F. Dercole – Politecnico di Milano – ver. 28/01/2013 1/22

COMPLETE SYNCHRONIZATION OF NETWORKS OF DYNAMICAL SYSTEMS:

• Master-slave synchronization • Synchronization via diffusion

o Local stability of synchronization and the

Master Stability Function (MSF) approach

o Global stability of synchronization and the

Connection Graph Stability (CGS) method

o Comparison and extensions

• References and further reading

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MASTER-SLAVE SYNCHRONIZATION The master node represents an isolated oscillator (periodic or chaotic)

)(wgw =& , mRw∈ , 2≥m

that generates an input signal )(wu ϕ= .

Slave nodes represent identical and non autonomous dynamical systems

),( )()( uxfx ii =& , ni Rx ∈)( , 2≥n , Ni ,,2,1 K=

influenced by the input u .

The unforced slave node )0,(xfx =& is an oscillator (periodic or chaotic).

The n -dimensional synchronization (linear) manifold

{ })()2()1()()1( :,, NN xxxxx ====Σ LK

is obviously invariant! The synchronous solution )()()()( )()2()1( txtxtxtx N ==== L , 0≥t , is a solution of the isolated ),( uxfx =&

Given a non synchronous initial condition { })0(),0(,),0( )()2()1( Nxxx K (and any )0(w ), does the solution tend toward a synchronous solution?

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Consider a single master-slave pair, say slave node i : )(wgw =& , ))(,( )()( wxfx ii ϕ=&

Let ⎥⎦

⎤⎢⎣

⎡= )(ix

wz and ⎥

⎦

⎤⎢⎣

⎡=

))(,()(

)( )( wxfwg

zF i ϕ, so we have the autonomous system )(zFz =&

The LE of the system are the LE of the master (LEM) together with the LE of the slave while forced by the master (LESM). These are the LE of the linear system

)()()( ))(,( iii xwxJx δϕδ =& , xuxfuxJ ∂∂= /),(),( Note: LESM are (obviously) different from the LE of the unforced slave (LES), e.g., if the unforced system is chaotic, the largest LES is positive, but LESM can well be all negative!

Main result: if all LESM are negative, then 0)()( →tx iδ from any small )0()(ixδ

Consequence: any group of slaves with sufficiently similar initial condition synchronize.

Indeed, L&& +−=−=− )())(,()))(,(())(,( )()()()()()()( ijiijij xxwxJwxfwxfxx ϕϕϕ

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Example: The “Moran” effect in ecology

Can common environmental fluctuations cause the synchronization of isolated communities? Moran [1953]: isolated communities are described by identical stable linear systems

uAxx ii += )()(&

so that

)( )()()()( ijij xxAxx −=− &&

i.e.

0)()( →− ij xx

from any initial condition and for any input u!

Lynx fur returns in five regiond of Dungeness crabs catches at eight locations Northern Canada of the Pacific coast

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21 ww +

1w2w

3w

The nonlinear explanation [Colombo et al. 2008 AN]:

weather

2121

2121

sup)(

wwwwwwww

wu+−+

+−+== εϕ

input

)1(0 upp +=

),( )()( uxfx ii =&

nodes

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Synchronization occurs if the largest LESM is negative, i.e if the populations do not contribute to chaos (no “biochaos”). Largest LESM

Blue: negative Red: positive

amplitude of input

bio

logic

al p

aram

eter

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SYNCHRONIZATION VIA DIFFUSION

The type of connection

Isolated nodes represent identical and autonomous dynamical systems

)( )()( ii xfx =& , ni Rx ∈)( , 2≥n , Ni ,,2,1 K=

The isolated system )(xfx =& is an oscillator (periodic or chaotic).

The network is undirected, unweighted and connected.

The coupling is of diffusive (i.e. linear) nature, i.e.

∑∑==

−+=1:

)(

1:

)()()( )(ijij aj

i

aj

jii xHdxHdxfx& ∑=

−=N

j

jij

i xHldxf1

)()( )(

where

• ][ ijaA = and ][ ijlL = are the adiacency and laplacian ( NN × ) matrices

• H is an nn× nonnegative matrix defining the diffusion profile

• 0≥d modulates the coupling strength

input output = d ki H x(i)

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The type of synchronization

We consider the strongest (and simplest) case: complete synchronization. The n -dimensional synchronization (linear) manifold

{ }xxxxxx NN =====Σ )()2()1()()1( :,, LK

is invariant, in fact

)()(1

)( xfxHldxfxN

jij

i ∑=

=−=& (recall that L is zero row-sums!)

is independent of i (because we consider identical dynamical systems!).

