Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

Подождите немного. Документ загружается.

complex Hermitian N  N matrices. By putting a

-invariant Gaussian probability measure

exp trH

=2



dH ð 2 RÞ

on that space, one gets what is called the GUE – the

Gaussian unitary ensemble – which defines the

Wigner–Dyson universality class of unitary symmetry.

Classes AI and AII

Consider next the case G

6¼;, with antiunitary

generator T. Let V



= TV



be any G

-isotypic

component of V invariant under T (the type-I

situation). The mapping U 7!TUT

1

= (U) then is

an automorphism of the groups U(V



), G

and

K = Z



ﬃ U



.IfK



is the subgroup of fixed points

of  in K, the space of good time evolutions can be

identified with the symmetric space K=K



by the

Cartan embedding. Our task is to determine K



To simplify the notation let us write V



 V, R





R, and L



 L. We now ask what happens with

T : V !V in the process of transfer to L  R ﬃ V.

The answer, so we claim, is that T transfers to a

pure tensor made from antiunitary maps  : L !L

and  : R !R,

T ¼   

To prove this claim, let C be the antilinear map

from V to the dual vector space V



by v 7!hv,i.

Because the elements of G

are represented by

unitaries, the C-linear operator CT : V !V



inter-

twines G

-actions:

CT aðgÞ¼g

1



CT ðg 2 G

where a is the automorphism a(g) = T

1

gT. From

the irreducibility of R it follows that the space of

intertwiners R !R



is one dimensional here (Schur’s

lemma). Therefore, CT : L  R !L



 R



must be a

pure tensor (as opposed to a sum of such tensors),

and since C is clearly a pure tensor, so is T . This

completes the proof.

By the involutive property T

= "

= 1),

the two antiunitary factors of T =    cannot

but square to 

= "



and 

= "



where



, "



= 1 are related by "





= "

. The factor 

determines a nondegenerate complex bilinear form

Q : L  L !C by

Qðl

; l

Þ¼ l

; l

ðl

; l

2 LÞ

Since  is antiunitary one has the exchange

symmetry

Qðl

; l

Þ¼ 

;l

¼ "



Qðl

; l

Thus, the complex bilinear form (or pairing) Q is

symmetric for "



= þ1 and alternating for "



= 1.

Knowing the sign of "



= 1 we know the group



. Indeed, an element k 2 K



commutes with T and

after transfer from V to L still commutes with . But

since K



is a subgroup of K = U



, this means that

k 2 K



preserves Q. In the case of "



= þ1, what is

preserved is a symmetric pairing, and therefore K



ﬃ



. For "



= 1, the multiplicity m



must be even

and K



preserves an alternating pairing (or symplec-

tic structure); in that case K



ﬃ USp



, the unitary

symplectic group.

Thus, there is a dichotomy for the sets of good

time evolutions M ﬃ K=K



Class AI : K=K



ﬃ U

ðN ¼ m



Class AII : K=K



ﬃ U

=USp

ð2N ¼ m



Again we are referring to symmetric spaces by the

names they – or rather their simple parts SU

=SO

and SU

=USp

– have in the Cartan classi fication.

In general, there is no immediate means of

predicting the parity "



, and one has no choice but

to go through the steps of constructing .If

 : R !R ha ppens to be G

-invariant, however, the

situation simplifies. In that case  determines a

-invariant pairing R  R !C (in the same way as

 determines Q : L  L !C above). On general

grounds, an irreducible G

-representation space

admits at most one such pairing. If that pairing is

symmetric, then, as we have seen, "



= 1; if it is

alternating, then "



= 1. The parity "



is given by





= "

Example Consider any physical system with spin-

rotation symmetry (G

= SU

) and time-reversal

symmetry. The physical operation of time reversal,

T, commutes with spin rotations and, hence, here

is a case where the factor  in T =    is

-invariant. On fundamental physics grounds one

has T

= (1)

on states with spin S. The spin-S

representation of SU

is known to carry an invariant

pairing which is symmetric or skew depending on

whether the integer 2S is even or odd. Therefore,

= "



and "



= þ1 in all cases.

Thus, T-invariant systems with no symmetries

other than energy and spin invariably are class AI.

By breaking spin-rotation symmetry (G

= {Id},



= þ1) while maintaining T-symmetry for states

with half-integer spin (say single electrons, which

carry spin S = 1=2), one gets "



= 1, thereby

realizing class AII.

The Hamiltonians By passing to the tangent space

of K=K



at unity one obtains Hermitian matrices

with entries that are real numbers (class AI) or real

quaternions (class AII). When K



-invariant Gaussian

208 Symmetry Classes in Random Matrix Theory

probability measures (called GOE resp., GSE) are

put on these spaces, one gets the Wigner–D yson

universality classes of orthogonal resp., symplectic

symmetry. In mesoscopic physics, these are realized

in disordered metals with time-reve rsal invariance

(absence of magnetic fields and magnetic impuri-

ties). Spin-rotation symmetry is broken by strong

spin–orbit scatterers such as gold impurities.

