General stability analysis of synchronized dynamics in coupled systems

General stability analysis of synchronized dynamics in coupled systems

Yonghong Chen,^1,3Govindan Rangarajan,²and Mingzhou Ding³

1Institute of Nonlinear Dynamics, Xi’an Jiaotong University, Xi’an 710049, Peoples Republic of China

2Department of Mathematics and Center for Theoretical Studies, Indian Institute of Science, Bangalore 560012, India

3Center for Complex Systems and Brain Sciences, Florida Atlantic University, Boca Raton, Florida 33431

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 Gershgo¨rin 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 sub- jects of great interest in a variety of fields ranging from bi- ology 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 syn- chronized 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 cou- pling 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 stabil- ity 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 combin- ing the master stability function with the Gershgo¨rin 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 _j

兺

_⫽^N₁ ^{Gi jH„xj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^Mis the coupling function. We define G⫽关G_{i j}兴 as the coupling matrix, where G_{i j} gives the cou- pling strength from map j to map i. The condition 兺jG_{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 _j

兺

_⫽^N₁ ^{Gi jDH„x共n兲…•zj共n兲,}

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) ••• z^N(n)…, Eq.共2兲can be recast as

S共n⫹1兲⫽J„x共n兲…•S共n兲⫹1

NDH„x共n兲…•S共n兲•G^T. 共3兲 According to the theory of Jordan canonical forms, the sta- bility 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

u共n⫹1兲⫽

冉

^{J„x共n兲…⫹N1␭DH„x共n兲…}

冊

^{•u共n兲. 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 MⰆN. It is worth mentioning that this eigenvalue based analysis is valid even if the cou- pling 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)^Tdue to the synchro- nization constraint 兺j⫽1

N 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 h₁⫽h_max⭓h₂⭓•••⭓h_M. These exponents describe the dynamics within the synchro- nization manifold defined by xⁱ⫽x᭙ i.

The subspace spanned by the remaining eigenvectors is transverse to the synchronization manifold, in which the dy- namics 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␮maxas 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 eigen- value 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.

Clearly, if all the transverse eigenvalues of G lie within

⍀, then the synchronized state is stable. Here we seek con- straints applicable directly to the coupling strengths. This problem is dealt with by combining the master stability func- tion with the Gershgo¨rin disk theory.

The Gershgo¨rin disk theorem关10兴states that all the eigen- values of an n⫻n matrix A⫽关a_{i j}兴are located in the union of n disks 共called Gershgo¨rin disks兲, where each disk is given by

再

^{z苸C:兩z⫺aii兩⬍}

^兺

^{j⫽i 兩aji兩}

冎

^,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 knowl- edge 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

⫽(1,e_N^T_⫺1)^T. Rewrite G in the following block form:

G⫽

冉

^{Gs11 GrNT}⫺1

冊

^共6兲

with r⫽(G₁₂, . . . ,G_1N)^T, s⫽(G₂₁, . . . ,G_N1)^T, and G_N⫺1⫽

冉

^{GG⯗N222 ••••••⯗ GG⯗2NNN}

冊

^{. 共7兲}

Choose a matrix P in the form

P⫽

冉

^eN1⫺1 I_N⁰_⫺^T₁

冊

^{. 共8兲}

Here I_N⫺1is the (N⫺1)⫻(N⫺1) identity matrix. Similarity transformation of G by P yields

P^À1•G"P⫽

冉

^{˜␭0 GN}⫺1⫺^rTe_N⫺1r^T

冊

^{. 共9兲}

Since P^⫺1GP and G have identical eigenvalue spectra, the (N⫺1)⫻(N⫺1) matrix

D¹⫽G_N⫺1⫺e_N⫺1r^T 共10兲 assumes the eigenvalues of G sans˜ . We can obtain N dif-␭ ferent versions of the reduced matrix, which we denote by D^k (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 D^k⫽关d_{i j}^k兴, where d_{i j}^k⫽G_{i j}⫺G_{k j}. From the Gershgo¨rin theorem the sta- bility conditions of the synchronized dynamics are expressed as follows:

共1兲The center of every Gershgo¨rin disk of D^klies inside the stability zone⍀. That is, (G_ii⫺G_ki,0)苸⍀.

