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

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Theorem 6 The map defined in eqns [7] and [6] is

a linear isomorphism. The inverse map is given by

BðHH

Þ3 7!T



2L ½8

with



; T



ðÞe



i¼d tr ðje



ihe



j



½9

where e

, ..., e

denote an (arbitrary) ortho-

normal basis of H

and the transposition of  is

defined with respect to the basis e



,  = 1, ..., d used

to define  in [5].

From the definition of R

in eqn [6], it is obvious

that R

is positive, if T is CP. To see that the

converse is also true is not as trivial (because a

transposition is involved), but it requires only a

short calculation, which is omitted here. Hence, we

get:

Corollary 7 The operator R

is positive, iff the

mapTisCP.

Examples

Let us return now to the general case (i.e., arbitrary

input and output algebras) and discuss several

examples.

Channels Under Symmetry

It is often useful to consider channels with special

symmetry properties. To be more precise, consider

a group G and two unitary representations 

, 

on the Hilbert spaces H

and H

, respectively.

A channel T : B (H

) !B(H

) is called covariant

(with respect to 

and 

)if

T½

ðUÞ A

ðUÞ



¼

ðUÞ T½A

ðUÞ



8A 2BðH

Þ8U 2 G ½10

holds. The general structure of covariant channels

is governed by a fairly powerful variant of Stine-

springs theorem (Keyl and Werner 1999).

Theorem 8 Let G be a group with finite-dimen-

sional unitary representations 

: G ! U(H

) and

T : B(H

) !B(H

) a 

, 

-covariant channel.

(i) Then there is a finite-dimensional unitary

representation

 : G ! U(K) and an operator

V : H

Kwith V

(U) = 

(U) 

(U )V

and T(A) = V



A  1V.

(ii) If T =



is a decomposition of T in CP and

covariant summands, there is a decomposition

1 =



of the identity operator on K into

positive operators F



2B(K) with [F



(g)] = 0

such that T



(X) = V



(X  F



)V.

The most prominent examples of covariant

channels arise with H

= H

= C

, G = U(d)and



(U) = 

(U) = U . All channels of this type are of

the form

TðAÞ¼ð1  #ÞA þ #d

1

trðAÞ1

with # 2½0; d

=ðd

 1Þ ½11

and are known as ‘‘depolarizing channels.’’ They

often serve as a standard model for noise. Two

particular cases are the ideal channel arising with

# = 0, and the completely depolarizing channel

(# = 1) which erases all information. If we choose



(U) =



U (where the bar denotes complex conju-

gate) instead of 

(U) = U, we get

TðAÞ¼

d þ 1

trðAÞ1 þ A



1  #

d  1

trðAÞ1  A



;#2½0; 1½12

If we map these channels to states of bipartite

systems (using the Jamiołkowski isomorphism from

the last section), we get ‘‘Isotropic states’’ from

eqn [11] and ‘‘Werner states’’ from [12].

Classical Channels

The classical analog to a quantum operation is a

channel T : C(X) !C(Y) which describes the trans-

mission or manipulation of classical information. As

already mentioned in the subsection ‘‘Completely

positive maps,’’ positivity and complete positivity

are equivalent in this case. Hence, we have to

assume only that T is positive and unital. Obviously,

T is characterized by its matrix elements

= 

(Te

), where 



(X) denotes the Dirac

measure at y 2 Y and e

2C(X) is the canonical

basis in C(X). More precisely, 

and e

denote,

respectively, the probability distribution and the

function on X, given by



¼ð

x2X

and e

ðyÞ¼

½13

We will keep this notation up to the end of this

article. Positivity and normalization of T imply that

0  T

 1and

1 ¼ 

ð1Þ¼

Tð1ÞðÞ

¼ 

!"#

½14

holds. Hence the family (T

)

x2X

is a probability

distribution on X and T

is, therefore, the transition

probability to get the information x 2 X at the

output side of the channel if y 2 Y was sent.

Channels in Quantum Information Theory 475

Observables

Let us consider now a channel which transforms

quantum information B(H) into classical information

C(X). Since positivity and complete positivity are

again equivalent, we just have to look at a positive

and unital map E : C(X) !B(H). With the canonical

basis e

, x 2 X,ofC(X ), we get a family

= E(e

), x 2 X, of positive operators E

2B(H)

with

x2X

= 1.Hence,theE

form a positive

operator valued (POV) measure, i.e., an observable.