Warning

The word ‘‘symmetry class’’ is not synonymous with

‘‘universality class.’’ Indeed, inside a symmetry class

many different types of physical behavior are

possible. For example, random matrix models for

disordered metallic grains with time-reversal sym-

metry belong to the symmetry class of the example

above (class AI), and so do Ander son tight-binding

models with real hopping. The former are known to

exhibit energy level statistics of universal GOE type,

whereas the lat ter have localized eigenfunctions and

hence level statistics which is expected to approach

the Poisson limit when the system size goes to

infinity.

Disordered Superconductors

When Dirac first wrote down his famous equation in

1928, he assumed that he was writing an equation

for the wave function of the electron. Later, because

of the instability caused by negative-energy solu-

tions, the Dirac equation was reinterpreted (via

second quantization) as an equation for the ferm-

ionic field operators of a quantum field theory. A

similar change of viewpoint is carried out in reverse

in the Hartree–Fock–Bogoliubov mean-field descrip-

tion of quasiparticle excitations in superconductors.

There, one starts from the equations of moti on for

linear superpositions of the electro n creation and

annihilation operators, and reinterprets them as a

unitary quantum dynamics for what might be called

the quasiparticle ‘‘wave function.’’

In both cases – the Dirac equation and the

quasiparticle dynamics of a superconductor – there

enters a structure not present in the standard

quantum mechanics underlying Dyson’s classifica-

tion: the field operators for fermionic particles are

subject to a set of relations called the ‘‘canonical

anticommutation relations,’’ and these are preserved

by the quantum dynamics. Therefore, whenever

second quantization is undone (assuming it can be

undone) to return from field operators to wave

functions, the wave-function dynamics is required to

preserve some extra structure. This puts a linear

constraint on the good Hamiltonians H. For our

purposes, the best viewpoint to take is to attribute

the extra invariant structure to the Hilbert space V,

thereby turning it into a Nambu space.

Nambu Space

Adopting the standard physics conventions of

second quantization, consider some set of single-

particle creation and annihilation operators c

and

, where i = 1, 2, ... labels an orthon ormal system

of single-particle states. Such operators are subject

to the canonical anticommutation relations (CARs)

þ c

¼ 

þ c

¼ 0 ¼ c

þ c

½1

When written in terms of c

þ c

and i(c

 c

), these

become the standard defining relations of a Clifford

algebra over R. Field operators are linear combina-

tions y =

þ f

) with complex coefficients u

and f

Now take H to be some Hamiltonian which is

quadratic in the creation and annihilation operators:

H ¼

i;j





and let H act on field operators y by the

commutator: H  y  [H, y ]. The time evolution of

y is then determined by the Heisenberg equation of

motion

ih

¼ H  y ½2

which integrat es to y (t ) = e

itH=h

 y (0), and is easily

verified to preserve the CARs [1].

The dynamical equation [2] is equivalent to a

system of linear differential equations for the

amplitudes u

and f

. If these are assembled into

vectors, and the W

and Z

into matrices, eqn [2]

becomes

ih









The Hamiltonian matrix on the right-hand side has

some special properties due to Z

= Z

(from

= c

)andW



(from H being self-

adjoint as an operator in Fock space). To keep

track of these properties while imposing some

unitary and antiunitary symmetries, it is best to put

everything in invariant form.

So, let U be the unitary vector space of annihila-

tion operators u =

, and view the creation

operators f =

as lying in the dual vector space



. The field operators y = u þ f then are elements

of the direct sum U  U



=: V, called ‘‘Nambu

Symmetry Classes in Random Matrix Theory 209

space.’’ On V there exists a canonical unitary

structure expressed by

y ;



A second canonical structure on V = U  U



is given

by the symmetric complex bilinear form

fy ;

y g¼

þ f

Þ¼

f ðuÞþf ð

uÞ

where the last expression uses the meaning of f as a

linear function f : U !C. Note that {y ,

y } agrees

with the anticommutator of the field operators,

y þ

yy.

Now recall that the quantum dynamics is deter-

mined by a Ham iltonian H that acts on y by the

commutator H  y = [H, y ]. The one-parameter

groups t 7!e

itH= h

generated by this action (the time

evolutions) preserve the symmetric pairing:

fy ;

y g¼fe

itH=h

y ; e

itH=h

y g

since the anticommutation relations [1] do not

change with time. They also preserve the unitary

structure,

y ;

¼ e

itH= h

y ; e

itH= h

because probability in Nambu space is conserved.

(Physically speaking, this holds true as long as H is

quadratic, i.e., many-body interactions are negligible.)

One can now pose Dyson’s question again: given

Nambu space V and a symmetry group G acting on

it, what is the set of time evolution operators that

preserve the structure of V and are compatible with

G ? From the section ‘‘The framew ork,’’ we know

the answer to be some symmetric space, but which

are the symmetric spaces that occur?

