General stability analysis of synchronized dynamics in coupled systems
Yonghong Chen,1,3 Govindan Rangarajan,2 and Mingzhou Ding3
1
Institute of Nonlinear Dynamics, Xi’an Jiaotong University, Xi’an 710049, Peoples Republic of China
Department of Mathematics and Center for Theoretical Studies, Indian Institute of Science, Bangalore 560012, India
3
Center for Complex Systems and Brain Sciences, Florida Atlantic University, Boca Raton, Florida 33431
2
We consider the stability of synchronized states 共including equilibrium point, periodic orbit, or chaotic
attractor兲 in arbitrarily coupled dynamical systems 共maps or ordinary differential equations兲. We develop a
general approach, based on the master stability function and Gershgörin disk theory, to yield constraints on the
coupling strengths to ensure the stability of synchronized dynamics. Systems with specific coupling schemes
are used as examples to illustrate our general method.
Large networks of coupled dynamical systems that exhibit
synchronized static, periodic, or chaotic dynamics are subjects of great interest in a variety of fields ranging from biology 关1兴 to semiconductor lasers 关2兴 to electronic circuits
关3兴. For a given problem it is essential to know the extent to
which the coupling strengths can be varied so that the synchronized state remains stable. Early attempts 关4,5兴 at this
question have typically looked either at systems of very
small size or at very specific coupling schemes 共diffusive
coupling, global all to all coupling, etc., with a single coupling strength兲. Recent work 关6,7兴 introduced the notion of a
master stability function that enables the analysis of general
coupling topologies. This function defines a region of stability in terms of the eigenvalues of the coupling matrix. In this
paper we present a general method that provides explicit
constraints on the coupling strengths themselves by combining the master stability function with the Gershgörin disk
theory. Our approach is applicable to both coupled maps and
coupled ordinary differential equations 共ODEs兲. Commonly
studied coupling schemes are used as illustrative examples.
Coupled maps. The system we consider is represented by
xi 共 n⫹1 兲 ⫽f„xi 共 n 兲 …⫹
1
N
N
兺
j⫽1
G i j H„x j 共 n 兲 …,
共1兲
where xi (n) is the M-dimensional state vector of the ith map
at time n and H:R M →R M is the coupling function. We define
G⫽ 关 G i j 兴 as the coupling matrix, where G i j gives the coupling strength from map j to map i. The condition 兺 j G i j
⫽0 is imposed to ensure that synchronized dynamics is a
solution to Eq. 共1兲.
Linearizing Eq. 共1兲 around the synchronized state x(n),
which evolves according to x(n⫹1)⫽f„x(n)…, we have
zi 共 n⫹1 兲 ⫽J„x共 n 兲 …•zi 共 n 兲 ⫹
1
N
N
兺 G i j DH„x共 n 兲 …•zj 共 n 兲 ,
j⫽1
共2兲
where zi (n) denotes the ith map’s deviations from x(n), J
(•) is the M ⫻M Jacobian matrix for f, and DH(•) is the
Jacobian of the coupling function H. In terms of the M ⫻N
matrix S(n)⫽„z1 (n) z2 (n) ••• zN (n)…, Eq. 共2兲 can be recast
as
S共 n⫹1 兲 ⫽J„x共 n 兲 …•S共 n 兲 ⫹
1
DH„x共 n 兲 …•S共 n 兲 •GT . 共3兲
N
According to the theory of Jordan canonical forms, the stability of Eq. 共3兲 is determined by the eigenvalue ␭ of G.
Denote the corresponding eigenvector by e and let u(n)
⫽S(n)e. Then
冉
冊
1
u共 n⫹1 兲 ⫽ J„x共 n 兲 …⫹ ␭DH„x共 n 兲 … •u共 n 兲 .
N
共4兲
So the stability problem originally formulated in the M ⫻N
space has been reduced to a problem in an M ⫻M space
where it is often the case that M ⰆN. It is worth mentioning
that this eigenvalue based analysis is valid even if the coupling matrix G is defective 关8兴.
We note that ␭⫽0 is always an eigenvalue of G and its
corresponding eigenvector is (1 1•••1) T due to the synchronization constraint 兺 Nj⫽1 G i j ⫽0. In this case, Eq. 共4兲 can be
used to generate the Lyapunov exponents for the individual
system, which we denote by h 1 ⫽h max ⭓h 2 ⭓•••⭓h M .