If, on the other hand, a POV measure E

2B(H), x 2

X, is given, we can define a quantum-to-classical

channel E : C(X) !B(H)by

Eðf Þ¼

x2X

f ðxÞE

½15

This shows that the observable E

, x 2 X, and the

channel E can be identified.

Preparations

Let us now exchange the role of C(X) and B(H); in

other words, let us consider a channel R : B(H) !

C(X) with a classical input and a quantum output

algebra. In the Schro¨ dinger picture, we get a family of

density matrices 

:= R



(

) 2B



(H), x 2 X, where





(X) denotes again the Dirac measure on X.

Hence, we get a parameter-dependent preparation

that can be used to encode the classical information

x 2 X into the quantum information 



(H).

Instruments

An observable describes only the statistics of

measuring results, but does not contain information

about the state of the system after the measurement.

To get a description which fills this gap, we have

to consider channels which operate on quantum

systems and produce hybrid systems as output, that is,

T : B(H) C(X) !B(K). Following Davies (1976),

we will call such an object an instrument. From T we

can derive the subchannel

CðXÞ3f 7!Tð1  f Þ 2 BðKÞ ½16

which is the observable measured by T, that is,

tr(T(1  e

)) is the probability to measure x 2 X on

systems in the state . On the other hand, we get for

each x 2 X a quantum channel (which is not unital)

BðHÞ 3 A 7!T

ðAÞ¼TðA  e

Þ 2 BðKÞ ½17

It describes the operation performed by the instru-

ment T if x 2 X was measured. More precisely, if a

measurement on systems in the state  gives the

result x 2 X, we get (up to normalization) the state



() after the measurement, while

tr T



ðÞ



¼ tr T



ðÞ1



¼ tr Tð1  e

ÞðÞ½18

is (again) the probability to measure x 2 X on .

The instrument T can be expressed in terms of the

operations T

TðA  f Þ¼

f ðxÞT

ðAÞ½19

Hence, we can identify T with the family T

, x 2 X.

Finally, we can consider the second marginal of T

BðHÞ 3 A 7!TðA  1Þ¼

x2X

ðAÞ 2 BðKÞ ½20

It describes the operation we get if the outcome of

the measurement is ignored.

The best-known example of an instrument is a von

Neumann–Lu¨ ders measurement associated with a PV

measure given by family of projections E

, x = 1,

..., d; for example, the eigenprojections of a self-

adjoint operator A 2B(H). It is defined as the channel

T : BðHÞ  CðXÞ ! BðHÞ

with X ¼f1; ...; dg and T

ðAÞ¼E

½21

Hence, we get the final state tr(E

)

1

E

if we

measure the value x 2 X on systems initially in the

state  – this is well known from quantum mechanics.

Parameter-Dependent Operations

Let us change now the role of B(H) C(X) and

B(K ); in other words, consider a channel T : B(K) !

B(H) C(X) with hybrid input and quantum output.

It describes a device which changes the state of a

system depending on the additional classical infor-

mation. As for an instrument, T decomposes into a

family of (unital!) channels T

: B(K) !B(H) such

that we get T



(  p) =



() in the Schro¨ din-

ger picture. Physically, T describes a parameter-

dependent operation: depending on the classical

information x 2 X, the quantum information  2

B(K ) is transformed by the operation T

Finally, we can consider a channel T : B(H) 

C(X) !B(K) C(Y) with hybrid input and output

to get a parameter-dependent instrument: similarly

to the above discussion, we can define a family of

instruments T

: B(H) C(X) !B(K), y 2 Y, by the

equation T



(  p) =



(). Physically, T

describes the following device: it receives the

classical information y 2 Y and a quantum system

in the state  2B



(K) as input. Depending on y,a

measurement with the instrument T

is performed,

which in turn produces the measuring value x 2 X

and leaves the quantum system in the state (up to

normalization) T



y, x

(); with T

y, x

given as in eqn

[17] by T

y, x

(A) = T

(A  e

476 Channels in Quantum Information Theory

See also: Capacities Enhanced by Entanglement;

Capacity for Quantum Information; Entanglement;

Optimal Cloning of Quantum States; Positive Maps on

C*-Algebras; Quantum Channels: Classical Capacity;

Quantum Dynamical Semigroups; Quantum Entropy;

Quantum Spin Systems; Source Coding in Quantum

Information Theory.

Further Reading

Arveson W (1969) Subalgebras of C



-algebras. Acta Mathematica

123: 141–224.