Note: the synchronous solution )()()()( )()2()1( txtxtxtx N ==== L , 0≥t , is a solution of the isolated system )(xfx =& and is assumed to converge to a synchronous attractor A (periodic or chaotic). Note: this framework seems rather restricted, but there are many interesting applications (in physics, biology, ecology, mechanics,…see [Boccaletti et al. 2006 PR and refs therein]). There are also several extensions (see last page).

)()( tx i

)()( tx j

Σ

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An example [Belykh et al. 2009 JBD]

Each node represents a geographic location (island, patch) that is the habitat of a three-trophic food chain (resource R, consumer C, predator P).The isolated demographic dynamics are described with the classical Rosenzweig-MacArthur model:

CRba

RakRrRR

11

11

1+

−⎟⎠⎞

⎜⎝⎛ −=&

PCba

CaCdCRba

RaeC22

21

11

11 11 +

−−+

=&

PdPCba

CaeP 222

22 1

−+

=&

H is diagonal. It says which are the species that disperse and sets the relative dispersal rates, e.g.

• [ ]001diagHH =′= , only R disperses (e.g. seeds transported by the wind)

• [ ]010diagHH =′′= , only C disperses (e.g. planktonic feeders predated by trouts)

• [ ]100diagHH =′′′= , only P disperses (e.g. herbivores)

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LOCAL STABILITY OF SYNCHRONIZATION AND THE MSF APPROACH Local stability and Lyapunov exponents (LEs)

By linearizing around a synchronous chaotic solution )(tx , one can compute nN LEs, n of which are the exponents

000 1 <≥≥≥≥< nLL LL

of the chaotic attractor A of the isolated system )(xfx =& (the LEs corresponding to perturbations within the synchronization manifold Σ). The )1( −Nn remaining LEs are called transversal to the synchronization manifold. The local stability of synchronization “requires” all transversal LEs to be negative. Notes:

• the condition is not necessary, as LEs are based on linearization • the condition is not even sufficient! Some of the transversal LEs

computed for an unstable periodic synchronous solution γ within the chaotic attractor A might be positive − then, synchronization is attracting in the Milnor sense

• “in practice”, it can be used as an iff condition )()( tx i

)()( tx j

Σ

• Aγ

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The MSF approach [Pecora and Carroll 1998 PRL, Jansen and Lloyd 2000 JMB]

It exploits the spectral properties of the laplacian matrix L

• L is real and symmetric ⇒ real eigenvalues and diagonalizable (N linearly independent and orthogonal eigenvectors)

[ ]iT diagTLT λ= , with T = [eigenvectors]T

• L is zero row-sums ⇒ 0 is eigenvalue associated to the eigenvector [ ]T1,,1K

• L− is Metzler (i.e. 0≥− ijl for ji ≠ ) and irreducible (⇒ Perron–Frobenius theory)

⇒ the dominant eigenvalue dλ is unique and real

⇒ 0 = min{row-sums} ≤≤ dλ max{row-sums} = 0 ⇒ 0=dλ

Thus, the spectrum of L can be ordered as: Nλλλ ≤≤<= K210

Examples

the two-node net complete nets star nets scale-free nets Watt-Strogatz loops

NN === λλ L2

22 =λ

112 === −Nλλ L12 ≈λ 02 →λ ,

2λ

λ N∼ 2N as ∞→N NN =λ

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The MSF also exploits the following “tricky” change of coordinates:

)()()( txx ii −=δ , Ni ,,2,1 K=

The network model gets transformed into

+−= )()()( ))(( ii

ii vHdvtxJv λ& h.o.t., Ni ,,2,1 K= ,

where xfxJ ∂∂= /)( is the Jacobian of the isolated system.