These exponents describe the dynamics within the synchronization manifold defined by xi ⫽x ᭙ i.
The subspace spanned by the remaining eigenvectors is
transverse to the synchronization manifold, in which the dynamics will be stable if the transverse Lyapunov exponents
are all negative 关9兴. To examine this problem, we treat ␭ in
Eq. 共4兲 as a complex parameter and calculate the maximum
Lyapunov exponent ␮ max as a function of ␭. This function is
referred to as the master stability function by Pecora and
Carroll 关6兴. The region in the 关 Re(␭),Im(␭) 兴 plane where
␮ max ⬍0 defines a stability zone denoted by ⍀. Figure 1
shows a schematic of two possible configurations of ⍀.
Whether ⍀ is an unbounded area 关Fig. 1共a兲兴 or a bounded
one 关Fig. 1共b兲兴 is contingent on the coupling scheme and
other system parameters. The origin, which is the zero eigenvalue of G, may or may not lie in the stability zone. For
example, for equilibrium or periodic state in coupled maps,
the origin is in ⍀, but for chaos, it lies outside of ⍀. We note
that, typically, ⍀ is obtained numerically. In some instances
analytical results are possible 共see below兲.
G⫽
冉
G 11
rT
s
GN⫺1
冊
共6兲
with r⫽(G 12 , . . . ,G 1N ) T , s⫽(G 21 , . . . ,G N1 ) T , and
GN⫺1 ⫽
冉
•••
G 22
⯗
G N2
G 2N
⯗
⯗
•••
G NN
冊
共7兲
.
Choose a matrix P in the form
P⫽
冉
1
0T
eN⫺1
IN⫺1
冊
共8兲
.
Here IN⫺1 is the (N⫺1)⫻(N⫺1) identity matrix. Similarity
transformation of G by P yields
À1
P •G"P⫽
冉
˜␭
rT
0
GN⫺1 ⫺eN⫺1 rT
冊
.
共9兲
Since P⫺1 GP and G have identical eigenvalue spectra, the
(N⫺1)⫻(N⫺1) matrix
D1 ⫽GN⫺1 ⫺eN⫺1 rT
FIG. 1. Schematic illustrations of the stability zone: 共a兲 unbounded area, 共b兲 bounded area.
Clearly, if all the transverse eigenvalues of G lie within
⍀, then the synchronized state is stable. Here we seek constraints applicable directly to the coupling strengths. This
problem is dealt with by combining the master stability function with the Gershgörin disk theory.
The Gershgörin disk theorem 关10兴 states that all the eigenvalues of an n⫻n matrix A⫽ 关 a i j 兴 are located in the union of
n disks 共called Gershgörin disks兲, where each disk is given
by
再
z苸C: 兩 z⫺a ii 兩 ⬍
兺
j⫽i
冎
兩 a ji 兩 ,i⫽1,2, . . . ,n.
共5兲
To apply this theorem to the transverse eigenvalues, we need
to remove ␭⫽0. We apply an order reduction technique in
matrix theory 关11兴, which leads to a (N⫺1)⫻(N⫺1) matrix
D whose eigenvalues are the same as the eigenvalues of G
except for ␭⫽0.
Suppose that, for a given matrix G, we have the knowledge of one of its eigenvalues ˜␭ and the eigenvector e.
Through proper normalization we can make any component
of e equal to 1. Here, without loss of generality, we assume
that the first component is made equal to 1, namely, e
T
) T . Rewrite G in the following block form:
⫽(1,eN⫺1
共10兲
assumes the eigenvalues of G sans ˜␭ . We can obtain N different versions of the reduced matrix, which we denote by
Dk (k⫽1,2, . . . ,N), depending on which component of e is
made equal to 1.
Applying the above technique to the coupling matrix G
by letting ˜␭ ⫽0 and e⫽(1 1 . . . 1) T we get Dk ⫽ 关 d ki j 兴 ,
where d ki j ⫽G i j ⫺G k j . From the Gershgörin theorem the stability conditions of the synchronized dynamics are expressed
as follows:
共1兲 The center of every Gershgörin disk of Dk lies inside
the stability zone ⍀. That is, (G ii ⫺G ki ,0)苸⍀.