Class D

Consider a superconductor with no symmetries in its

quasiparticle dynami cs, so G = {Id}. (A concrete

example would be a disordered spin-triplet super-

conductor in the vortex phase.) The time evolutions

= e

itH= h

are then constrained only by invariance

of the unitary structure and the symmetric pairing

{ , } of Nambu space. These two structures are

consistent; they are related by particle–hole con-

jugation C:

fy ;

y g¼ Cy ;

which is an antiunitary operator with square C

= þId.

Let V

 V denote the real vector space of fixed

points of C. (The field operators in V

are called of

‘‘Majorana’’ type in physics.) The condition

{y ,

y } = {U

y , U

y } selects a complex orthogonal

group SO(V), and imposing unitarity yields a real

orthogonal subgroup SO(V

) with dim V

2 2N –

a symmetric space of the D family.

When expressed in some basis of Majorana

fermions (meaning a basis of V

), the matrix of

the time evolution generator iH 2 so(V

) is real

skew, and that of H imaginary skew. The simplest

random matrix model for class D, the SO-invariant

Gaussian ensemble of imaginary skew matrices, is

analyzed in the second edition of Mehta’s (1991)

book. From the expressions given by Mehta it is

seen that the level correlation functions at high

energy coincide with those of the Wigner–Dyson

universality class of unitary symmetry. The level

correlations at low energy, however, show different

behavior defining a separate universality class.

This universal behavior at low energies has immedi-

ate physical relevance, as it is precisely the low-

energy quasiparticles that determine the thermal

transport properties of the superconductor at low

temperatures.

Class DIII

Let now magnetic fields and magnetic impurities

be absent, so that time reversal T is a symmetry of

the quasiparticle system: G = {Id, T}. Following the

section ‘‘The framew ork,’’ the set of goo d time

evolutions is M ﬃ K=K



with K = SO(V

)andK



the set of fixed points of U 7!(U) = TUT

1

in K.

What is K



The square of the time-reversal operator is T

= Id

(for particles with spin 1/2), and commutes with

particle–hole conjugation C, which makes P := iCT a

useful operator to consider. Since C by definition

commutes with the action of K,andhencealsowith

that of K



, the subgroup K



has an equivalent

description as



¼fk 2 UðVÞjk ¼ PkP

1

¼ ðkÞg

The operator P is easily seen to have the following

properties: (1) P is unitary, (2) P

= Id, and (3) tr

P = 0. Consequently, P possesses two eigenspaces



of equal dimension, and the condition k = PkP

1

fixes a subgroup U(V

)  U(V



)ofU(V). Since P

contains a factor i =

ﬃﬃﬃﬃﬃﬃ

1

in its definition, it antic-

ommutes with the antilinear operator T. Therefore,

the au tomorphism  exchanges U(V

) with U(V



and the fixed-point set K



is the same as U(V

) ﬃ

. Thus,

M ¼ K=K



ﬃ SO

ðdim V

¼ 2NÞ

a symmetric space in the DIII family. Note that

for particles with spin 1/2 the dimension of V

has

to be even.

210 Symmetry Classes in Random Matrix Theory

By realizing the algebra of involutions C, T as

Cy = (1

 i

)



y and Ty = (1

 i

)



y , the

Hamiltonians H in class DIII are brought into the

standard form

H ¼

0 Z





Z 0



where the 2N  2N matrix Z is complex and skew.

Class C

Next let the spin of the quasiparticles be

conserved, as is the case for a spin-singlet super-

conductor with no spin–orbit scatterers present, and

let time-reversal invariance be broken by a magnetic

field. The symmetry group of the quasiparticle

system then is the spin-rotation group: G = G

Spin

= SU

Nambu space V can be arranged to be a tensor

product V = L  R so that G

acts trivially on L and

by the spinor representation on the spinor space R 

. Since two spinors combine to give a scalar, the

latter comes with an alternating bilinear form a : R 

R !C. In a suitable basis, the anticommutation

relations [1] factor on particle–hole and spin indices.

The symmetric bilinear form {,} of V correspondingly

factors under the tensor product decomposition

V = L  R as

 r

; l

 r

g¼½l

; l

aðr

; r

where [ , ] is an alternating form on L, giving L the

structure of a complex symplec tic vector space.

The good set M now consists of the time

evolutions that, in addition to preserving the

structure of Nambu space, commute with the spin-

rotation group SU

M ¼fU 2 UðVÞjUC ¼ CU; 8g 2 SU

: gU ¼ Ugg

By the last condition, all time evolutions act trivially

on the factor R. The condition UC = CU, which

expresses invariance of the symmetric form of V,

then implies that time evolut ions preserve the

alternating form of L. Time evolutions therefore

are unitary symplectic transformations of L, hence

M = USp(L) ﬃ USp

– a symmetric space of the C

family. The Hamiltonian matrices in class C have

the standard form

H ¼







with W being Hermitian and Z complex and

symmetric.

Class CI

The next class is obtained by taking the time

reversal T as well as the spin rotations g 2 SU

be symmetries of the quasiparticle system.