Davies EB (1976) Quantum Theory of Open Systems. London:

Academic Press.

Jamiołkowski A (1972) Linear transformations which preserve

trace and positive semidefiniteness of operators. Reports on

Mathematical Physics 3: 275–278.

Keyl M and Werner RF (1999) Optimal cloning of pure states, testing

single clones. Journal of Mathematical Physics 40: 3283–3299.

Kraus K (1983) States Effects and Operations. Berlin: Springer.

Paulsen VI (2002) Completely Bounded Maps and Dilations.

Cambridge: Cambridge University Press.

Stinespring WF (1955) Positive functions on C



-algebras.

Proceedings of the American Mathematical Society 6: 211–216.

Chaos and Attractors

R Gilmore, Drexel University, Philadelphia, PA, USA

Introduction

Chaos is a type of behavior that can be exhibited by

a large class of physical systems and their mathe-

matical models. These systems are deterministic.

They are modeled by sets of coupled nonlinear

ordinary differential equations (ODEs):

¼ f

ðx; cÞ½1

called dynamical systems. The coordinates x desig-

nate points in a state space or phase space.

Typically, x 2 R

or some n-dimensional manifold

for some n  3, and c 2 R

are called control

parameters. They describe parameters that can be

controlled in physical systems, such as pumping

rates in lasers or flow rates in chemical mixing

reactions. The most important mathematical prop-

erty of dynamical systems is the uniqueness theorem,

which states that there is a unique trajectory through

every point at which f (x; c) is continuous and

Lipschitz and f (x; c) 6¼ 0. In particular, two distinct

periodic orbits cannot have any points in common.

The properties of dynamical systems are gov-

erned, in lowest order, by the number, stability, and

distribution of their fixed points, defined by

= f

(x; c) = 0. It can happen that a dynamical

system has no stable fixed points and no stable

limit cycles (x(t) = x(t þ T), some T > 0, all t). In

such cases, if the solution is bounded and recurrent

but not periodic, it represents an unfamiliar type of

attractor. If the system exhibits ‘‘sensitivity to initial

conditions’’ (jx(t)  y(t)je

t

jx(0)  y(0)j for

jx(0)  y(0)j=  and >0 for most x(0)), the

solution set is called a ‘‘chaotic attractor.’’ If the

attractor has fractal structure, it is called a ‘‘strange

attractor.’’

Tools to study strange attractors have been

developed that depend on three types of mathe-

matics: geometry, dynamics, and topology.

Geometric tools attempt to study the metric

relations among points in a strange attractor.

These include a spectrum of fractal dimensions.

These real numbers are difficult to compute, require

very long, very clean data sets, provide a number

without error estimates for which there is no

underlying statistical theory, and provide very little

information about the attractor.

Dynamical tools include estimation of Lyapunov

exponents and a Lyapunov dimension. They include

globally averaged exponents and local Lyapunov

exponents. These are eigenvalues related to the

different stretching (>0) and squeezing (<0)

eigendirections in the phase space. To each globally

averaged Lyapunov exponent 

, 

 



there corresponds a ‘‘partial dimension’’ 

,0 

 1,

with 

= 1if

 0. The Lyapunov dimension is

the sum of the partial dimensions d

i = 1



That the partial dimension 

= 1for

 0 indicates

that the flow is smooth in the stretching (

> 0) and

flow directions and fractal in the squeezing (

< 0)

directions with 

< 1. Dynamical indices provide

some useful information about a strange attractor.

In particular, they can be used to estimate some

fractal properties of a strange attractor, but not vice

versa.

Topological tools are very powerful for a

restricted class of dynamical systems. These are

dynamical systems in three dimensions (n = 3). For

such systems there are three Lyapunov exponents



>

, with 

> 0 describing the stretching

direction and responsible for ‘‘sensitivity to initial

conditions,’’ 

= 0 describing the direction of the

flow, and 

< 0 describing the squeezing direction

Chaos and Attractors 477

and responsible for ‘‘recurrence.’’ Strange attractors

are generated by dissipative dynamical systems,

which satisfy the additional condition 

þ 



< 0. For such attractors, 

= 

= 1and



= 

=j

j by the Kaplan–Yorke conjecture, so

that d

= 2 þ 

= 2 þ 

=j

A number of tools from classical topology have

been exploited to probe the structure of strange

attractors in three dimensions. These include the

Gauss linking number, the Euler characteristic, the

Poincare´–Hopf index theorem, and braid theory.