共2兲 The radius of every Gershgörin disk of Dk satisfies the
inequality
N
兺
j⫽1,j⫽i
兩 G ji ⫺G ki 兩 ⬍ ␦ 共 G ii ⫺G ki 兲 ,i⫽1,2, . . . ,N and i⫽k.
Here ␦ (x) is the distance from point x on the real axis to the
boundary of the stability zone ⍀.
As k varies from 1 to N, we obtain N sets of stability
conditions. Each one provides sufficient conditions constraining the coupling strengths.
Coupled ODEs. The above procedure for obtaining stability bounds can also be applied to coupled identical ODEs
written as
1
ẋ ⫽F共 x 兲 ⫹
N
i
i
N
兺
j⫽1
G i j H共 x j 兲 ,
共11兲
where xi is the M-dimensional vector of the ith node. The
dynamics of the individual node is ẋ⫽F(x). Linearizing
around the synchronized state, we get
˙
zi ⫽J共 x兲 •zi ⫹
1
N
兺 G i j DH共 x兲 •zj ,
j⫽1
共12兲
i
where z denotes deviations from x, J(•) and DH(•) are the
M ⫻M Jacobian matrices for the functions F and H, respectively. Adopting the Jordan canonical form, we obtain
冋
u̇⫽ J共 x兲 ⫹
G i j ⬎h max, ᭙ i, j.
N
册
1
␭DH共 x兲 u,
N
共13兲
where ␭ is an eigenvalue of G. Performing the same analysis
as for coupled maps, we obtain the same stability conditions
as given above.
Examples. We now illustrate the general approach by applying the above results to two examples where analytical
results are possible. In the first example we consider the
coupled differential equation systems with H(x)⫽x 关7兴. It is
easy to see that DH is an M ⫻M identity matrix. The
Lyapunov exponents for Eq. 共13兲 are easily calculated since
the identity matrix commutes with J(x). Denoting them by
␮ 1 (␭), ␮ 2 (␭), . . . , ␮ M (␭), we have
We show here that Eq. 共19兲 is a consequence of the more
general stability conditions given in Eq. 共18兲. This can be
seen as follows. First consider k⫽1. Substituting G ii ⫽
⫺ 兺 Nj⫽1,j⫽i G ji 共synchronization condition兲 and simplifying
we get the following equation:
N
兺
N
j⫽2,j⫽i
兩 G ji ⫺G 1i 兩 ⫺
共15兲
In other words, the stability zone ⍀ is the region defined by
Re(␭)⬍⫺Nh max . The distance function from the center of
each Gershgörin disk to the stability boundary is given by
␦ (G ii ⫺G ki )⫽⫺Nh max ⫺(G ii ⫺G ki )
(i⫽1, . . . ,N,i⫽k).
Thus, the kth set of stability conditions is
共 G ii ⫺G ki 兲 ⬍⫺Nh max ,
共16兲
N
兺
j⫽1,j⫽i
兩 G ji ⫺G ki 兩 ⬍⫺Nh max ⫺ 共 G ii ⫺G ki 兲 ,
i⫽1,2, . . . ,N, i⫽k.
N
兺
N
共17兲
N
兺
兺
j⫽1,j⫽i⫽2
G ji ⫺2G 2i ⬍⫺Nh max ,
i⫽2
⯗
N⫺1
兺
j⫽1,j⫽i
共21兲
N⫺1
兩 G ji ⫺G Ni 兩 ⫺
兺
j⫽1,j⫽i
G ji ⫺2G Ni ⬍⫺Nh max , i⫽N.
If we take the average of the inequalities over k, cancellation
takes place, resulting in a simplified inequality that will be
satisfied if the sufficient condition given in Eq. 共19兲 is met.
In other words, the previously derived stability condition is
obtained as a special case when we require the coupling
strengths to meet the N stability conditions simultaneously.
In the second example, we consider a coupled map with
H⫽f 关5兴. Under this assumption, DH⫽J and the linearized
equation 关cf. Eq. 共4兲兴 reduces to
共22兲
The Lyapunov exponents for Eq. 共22兲 are easily calculated
analytically. Denoting them by ␮ 1 (␭), ␮ 2 (␭), . . . , ␮ M (␭),
we have
␮ i 共 ␭ 兲 ⫽h i ⫹ln兩 ␭/N⫹1 兩 , i⫽1,2, . . . ,M .