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1=i ( 01 =λ ) ⇒ )1()1( ))(( vtxJv =& yields the LEs nLL ,,1 K of the synchronous attractor A

Ni ,,2 K= ( 0>iλ ) ⇒ local stability of synchronization “requires” the largest LE of )()()( ))(( i

iii vHdvtxJv λ−=& to be negative

Instead of the above )1( −N linear n-dimensional systems, we can discuss the

Master Stability Equation (MSE): vHvtxJv ε−= ))((& , nRv∈ , 0≥ε

and, in particular, draw the

Master Stability Function (MSF): == )(εMSFMSF the largest LE of the MSE Notes:

• 0)0( >MSF ( 0= ) if the synchronous attractor A is chaotic (periodic) • the MSF only depends on the isolated system (f ) and on the diffusion profile (H ),

it is independent on the topology of the network. • drawing the MSF is computationally much heavier than computing 1−N of its values!

local stability of synchronization “requires” 0)( <idMSF λ , Ni ,,2 K=

MSF

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Types of MSF

type I synchronization is not possible on any network type II synchronization is possible on any network, provided

2/λε sd >

e.g.: sync is easier for large N on complete nets ( N=2λ )

sync is hard for large N on Watts-Strogatz loops ( 02 →λ )

type III (= non type I or II) sync is possible (for suitable d) for networks with

12 ελ >d and 2ελ <Nd ⇔ 122 // εελλ <N

sync is easier in nets with small 2/λλN

e.g.: sync can be lost for increasing d sync is hard for large N on Watts-Strogatz loops ( 2/ λλ N ∼ 2N )

MSF

MSF

MSF

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A “trivial” type II example

H = I ⇒ εε −= )0()( MSFMSF

A type III example

MSFs for coupled Rössler oscillators

bold/regular lines: chaotic/periodic regime solid lines: x1-coupling ⇒ type III dashed lines: x2-coupling ⇒ type II Small-world networks

Note: sync is favored by rewiring/adding (for type III MSF 2/λλN drops by increasing p).

ε

02 =− λN = 500 p = 0.1

1ε 2εMSF

ε

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The ecological example

(I) [ ]001diagHH =′= (II) [ ]010diagHH =′′= (III) [ ]100diagHH =′′′=

Note: sync is favored if intermediate trophic levels diffuse more than bottom/top levels.

MSF

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GLOBAL STABILITY OF SYNCHRONIZATION AND THE CGS METHOD Note: obviously global stability implies local stability. A “strong” assumption

Given the isolated system (f ) and the type of coupling (H ), we assume that the network of two coupled systems “globally synchronizes” for all 02 >> dd . Notes:

• it is a sort of “MSF-type-II” assumption • this must be shown for each given pair (f , H ) • it is the most difficult step, but can be easily tested numerically!

The ecological example [Belykh et al. 2009 JBD, Appendix]

• dispersion profile [ ]111diagH = (the analysis is similar for other profiles) • boundedness of solutions: it can be shown that the region of nonnegative

)2,1(R , )2,1(C , )2,1(P bounded by the plane 0)2()2()2()1()1()1( =−+++++ cPCRPCR is an

absorbing domain for L=> *cc

• Lyapunov function: ( ) 8/)()()( 2)1()2(2)1()2(2)1()2( PPCCRR −+−+−=Φ By exploiting the above bound, it can be shown that 0<Φ& when L=> 2dd

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Notes:

• the EIG condition is similar to the MSF type-II condition ( 2/λε sd > )

• CGS avoids spectra computations and directly links sync with network topology

• the choice of ijP ’s is not unique; shortest paths do not always yield the best bound

• overloaded links (involved by many ijP ’s) make sync difficult (high kz )

• adding a link with constant N can only make sync easier

• the computation of kk zmax for various network topologies can be found in [Belykh et al. 2004 PD, 2005 CHAOS, 2005 IJBC, 2006 PD]

The eigenvalue (EIG) method [Wu and Chua 1996 I3ESC-I]

Synchronization is globally stable if 22 /2 λdd >

The CGS method [Belykh et al. 2004 PD]

• select one path ijP from each node i to each node j ( ji < )

• compute the sum kz of the lengths of all paths containing link k , mk ,,1K=

Synchronization is globally stable if kmk

zNdd

≤≤>

12 max2

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Sketch of the proof (for diagonal H)