共2兲The radius of every Gershgo¨rin disk of D^ksatisfies the inequality

j⫽

兺

1,j⫽i N

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 con- straining the coupling strengths.

Coupled ODEs. The above procedure for obtaining stabil- ity bounds can also be applied to coupled identical ODEs written as

x˙ⁱ⫽F共xⁱ兲⫹1

N

兺

_j⫽^N₁ ^{Gi jH共xj兲, 共11兲}

where xⁱ is the M-dimensional vector of the ith node. The dynamics of the individual node is x˙⫽F(x). Linearizing around the synchronized state, we get

FIG. 1. Schematic illustrations of the stability zone: 共a兲 un- bounded area,共b兲bounded area.

zⁱ

˙⫽J共x兲•zⁱ⫹1 N

兺

j⫽1

N

G_{i j}DH共x兲•z^j, 共12兲 where zⁱdenotes deviations from x, J(•) and DH(•) are the M⫻M Jacobian matrices for the functions F and H, respec- tively. Adopting the Jordan canonical form, we obtain

u˙⫽

冋

^{J共x兲⫹N1 ␭DH共x兲}

册

^{u, 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 ap- plying 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

␮i共␭兲⫽h_i⫹1

NRe共␭兲,i⫽1,2, . . . , M . 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:

␮max共␭兲⫽h_max⫹1

NRe共␭兲⬍0. 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 Gershgo¨rin disk to the stability boundary is given by

␦^(Gii⫺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

j⫽

兺

1,j⫽i N

兩G_ji⫺G_ki兩⬍⫺Nh_max⫺共G_ii⫺G_ki兲,

i⫽1,2, . . . ,N, i⫽k. 共17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⫽

兺

1,j⫽i N

兩G_ji⫺G_ki兩⫹共G_ii⫺G_ki兲⬍⫺Nh_max,

i⫽1,2, . . . ,N, i⫽k. 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 con- straint

G_{i j}⬎h_max, ᭙ i, j . 共19兲 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j⫽1,j⫽i

N G_ji 共synchronization condition兲 and simplifying we get the following equation:

j⫽

兺

2,j⫽i N

兩G_ji⫺G_1i兩⫺_j⫽

兺

_2,j^N_⫽i ^{Gji⫺2G1i⬍⫺Nhmax,i⫽1.}

共20兲 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:

j⫽1,j

兺

⫽i⫽2 N

兩G_ji⫺G_2i兩⫺j⫽1,j

兺

⫽i⫽2 N

G_ji⫺2G_2i⬍⫺Nh_max,

i⫽2

⯗ 共21兲

j⫽

兺

1,j⫽i N⫺1

兩G_ji⫺G_Ni兩⫺_j⫽^N

兺

_1,j^⫺1_⫽i ^{Gji⫺2GNi⬍⫺Nhmax, 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

u共n⫹1兲⫽共␭/N⫹1兲J„x共n兲…u共n兲. 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␭⫹N兩⬍Nexp共⫺h_max兲. 共24兲

The distance from the center of each Gershgo¨rin disk to the boundary is easily calculated to be ␦^(Gii⫺G_ki)⫽Nexp (⫺h_max)⫺兩N⫹G_ii⫺G_ki兩(i⫽1, . . . ,N, i⫽k). Thus, the condi- tions of stability are

j⫽

兺

1,j⫽i N

兩G_ji⫺G_ki兩⫹兩N⫹共G_ii⫺G_ki兲兩⬍Nexp共⫺h_max兲,

i⫽1, . . . ,N, i⫽k, k⫽1 or 2 or••• or N. 共25兲 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

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 can again be derived from the general stability condi- tion 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.

This work was supported by U.S. ONR Grant No.

N00014-99-1-0062.

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