By arguments that should be familiar by now, the

set of good time evolutions is a symmetric space

M ﬃ K=K



with K = USp(L) and K



the set of fixed

points of  in K. Once again, the question to be

answered is: what is K



? The situation here is very

similar to the one for class DIII, with L and USp(L)

taking the roles of V and SO(V

). By adapting the

previous argument to the present case, one shows

that K



is the same as U(L

) ﬃ U

, where L

is the

positive eigenspace of P = iCT viewed as a unitary

operator on L. Thus,

M ¼ K=K



ﬃ USp

Dirac Fermions: The Chiral Classes

Three large families of symmetric spaces remain to

be implemented. Although these, too, occur in

mesoscopic physics, their most natural realization

is by 4D Dirac fermions in a random gauge field

background.

Consider the Lagrangian L for the Euclidean

spacetime version of QCD with N

 3 colors of

quarks coupled to an SU

gauge field A



L ¼ i



y 



ð@



 A



Þy þ im



The massless Dirac operator D = i



 A



) anti-

commutes with 

= 



. Therefore, in a basis

of eigenstates of 

the matrix of D takes the form

D ¼

0 Z



½3

If the gauge field carries topological charge  2 Z,

the Dirac operator D has at least jj zero modes by the

index theorem. To make a simple model of the

challenging situation where A



is distributed according

to Yang–Mills measure, one takes the matrices Z to be

complex rectangular, of size p  q with p  q = ,and

puts a Gaussian probability measure on that space.

This random matrix model for D captures the

universal features of the QCD Dirac spectrum in the

massless limit.

The exponential of the trunca ted Dirac operator,

itD

(where t is not the time), lies in a space

equivalent to U

pþq

 U

– a symmetric space of

the AIII family. We therefore say that the universal

behavior of the QCD Dirac spectrum is that of

symmetry class AIII.

But hold on! Why are we entitled to speak of a

symmetry class here? By definition, symmetries

Symmetry Classes in Random Matrix Theory 211

always commute with the Hamiltonian, never do

they anticomm ute! (The relation D = 

D 

is not

a symmetry in the sense of Dyson, nor is it a

symmetry in our sense.)

Class AIII

To incorporate the massless QCD Dirac operator

into the present classification scheme, we adapt it

to the Nambu space setting. This is done by

reorganizing the four-component Dirac spinor

y ,



y as an eight-component Majorana spinor ,

to write

m¼0





ð@



 A



Þ

The 8  8 matrices 



are real symmetric besides

satisfying the Clifford relations 







þ 







= 2



A possible tensor product realization is



¼ 1  

 1; 

¼ 

 



¼ 

 

 1; 

¼ 

 

The gauge field in this Majorana representation

is A



= 1  1  (A

()



 A

(þ)





) where A

()



= (1=2)



 A



) are the symmetric and skew parts of



2 su (N

The operator H = i



 A



) is imaginary

skew, therefore e

itH

is real orthogonal. This means

that there exists a Nambu space V with unitary

structure h, i and symmetric pairing { , }, both of

which are preserved by the action of e

itH

. No change

of physical meaning or interpre tation is implied by

the identic al rewriting from Dirac D to Majorana H.

The fact that Dirac fermions are not truly Majorana

is encoded in a U

-symmetry He

iQ

= e

iQ

H gener-

ated by Q = 1  1  

Now comes the essential point: since H obeys



H = H, the chiral ‘‘symmetry’’ H = 

H

with



= 1  

 1 can be recast as a true symmetry:

H ¼þ



H 

¼ THT

1

with antilinear T :  7!



. Thus, the massless

QCD Dirac operator is indeed associated with a

symmetry class in the present, post-Dyson sense:

that is class AIII, realized by self-adjoint operators

on Nambu space with Dirac U

-symmetry and an

antiunitary symmetry T.

Classes BDI and CII

Consider Hamiltonians D still of the form [3] but

now with matrix entries taken from either the real

numbers or the real quaternions. Their one-parameter

groups e

itD

belong to two further families of

symmetric spaces, namely the classes BDIandCII

of Table 1. These large families are known to be

realized as symmetry classes by the massless Dirac

operator with gauge group SU

(for BDI), or with

fermions in the adjoint representation (for CII). For

the details we must refer to Verbaarschot’s (1994)

paper and the recent article by Heinzner et al. (2005).

See also: Classical Groups and Homogeneous Spaces;

Compact Groups and Their Representations;

Determinantal Random Fields; Dirac Fields in Gravitation

and Nonabelian Gauge Theory; Dirac Operator and Dirac

Field; High T

Superconductor Theory; Integrable

Systems in Random Matrix Theory; Lie Groups: General

Theory; Random Matrix Theory in Physics; Random

Partitions; Supersymmetry Methods in Random Matrix

Theory; Symmetries and Conservation Laws.

Further Reading

Altland A, Simons BD, and Zirnbauer MR (2002) Theories of

low-energy quasi-particle states in disordered d-wave super-

conductors. Physics Reports 359: 283–354.

