 2: The MSF approach requires a priori knowledge
of the existence of the synchronization manifold. Contraction
theory, instead, does not require the knowledge of a specific
attractor to perform the stability analysis.
A. Lyapunov exponents and Contraction theory
Corollary 1: If Theorem 1 holds with C≡Rmthen the
transverse Lyapunov exponents to all the system trajectories
are negative.
Proof: Given a generic trajectory, xs(t), from Definition
1 and Definition 2, in [20] (page 382) we have:
˙
Jt=A(t)Jt,(18)
where Jt=∂φ(xs0)
∂x and A(t) = ∂f
∂x . To prove the theorem, we
show that if the contracting property holds, then the volume
of any given ball of initial conditions in state space shrinks.
To do this, we use the Liouville’s formula, given by:
∆′
t=T r (A(t)) ∆t
∆0= 1 ,(19)
where ∆t=det(Jt). From (19) the following result can be
obtained:
∆t= exp
t
Z
0
T r (A(t)) dt
.(20)
If the system is contracting, then:
1
2 ∂f
∂x
T
+∂f
∂x !≤ −βI. (21)
This hypothesis holds for all x(t)and for all t∈R+:
particularly it will be true for xs. From [25] (page 398) we
know that the trace, the determinant and all principal minors
of a negative definite matrix are negative. Thus, the integral
in (20) is negative, meaning that the volume of any given ball
in the phase space decreases.
V. SYNCHRONIZATION
A. Syncronization of all to all networks
Theorem 2: Consider a network of Nnodes with an all-to-
all topology. If the network dynamics are contracting for some
range of the coupling strength, A, then the master stability
function will be negative in the same range.
Proof: The virtual system corresponding to a network of
Nelements with all-to-all topology is:
˙x=f(x)−αNh (x) + αh (x1) + ... +αh (xN).(22)
System (22) is contracting if:
∂f
∂x −αN ∂h
∂x ≤ −βI, (23)
with β > 0. Since an all-to-all network can be viewed as a
complete graph, and since the Laplacian matrix for such a
graph has one zero eigenvalue (the first), while the others are
all equal to N, as shown in [26] (page 280 Lemma 13.1.3),
it is possible to rewrite (10) as:
˙
ξk=∂f (xs)
∂x −αN ∂h(xs)
∂x ξk,(24)
for each transverse mode, i.e. k6= 0. The matrix in (24)
has the same expression of the Jacobian (23). As the virtual
system is contracting in some range Aof the coupling strength
by hypothesys, then system (24) will be contracting over the
same range. Following Corollary 1, the maximum Lyapunov
exponent of (24) is negative and the theorem remains proved.
Example 2: Theorem 2 is applied to reinterpret the behavior
of two Rossler systems coupled on all state variables. We have
already pointed out that the virtual system corresponding to
the network of two coupled Rossler systems is contracting.
From (10), the variational equations for the modes of the
synchronous state of the network are:
˙
ξ0=
0−1−1
1 0.2 0
zs0xs−2.5
ξ0,(25)
˙
ξ1=
0−1−1
1 0.2 0
zs0xs−2.5
−2α
1 0 0
0 1 0
0 0 1
ξ1.
(26)
Since the matrix in (26) is equal to the Jacobian matrix of
the virtual system and it is contracting for all (x, y, z )for
large enough α, this will be true for (xs, ys, zs). Thus, the
dynamics of the transverse modes are contracting if αis
sufficiently large, implying the negativeness of the master
stability function in the same range of the parameter α.
Remark 3: Note that the higher is N(set equal to 2 in (26)),
the lower will be the value of αthat synchronizes the network,
confirming what stated in the literature [27].
B. Synchronization of networks with general topology
Theorem 3: Consider a network with Nidentical nodes.
If: (i) the network topology contains only one connected
component; (ii) the coupling functions are strictly increasing;
(iii) the virtual system is contracting for some range of the
coupling strength, A; then the master stability function will
be negative if the coupling strength αis sufficiently large.
Proof: For a network with general topology, the virtual
system can be chosen as:
˙x=f(x)−αX
j∈Nx
Lxjh(xj)−deg (x)αh (x),(27)
where Nxdenotes the set of nodes adjacent to the virtual
node xand deg (x)indicates the degree of the virtual node.