共23兲
For stability, we require ␮ max (␭)⫽h max ⫹ln兩␭/N⫹1兩⬍0. In
other words, the stability zone is defined by
兩 G ji ⫺G ki 兩 ⫹ 共 G ii ⫺G ki 兲 ⬍⫺Nh max ,
i⫽1,2, . . . ,N, i⫽k.
兩 G ji ⫺G 2i 兩 ⫺
u共 n⫹1 兲 ⫽ 共 ␭/N⫹1 兲 J„x共 n 兲 …u共 n 兲 .
It is obvious that the second inequality implies the first one.
So the stability condition for the synchronized state 共whether
an equilibrium, periodic, or chaotic state兲 is given by
j⫽1,j⫽i
G ji ⫺2G 1i ⬍⫺Nh max ,i⫽1.
If G ji ⫺G 1i is positive for all allowed i and j values, it is
easy to see that the above stability condition is satisfied
given the condition in Eq. 共19兲. However, if more than two
such terms are negative, we have a problem. We can get
around this by considering the other (N⫺1) sets of stability
conditions obtained by setting k⫽2,3, . . . ,N in Eq. 共18兲:
共14兲
For stability, we require the transverse Lyapunov exponents
(␭⫽0) to be negative. This is equivalent to the statement
that the maximum Lyapunov exponent is less than zero:
1
␮ max 共 ␭ 兲 ⫽h max ⫹ Re共 ␭ 兲 ⬍0.
N
兺
j⫽2,j⫽i
共20兲
j⫽1,j⫽i⫽2
1
␮ i 共 ␭ 兲 ⫽h i ⫹ Re共 ␭ 兲 ,i⫽1,2, . . . ,M .
N
共19兲
共18兲
When the coupling is symmetric, i.e., G i j ⫽G ji , Rangarajan
and Ding 关12兴, based on the use of Hermitian and positive
semidefinite matrices, derived a very simple stability constraint
兩 ␭⫹N 兩 ⬍Nexp共 ⫺h max 兲 .
共24兲
The distance from the center of each Gershgörin disk to the
boundary is easily calculated to be ␦ (G ii ⫺G ki )⫽Nexp
(⫺hmax)⫺兩N⫹Gii⫺Gki兩 (i⫽1, . . . ,N, i⫽k). Thus, the conditions of stability are
N
For each k from 1 to N, we obtain a set of sufficient stability
conditions.
In Ref. 关12兴, a simple stability bound for synchronized
chaos in the case of symmetric coupling was obtained as
This can again be derived from the general stability condition in Eq. 共25兲 with the averaging technique used above.
In summary, we have set up a general formalism to study
the stability of synchronized state in coupled identical maps
and ordinary differential equations. We have also considered
the often used coupling function for coupled maps and
coupled ODEs and given analytical results in these cases. We
have also shown that known stability bounds can be derived
from our more general results.
关 1⫺exp共 ⫺h max 兲兴 ⬍G i j ⬍ 关 1⫹exp共 ⫺h max 兲兴 , ᭙ i, j.
共26兲
This work was supported by U.S. ONR Grant No.
N00014-99-1-0062.
关1兴 R.R. Klevecz, J. Bolen, and O. Duran, Int. J. Bifurcation
Chaos Appl. Sci. Eng. 2, 941 共1992兲; D. Hansel and H. Sompolinsky, Phys. Rev. Lett. 68, 718 共1992兲; D. Hansel, Int. J.
Neural Syst. 7, 403 共1996兲; F. Pasemann, Physica D 128, 236
共1999兲.
关2兴 H.G. Winful and L. Rahman, Phys. Rev. Lett. 65, 1575 共1990兲;
R. Li and T. Erneux, Opt. Commun. 99, 196 共1993兲; Phys. Rev.
A 49, 1301 共1994兲; R. Roy and K.S. Thornburg, Jr., Phys. Rev.
Lett. 72, 2009 共1994兲; K. Otsuka, R. Kawai, S. Hwong, J. Ko,
and J. Chern, ibid. 84, 3049 共2000兲.