• Let 2/)()( )1()2()1()2(2 xxVxxV T −−=

be the Lyapunov function used to show global sync in the two-node net

• Consider the Lyapunov function ∑∑−

= >

−−=1

1

)()()()( )()(21 N

i ij

ijTijN xxVxxV for the network model

• It can be shown that NV& is negative definite if, for all ∑∑ ==== N

ii

iNi

iip wvX 2

)(2

)( βα , )()( i

ii vLv λ=

∑∑∑−

= >=

−>−1

1

2)()(2

1

2)()( )(2)(N

i ij

ip

jp

m

k

ip

jp xx

Ndxxd kk , np ,,1K=

• Variables )( )()( ip

jp xx − can be eliminated in the above inequality by substituting

∑∈

−=−ij

kk

Pk

ip

jp

ip

jp xxxx )( )()()()(

(sort of “Kirchhoff” rule)

and by bounding 2)()(2)()( )())(( ∑∑

∈∈−≤−

ij

kk

ij

kk

Pk

ip

jpij

Pk

ip

jp xxPxx (Cauchy-Schwarz inequality)

• Collecting terms yields kmk

zNdd

≤≤>

12 max2

22 pp

Tp XXLX λ≥=

2pXN=

)()( ii wNFw =it results:

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The ecological example

Synchronization is globally stable if 22 39

62 ddd =>

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COMPARISON AND EXTENSIONS General conclusion: synchronization (when possible) requires

• large coupling when connections are local (small 2λ , large kk zmax )

• small coupling when connections are global (large 2λ , small kk zmax )

MSF / CGS comparison and extensions

MSF Pros

• almost “iff” condition

• easy to apply

• also nonlinear (diffusive) coupling

Cons

• only local stability

• less extensions available directed and weighted nets – slightly nonidentical oscillators

CGS Pros

• global stability

• more extensions available directed and weighted nets – time-varying coupling – slightly nonidentical oscillators

Cons

• conservative “if” condition

• can be difficult to apply

• only linear (diffusive) coupling

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REFERENCES AND FURTHER READING

[Belykh et al. 2004 PD] V.N. Belykh, I. Belykh, and M. Hasler (2004) Connection graph stability method for synchronized coupled chaotic systems, Phys. D 195, 159–187.

[Belykh et al. 2005 CHAOS] I. Belykh, V.N. Belykh, and M. Hasler (2006) Synchronization in asymmetrically coupled networks with node balance, Chaos 16, 015102.

[Belykh et al. 2005 IJBC] I. Belykh, M. Hasler, M. Lauret, and H. Nijmeijer (2005) Synchronization and graph topology, Int. J. Bifurc. Chaos 11, 3423–3433.

[Belykh et al. 2006 PD] I. Belykh, V.N. Belykh, and M. Hasler (2006) Generalized connection graph method for synchronization in asymmetrical networks, Phys. D 224, 42–51.

[Belykh et al. 2009 JBD] I. Belykh, C. Piccardi, and S. Rinaldi (2009) Synchrony in tritrophic food chain metacommunities. J. Biol. Dynamics 3, 497–514.

[Boccaletti et al. 2006 PR] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.-U. Hwang (2006) Complex networks: structure and dynamics, Phys. Rep. 424, 175–308.

[Colombo et al. 2008 AN] A. Colombo, F. Dercole, and S. Rinaldi (2008) Remarks on metacommunities synchronization with application to prey-predator systems, Am. Nat. 171, 430–442.

[Jansen and Lloyd 2000 JMB] V.A.A. Jansen and A. L. Lloyd (2000) Local stability analysis of spatially homogeneous solutions for multi-patch systems. J. Math. Biol. 41,232–252.

[Pecora and Carroll 1998 PRL] L.M. Pecora and T.L. Carroll (1998) Master stability functions for synchronized coupled systems. Phus. Rev. Lett. 80, 2109–2112.

[Sun et al. 2009 EPL] J. Sun, E.M. Bollt, and T. Nishikawa (2009) Master stability functions for coupled nearly identical dynamical systems. Europhys. Lett. 85, 60011.

[Wu and Chua 1996 I3ESC-I] C.W. Wu and L.O. Chua (1996) On a conjecture regarding the synchronization in an array of linearly couplet dynamical systems. IEEE Trans. Circuits Syst.-I 43, 161–165.