The Jacobian of (27) is:
∂f
∂x −deg (x)α∂h
∂x .(28)
The coupling that guarantees the synchronization depends
on deg (x). Since the system is contracting, and since the
coupling function is positive definite, there will exist an α∗
such that ∀α > α∗the network synchronizes. Notice that α∗
can be thought of as the αthat synchronizes the network when
deg (x) = degmin (x);degmin (x)being the minimum degree
of the network nodes. By comparing (28) and (10), we find
that they will be formally equal if:
α∗=αλk.(29)
To check the stability of the synchronous state the transverse
modes have to be considered. Since the network topology is
connected, then λ26= 0 and, since λ2is the smallest nonzero
eigenvalue, we can conclude that the master stability function
will be negative ∀α≥α∗
λ2.
Remark 4: Theorem 3 shows that αand α∗are related by
means of λ2. In particular, if λ2>1then the MSF becomes
negative for a value of αlower than that required for the
virtual system to become contracting. The viceversa happens
if λ2<1. The two coupling strengths coincide if λ2= 1.
Remark 5: Hypothesis (ii) of Theorem 3 is satisfied in
many practical situations e.g. biological systems [28], neural
networks [29], multi agent systems [30]. Moreover, if hypothe-
sis (ii) is relaxed, it is trivial to prove that, if the virtual system
is contracting over some range A, there exists a bounded set
of values of the coupling strength which guarantee negativity
of the MSF, i.e. there will exist an α∗and α∗∗ such that
α∗≤α≤α∗∗. This corresponds to using the eigenratio
λ2/λNas the MSF.
VI. NUMERICAL VALIDATIONS
To validate Theorem 3 we used the classical oscillator
defined in [31] and [32] as:
˙x1=x1−x2−x1x2
1+x2
2
˙x2=x1+x2−x1x2
1+x2
2.(30)
We assume that the coupling between nodes is linear and act-
ing on both state variables. Using a virtual system constructed
as in (27), it is easy to see that the second and third hypotheses
of Theorem 3 are satisfied. We now consider two connected
network topologies of N= 1000 nodes in order to satisfy the
first hypothesis of Theorem 3.
A. Nearest neighbor network
This topology presents a small algebraic connectivity, thus,
the coupling strength α, computed as required by Theorem 3
is expected to be large (in this case α∼
=12000). In Figure 1
the behavior is shown of the network state variables when the
coupling strength is increased from 0to αat time t= 10s.
B. Small world network
The algebaic connectivity is increased by adding new links,
with uniform probability, to the nearest neighbor topology of
Section VI-A. Then, the coupling strength computed using
Theorem 3 decreases considerably with respect to the previous
case. In Figure 2 the behavior is shown of the states of the
network and the applied coupling strength (in this case α∼
=
800).
0 10 20 30 40 50 60
−5
0
5
0 10 20 30 40 50 60
0
5000
10000
15000
t
cx1
Fig. 1. Evolution of the first state component for all oscillators (top) when
the coupling strength (c) is varied between 0and αat t= 10s.
0 10 20 30 40 50 60
−5
0
5
0 10 20 30 40 50 60
0
200
400
600
800
t
cx1
Fig. 2. Evolution of the first state component for all oscillators (top) when
the coupling strength (c) is varied between 0and αat t= 10s.
VII. CONCLUSIONS
We discussed the relationship between two approaches to
study synchronization of networks of oscillators: contraction
theory and the MSF. Using the contraction principle we estab-
lished a link between contractivity of a dynamical system and
its Lyapunov exponents. This allowed us to propose a rigorous
formal relationship between contractivity and the MSF. In
particular, we found that contractivity of the virtual system
implies negativity of the MSF but that the viceversa does not
hold. The link between the contraction principle and the MSF
was further explored by considering the problem of synchro-
nizing a network with an all-to-all connection topology. It was
shown that, as predicted when the MSF is used, the contraction
theoretic approach also predicts the network synchronizability
to improve as the number of oscillators increases. The analysis
was then generalized to the case of a network with arbitrary
topology and validated numerically on a network of nonlinear
oscillators.
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