Altland A and Zirnbauer MR (1997) Non-standard symmetry

classes in mesoscopic normal-/superconducting hybrid systems.

Physical Review B 55: 1142–1161.

Dyson FJ (1962) The threefold way: algebraic structure of

symmetry groups and ensembles in quantum mechanics.

Journal of Mathematical Physics 3: 1199–1215.

Heinzner P, Huckleberry A, and Zirnbauer MR (2005) Symmetry

classes of disordered fermions. Communications in Mathe-

matical Physics 257: 725–771.

Helgason S (1978) Differential Geometry, Lie Groups and

Symmetric Spaces. New York: Academic Press.

Mehta ML (1991) Random Matrices. New York: Academic Press.

Weyl H (1939) The Classical Groups: Their Invariants and

Representations. Princeton: Princeton University Press.

Verbaarschot JJM (1994) The spectrum of the QCD Dirac

operator and chiral random matrix theory: the threefold

way. Physical Review Letters 72: 2531–2533.

212 Symmetry Classes in Random Matrix Theory

Synchronization of Chaos

M A Aziz-Alaoui, Universite

du Havre, Le Havre,

France

Introduction: Chaotic Systems Can

Synchronize

Synchronization is a ubiquitous phenomenon char-

acteristic of many processes in natural systems and

(nonlinear) science. It has permanently remained an

objective of intensive research and is today consid-

ered as one of the basic nonlinear phenomena

studied in mathematics, physics, engineering, or life

science. This word has a Greek root, syn = common

and chronos = time, which means to share the

common time or to occur at the same time, that is,

correlation or agreement in time of different

processes (Boccaletti et al. 2002). Thus, synchroni-

zation of two dynamical systems generally means

that one system somehow traces the motion of

another. Indeed, it is well known that many coupled

oscillators have the ability to adjust some common

relation that they have between them due to weak

interaction, which yields to a situation in which a

synchronization-like phenomenon takes place.

The original work on synchronization involved

periodic oscillators. Indeed, observations of (peri-

odic) synchronization phenomena in physics go back

at least as far as C Huygens (1673), who, during his

experiments on the development of improved pen-

dulum clocks, discovered that two very weakly

coupled pendulum clocks become synchronized in

phase: two clocks hanging from a common support

(on the same beam of his room) were found to

oscillate with exactly the same frequency and

opposite phase due to the (weak) coupling in terms

of the almost imperceptible oscillations of the beam

generated by the clocks.

Since this discovery, periodic synchronization has

found numerous applications in various domains,

for instance, in biological systems and living nature

where synchronization is encountered on different

levels. Examples range from the modeling of the

heart to the investigation of the circadian rhythm,

phase locking of respiration with a mechanical

ventilator, synchronization of oscillations of human

insulin secretion and glucose infusion, neuronal

information processing within a brain area and

communication between different brain areas. Also,

synchronization plays an important role in several

neurological diseases such as epilepsies and patho-

logical tremors, or in different forms of cooperative

behavior of insects, animals, or humans (Pikovsky

et al. 2001).

This process may also be encountered in celestial

mechanics, where it explains the locking of revolu-

tion period of planets and satellites.

Its view was strongly broadened with the devel-

opments in radio engineering and acoustics, due to

the work of Eccles and Vincent, 1920, who found

synchronization of a triode generator. Appleton,

Van der Pol, and Van der Mark, 1922–27, ha ve,

experimentally and theoretically, extended it and

worked on radio tube oscillators, where they

observed entrainment when driving such oscillators

sinusoidally, that is, the frequency of a generator

can be synchronized by a weak external signal of a

slightly different frequency.

But, even though original notion and theory of

synchronization implies periodicity of oscillators,

during the last decades, the notion of synchroniza-

tion has been generalized to the case of interacting

chaotic oscillator s. Indeed, the disco very of determi-

nistic chaos introduced new types of oscillating

systems, namely the chaotic generators.

Chaotic oscillators are found in many dynamical

systems of various origins; the behavior of such

systems is characterized by instability and, as a

result, limited predictabi lity in time.

Roughly speaking, a system is chaotic if it is

deterministic, has a long-term aperiodic behavior,

and exhibits sensitive dependence on initial condi-

tions on a closed invariant set (the chaos theory is

discussed in more detail elsewhere in this encyclo-

pedia) (see Chaos and Attractors).

Consequently, for a chaotic system, trajectories

starting arbitrarily close to each other diverge

exponentially with time, and quickly become uncor-

related. It follows that two identical chaotic syst ems

cannot synchronize. This means that they cannot

produce identic al chaotic signals, unless they are

initialized at exactly the same point, which is in

general physically impossible. Thus, at first sight,

synchronization of chaotic systems seems to be

rather surprising because one may intuitively (and

naively) expect that the sensitive dependence on

initial conditions would lead to an immediate

breakdown of any synchronization of coupled

chaotic systems . This scenario in fact led to the

belief that chaos is uncontrollable and thus unusa-

ble. Despite this, in the last decades, the search for

synchronization has moved to chaotic syst ems.