关3兴 C.W. Wu and L.O. Chua, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 42, 430 共1995兲; S. Jankowski, A. Londei,
C. Mazur, and A. Lozowski, Int. J. Electron. 79, 823 共1995兲;
G. Filatrella, B. Straughn, and P. Barbara, J. Appl. Phys. 90,
5675 共2001兲.
关4兴 H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 共1983兲;
I. Waller and R. Kapral, Phys. Lett. 105A, 163 共1984兲; V.S.
Afraimovich et al., Sov. Radiophys. 29, 795 共1986兲; G.L.
Oppo and R. Kapral, Phys. Rev. A 36, 5820 共1987兲; L.A. Bunimovich and Ya.G. Sinai, Nonlinearity 1, 491 共1988兲; J.F.
Heagy, L.M. Pecora, and T.L. Carroll, Phys. Rev. E 50, 1874
共1994兲; L. Kocarev and U. Parlitz, Phys. Rev. Lett. 76, 1816
共1996兲; R. Brown and N.F. Rulkov, ibid. 78, 4189 共1997兲; N.F.
Rulkov and M.M. Sushchik, Int. J. Bifurcation Chaos Appl.
Sci. Eng. 7, 625 共1997兲; P.C. Bresloff, Phys. Rev. E 60, 2160
共1999兲; P. Glendinning, Phys. Lett. A 259, 129 共1999兲; V.N.
Belykh, I.V. Belykh, and M. Hasler, Phys. Rev. E 62, 6332
共2000兲; V.N. Belykh, I.V. Belykh, and E. Mosekilde, ibid. 63,
036216 共2001兲.
关5兴 A.S. Pikovsky, Z. Phys. B: Condens. Matter 55, 149 共1984兲; K.
Kaneko, Prog. Theor. Phys. 72, 480 共1984兲; K. Kaneko,
Physica D 23, 436 共1986兲; N. Chatterjee and N. Gupte, Phys.
Rev. E 53, 4457 共1996兲; D.J. Gauthier and J.C. Bienfang, Phys.
Rev. Lett. 77, 1751 共1996兲; M. Ding and W. Yang, Phys. Rev.
E 56, 4009 共1997兲; K. Zhu and T. Chen, ibid. 63, 067201
共2001兲; J. Jost and M.P. Joy, ibid. 65, 016201 共2002兲.
关6兴 L.M. Pecora and T.L. Carroll, Phys. Rev. Lett. 80, 2109
共1998兲; K.S. Fink, G. Johnson, T.L. Carroll, and L.M. Pecora,
Phys. Rev. E 61, 5080 共2000兲; M. Barahona and L.M. Pecora,
Phys. Rev. Lett. 89, 054101 共2002兲.
关7兴 G. Hu, J. Yang, and W. Liu, Phys. Rev. E 58, 4440 共1998兲; M.
Zhan, G. Hu, and J. Yang, ibid. 62, 2963 共2000兲.
关8兴 M. W. Hirsch and S. Smale, Differential Equations, Dynamical
Systems, and Linear Algebra 共Academic Press, New York,
1974兲.
关9兴 We note that since the Lyapunov exponent h max is computed
from typical initial conditions, the stability here is with respect
to the blowout bifurcation. Bubbling transition can occur while
the parameters are still within the bound derived in this work.
For details on these phenomena see E. Ott and J.C. Sommerer,
Phys. Lett. A 188, 39 共1994兲; P. Ashwin, J. Buescu, and I.
Stewart, Phys. Lett. 193, 126 共1994兲; S.C. Venkataramani
et al., Phys. Rev. Lett. 77, 5361 共1996兲; Phys. Rev. E 54, 1346
共1996兲.
关10兴 R. A. Horn and C. R. Johnson, Matrix Analysis 共Cambridge
University Press, Cambridge, 1985兲.
关11兴 G. H. Golub and C. F. Van Loan, Matrix Computations 共Johns
Hopkins University Press, Baltimore, 1996兲.
关12兴 G. Rangarajan and M. Ding, Phys. Lett. A 296, 204 共2002兲.
兺 兩 G ji ⫺G ki兩 ⫹ 兩 N⫹ 共 G ii ⫺G ki 兲 兩 ⬍Nexp共 ⫺h max 兲 ,
j⫽1,j⫽i
i⫽1, . . . ,N, i⫽k, k⫽1 or 2 or ••• or N.
共25兲