Significant research has been done and, as a result,

Yamada and Fujisaka (1983), Afraimovich et al.

(1986),andPecora and Carroll (1990) showed that

Synchronization of Chaos 213

two chaotic systems could be synchronized by

coupling them: synchronization of chaos is actual

and chaos could then be exploitable. Ever since,

many rese archers have discussed the theory and the

design or applications of synchronized motion in

coupled chaotic systems. A broad variety of applica-

tions has emerged, for example, to increase the

power of lasers, to synchronize the output of

electronic circuit s, to control oscillation s in chemical

reactions, or to encode electronic messages for

secure communications.

The publication of the seminal paper of Pecora

and Caroll (1990) had a very strong impact in the

domain of chaos theory and chaos synchronization,

and their applications. It had stimulated very intense

research activities and the related studies continue to

attract great attention. Many authors have contrib-

uted to developing this domain, theoretically or

experimentally (Boccaletti et al. 2002, Pecorra et al.

1997, references therein).

However, the special features of chaotic systems

make it impossible to directly apply the methods

developed for synchronization of periodic oscilla-

tors. Moreover, in the topics of coupled chaotic

systems, many different phenomena, which are

usually referred to as synchronization, exist and

have been studied now for over a decade. Thus,

more precise descriptions of such systems are indeed

desirable.

Several different regimes of synchronization have

been investigated. In the following, the focus will be

on explaining the essentials on this large topic,

subdivided into four basic types of synchronization

of coupled or forced chaotic systems which have

been found and have received much attention, while

emphasizing on the first three:



identical (or complete) synchronization (IS),

which is defined as the coincidence of states of

interacting systems;



generalized synchronization (GS), which extends

the IS phenomenon and implies the presence of

some functional relation between two coupled

systems; if this relationship is the identity, we

recover the IS;



phase synchronization (PS), which means entrain-

ment of phases of chaotic oscillators, whereas

their amplitudes remain uncorrelated; and



lag synch ronization (LS), which appears as a

coincidence of time-shifted states of two systems.

Other regimes exis t, some of them will be briefly

pointed out at the end of this articl e; we also will

briefly discuss the very relevant issue of the stability

of synchronous motions.

Our discussion and examples given here are based

on unidirectionally continuous systems, most of the

exposed ideas can be easily extended to discrete

systems.

Let us also emphasize that the same year, 1990,

saw the publication of another seminal paper, by

Ott, Grebogi, and Yorke (OGY) on the control of

chaos (Ott et al. 1990). Recently, it has been

realized that synchronization and control of chaos

share a common root in nonlinear control theory.

Both topics were presented by many authors in a

unified framework. However, synchronization of

chaos has evolved in its own right, even if it is

nowadays known as a part of the nonlinear control

theory.

Synchronization and Stability

For the basic master–slave configuration, where an

autonomous chaotic system (the master)

¼ FðXÞ; X 2 R

½1

drives another system (the slave),

¼ GðX; YÞ; Y 2 R

½2

synchronization takes place when Y asymptotically

copies, in a certain manner, a subset X

of X. That

is, there exists a relation between the two coupled

systems, which could be a smooth invertible func-

tion , which transforms the trajectories on the

attractor of a first system into those on the attractor

of a second system. In other words, if we know,

after a transient regime, the state of the first system,

it allows us to predict the state of the second:

Y(t) = (X(t)). Generally, it is assumed that n  m;

however, for the sake of easy readability (even if this

is not a necessary restriction) the case n = m will

only be considered; thus, X

= X. Henceforth, if we

denote the difference Y  (X)byX

, in order to

arrive at a synchronized motion, it is expected that

jjX

jj!0; as t ! þ 1 ½ 3

If is the identity function, the process is called IS.

Definition of IS System [2] synchronizes with

system [1], if the set M = {(X, Y) 2 R

 R

, Y = X}

is an attracting set with a basin of attraction B(M  B)

such that lim

t!1

kX(t)  Y(t)k= 0, for all

(X(0), Y(0)) 2 B.

Thus, this regime corresponds to the situation

where all the variables of two (or more) coupled

chaotic systems converge.

214 Synchronization of Chaos

If is not the identity function, the phenomenon

is more general and is referred to as GS.

Definition of GS System [2] synchronizes with

system [1], in the generalized sense, if there exists a

transformation : R

!R

, a manifold M =

{(X, Y) 2 R

nþm

, Y = (X)} and a subset B (M  B),

such that for all (X

, Y

) 2 B, the trajectory based

on the initial conditions (X

, Y

) approaches M as

time goes to infinity. This is explained further in the

following.

Henceforth, in the case of IS, eqn [3] above means

that a certain hyperplane M, called synchronization

manifold, within R

, is asymptotically stable.

Consequently, for the sake of synchrony motion,

we have to prove that the origin of the transverse

system X

= Y  X is asymptotically stable. That is,

to prove that the motion transversal to the synchro-

nization manifold dies out.

However, significant progress has been made by

mathematicians and physicists in studyi ng the

stability of synchronous motions. Two main tools

are used in the literature for this aim: conditional

Lyapunov exp onents and asymptotic stability. In the

examples given below, we will essentially formulate

conditions for synchronization in terms of Lyapunov

exponents, which play a central role in chaos theory.

These quantities measure the sensitive dependence

on initial conditions for a dynamical system and also

quantify synchronization of ch aos.

The Lyapunov exponents associated with the

variational equation corresponding to the transverse

system X

¼ DFðXÞX

½4

where DF(X) is the Jacobian of the vector field

evaluated onto the driving traject ory X, are referred

to as transverse or conditional Lyapunov exponents

(CLEs).

In the case of IS, it appears that the condition L

max

0 is sufficient to insure synchronization, where L

max

the largest CLE. Indeed, eqn [4] gives the dynamics of

the motion transverse to the synchronization manifold;

therefore, CLEs indicate if this motion dies out or not,

and hence, whether the synchronization state is stable

or not. Consequently, if L

max

is negative, it insures the

stability of the synchronized state. This will be best

explained using two examples below.

Even if there exist other approaches for studying

synchronization, one may ask if this condition on

max

is true in general. To answer this question,

mathematicians have recently formulated it in terms

of properties of manifolds (or synchronization

hyperplanes). Some rigorous results on (generalized)

synchronization, when the system is smooth, are

given by Josic (2000). This approach relies on the

Fenichel theory of normally hyperbolic invariant

manifolds and quantities that resemble Lyapunov

exponents, and is referred to as differentiable GS.

However, many situations correspond to the case

where, in some region of values of parameters

coupling, the function is only continuous but not

smooth, that is, the graph of is a compl icated

geometrical object. This kind of synchronization

is called nonsmooth GS (Afraimov ich et al. 2001).

Furthermore, the mathematical theory of IS often

assumes the coupled oscillators to be identical, even

if, in practice, no two oscillators are exact copies of

each other. This leads to small differences in system

parameters and then to synchronization errors.

These errors have been studied by many authors

(see, e.g., Illing (2002), and references therein).

Identical Synchronization

Perhaps the best way to explain synchronization of

chaos is through IS, also referred to as conventional

or complet e synchroni zation (Boccaletti et al. 2002).

It is the simplest form of chaos synchronization and

generalizes to the complete replacement which is

explained below. It is also the most typical form of

chaotic synchronization often observable in two

identical systems.

There are various processes leadi ng to synchroni-

zation; depending on the particular coupling config-

uration used these processes could be very different.

So, one has to distinguish between the following two

main situations, even if they are, in some sense,

similar: the unidirectional and the bidirectional

coupling. Indeed, synchronization of chaotic systems

is often studied for schemes of the form

¼ FðXÞþkNðX  YÞ

¼ GðYÞþkMðX  YÞ

½5

where F and G act in R

,(X, Y) 2 (R

)

, is a scalar,

and M and N are coupling matrices belonging to

nn

.IfF = G the two subsystems X and Y are

identical. Moreover, when both matrices are non-

zero then the coupling is called bidirectional, while

it is referred to as unidirectional if one is the zero

matrix, and the other nonzero.

Constructing Pairs of Synchronized Systems:

Complete Replacement

Pecora and Carroll (1990) proposed the use of

stable subsystems of given chaotic systems to

Synchronization of Chaos 215

construct pairs of unidirectionally c oupled synchro-

nizing systems. Since then generalizations of this

approach have been developed and various meth-

ods now exist to synchronize systems (Wu 2002,

Hasler 1998).

One way to build a couple of synchronized

systems is then to use the basic construction method

introduced by Pecora and Carroll, who made an

important observation. They found that, when they

make a replica of part of a chaotic system and send

a system variable from the original system (trans-

mitter) to drive this replica (receiver), sometimes the

replica subsystem and the original chaotic one lock

in their steps and evolve together chaotically in

synchrony. This method can be described as follows.

Consider the autonomous n-dimensional dynamical

system,

¼ FðuÞ½6

divide this system into two subsystems (u = (v, w)),

¼ Gðv; wÞ

¼ Hðv; wÞ

½7

where v = (u

, ..., u

), w = (u

mþ1

, ..., u

), G =(F

, ...,

), and H = (F

mþ1

, ..., F

). Next , create a new

subsystem w

identical to the w-subsystem. This

yields a (2n m)-dimensional system:

¼ Gðv; wÞ

¼ Hðv; wÞ

¼ Hðv; w

½8

The first state-variable component v(t) of the (v, w)

system is then used as the input to the w

-system.

The coupling is unidirectional and the (v, w)

subsystem is referred to as the driving (or master)

system, the w

-subsystem as the respons e (or slave)

system. In this context, the following notions and

results are useful.

Definition If lim

t !þ1

(t)  w(t) k= 0andw

(t)

continues to remain in step with w(t) in the course

of the time, the two subsystems are said to be

synchronized.

Definition The Lyapunov exponents of the

response subsyst em (w

) for a particular driven

trajectory v(t) are called CLEs.

Let w(t) be a chaotic trajectory with initial

condition w(0), and w

(t) be a trajectory started at

a nearly point w

(0). The basic idea of the Pecora–

Carroll approach is to establish the asymptotic

stability of the solutions of w

-subsystem by means

of CLEs. They have shown the following result

(Pecora and Carroll 1990):

Theorem A necessary and sufficient condition for

the two subsystems, w andw

, to be synchronized is

that all of the CLEs be negative.

Note that only a finite number of possible

decompositions (or couplings) v–w exist; this is

bounded by the number of different possible

subsystems, namely N(N  1)=2. (For a description

and mathematical analysis of various coupling

schemes see Wu (2002).) Furthermore, by splitting

the main system [6] in a different way, (complete)

synchronization could not exist. Indeed, in general,

only a few of the possible response subsystems

possess negative CLEs, and may thus be used to

implement synchronizing systems using the Pecora–

Caroll method. In fact, it has been pointed out in the

literature that in some cases, the CLE criterion is not

as practical as some other criteria.

For simplicity, the idea will now be developed on

the following three-dimensional simple autonomous

system, which belongs to the class of dynamical

systems called generalized Lorenz systems (see

Derivie` re and Aziz-Alaoui (2003), and references

therein):

x ¼9x  9y

y ¼17x  y  xz

z ¼z þ xy

½9

(This should be compared with the well-known

Lorenz system:

x ¼10x þ 10y

y ¼ 28x  y  xz

z ¼

z þ xy

which differs in the signs of various terms and the

values of coefficients.) From previous observations,

it was shown that system [9] oscillates chaotically;

its Lyapunov exponents are þ0.601, 0.000, and

16.470; it exhibits the chaotic attractor of Figure 1,

with a three-dimensional feature very similar to that

of Lorenz attractor (in fact, it satisfies the condition

z < 0, but in our context it does not matter).

Let us divide system [9] into two subsystems

v = x

and w = (y

, z

). By creating a copy

216 Synchronization of Chaos

= (y

, z

)ofthew-subsystem, we obtain the

following five-dimensional dynamical system:

¼9x

 9y

¼17x

 y

 x

¼z

þ x

¼17x

 y

 x

¼z

þ x

½10

In numerical experiments, it was observed that the

motion quickly results in the two equalities,

lim

t !þ1

 y

j= 0 and lim

t !þ1

 z

j= 0, to

be satisfied, that is, lim

t !þ1

 wk= 0. These

equalities persist as the system evolves. Hence, the

two subsystem s w and w

are synchronized. Figure 2

illustrates this phenomenon.

It is also easy to verify that the synchronization

persists even if a slight change in the parameters of

the system is made. The CLEs of the linearization of

the system around the synchronous state, the

negativity of which determines the stability of the

synchronized solution, are also computed easily.

Pecora–Carroll similarly built the system [10] by

using the following steps. Starting with two copies

of system [9], a signal x(t) is transmitted from the

first to the second: in the second system all x-

components are replaced with the signal from the

first system, that is, x

is replaced by x

in the

second system. Finally, the dx

=dt equation is

eliminated, since it is exactly the same as dx

=dt

equation, and is superfluous. This then results in

system [10]. For this reason, Pecora–Carroll called

this construction a complete replacement. Thus, it is

natural to think of the x

variable as driving the

second system, but also to label the first system the

drive and the second system the response. In fact,

this method is a particular case of the unidirectional

coupling method explained below. Note also that

this method could be modifi ed by using a partial

substitution approach, in which a response variable

is replaced with the drive counterpart only in certain

locations (Pecora et al. 1997).

Unidirectional IS

The IS synchronization has also been called as one-

way diffusive coupling, drive–response coupling,

master–slave coupling, or negative feedback control.

System [5], F = G and N = 0, becomes unidirec-

tionally coupled, and reads

¼ FðXÞ

¼ FðYÞþkMðX  YÞ

½11

M is then a matrix that determines the linear

combination of X components that will be used

in the difference, and k determines the strength of

the coupling (see, for an interesting review on

this subject, Pecora et al. (1997)). In unidirectional

synchronization, the evolution of the first system

(the drive) is unaltered by the coupling, the second

system (the response) is then constrained to copy the

dynamics of the first. Let us consider an example

with two copies of system [9], and for

M¼

100

000

½12

that is, by adding a damping term to the first equation

of the response system, we get a following unidir-

ectionally coupled system, coupled through a linear

term k > 0 according to variables x

1, 2

¼9x

 9y

¼17x

 y

 x

¼z

þ x

¼9x

 9y

 kðx

 x

¼17x

 y

 x

¼z

þ x

½13

–30

–25

–20

–15

–10

–5

–15 –10 –5 0 5 10 15

–15

–10

–5

–15 –10 –5 0 5 10 15

Figure 1 The chaotic attractor of system [9]: x–y and x–z plane projections.

Synchronization of Chaos 217