More recent topological contributions include sev-

eral definitions for entropy, the development of a

theory for knot holders or braid holders (also called

branched manifolds), the Birman–Williams theorem

for these objects, and relative rotation rates, a

topological index for individual periodic orbits and

orbit pairs.

Three-dimensional strange attractors are

remarkably well understood; those in higher

dimensions are not. As a result, the description

that follows is largely restricted to strange attrac-

tors with d

< 3 that exist in R

or other three-

dimensional manifolds (e.g., R

 S

). The obstacle

to progress in higher dimensions is the lack of a

higher-dimensional analog of the Gauss linking

number for orbit pairs in R

Overview

The program described below has two objectives:

1. classify the global topological structure of strange

attractors in R

; and

2. determine the ‘‘perestroikas’’ (changes) that such

attractors can undergo as experimental condi-

tions or control parameters change.

Four levels of structure are required to complete

this program. Each is topological and discretely

quantifiable. This provides a beautiful interaction

between a rigidity of structure, demanded by

topological constraints, and freedom within this

rigidity. These four levels of structure are:

1. basis sets of orbits,

2. branched manifolds or knot holders,

3. bounding tori, and

4. embeddings of bounding tori.

Branched Manifolds: Stretching

and Squeezing

A strange attractor is generated by the repetition of

two mechanisms: stretching and squeezing. Stretch-

ing occurs in the directions identified by the positive

Lyapunov exponents and squeezing occurs in the

directions identified by the negative Lyapunov

exponents. In R

there is one stretching direction

and one squeezing direction.

A simple stretch-and-squeeze mechanism that

nature appears to be very fond of is illustrated in

Figure 1. In this illustration, a cube of initial

conditions at (a) is advected by the flow in a short

time to (b). During this process, the cube is

deformed by being stretched (

> 0). It also shrinks

in a transverse direction (

< 0). During the initial

phase of this deformation, two nearby points

typically separate exponentially in time. If they

were to continue to separate exponentially for all

times, the invariant set would not be bounded.

Therefore, this separation cannot continue indefi-

nitely, and in fact it must somehow reverse itself

after some time because the motion is recurrent. The

mechanism shown in Figure 1 involves folding,

which begins between (b) and (c) and continues

through to (d). Squeezing occurs where points from

distant parts of the attractor approach each other

exponentially, as at (d). Finally, the cube, shown

deformed at (d), returns to the neighborhood of

initial conditions (a). This process repeats itself and

builds up the strange attractor. As can be inferred

from this figure, the strange attractor constructed by

the repetitive process is smooth in the expanding

(

) and flow (

= 0) directions but fractal in the

squeezing (

) direction. The attractor’s fractal

dimension is 

þ 

= 2 þ 

= 2 þ 

=j

Figure 1 summarizes the boundedness and recur-

rence conditions that were introduced to define

strange attractors, and illustrates one stretching and

squeezing mechanism that occurs repetitively to

build up the fractal structure of the strange attractor

Boundary

layer

Squeeze

Stretch

(d)

(c)

(a)

(b)

Figure 1 A common stretch-and-fold mechanism generates

many experimentally observed strange attractors. The Topology

of Chaos; R Gilmore and M Lefranc; Copyright ª 2002, Wiley.

This material is used by permission of John Wiley & Sons, Inc.

478 Chaos and Attractors

and to organize all the (unstable) periodic orbits in it

in a unique way. The particular mechanism shown

in Figure 1 is called a stretch-and-fold mechanism.

Other mechanisms involve stretch and roll, and tear

and squeeze.

The stretch-and-squeeze mechanisms are well

summarized by the cartoons shown in Figure 2.On

the left, a cube of initial conditions (top) is deformed

under the flow. The flow is downward. Stretching

occurs in one direction (horizontal) and shrinking

occurs in a transverse direction (perpendicular to the

page). In the limit of extreme shrinking (

‘‘1”), the dynamics of the stretching part of the

flow is represented by the two-dimensional surface

shown on the bottom left. This surface fails to be a

manifold because of the singularity, called a splitting

point. This singularity represents an initial condition

that flows to an unstable fixed point with at least

one stable direction. On the right (squeezing), two

distant cubes of initial conditions (top) in the flow

are deformed and brought to each other’s proximity

under the flow (middle). In the limit of extreme

dissipation, two two-dimensional surfaces represent-

ing inflows are joined at a branch line to a single

surface representing an outflow. This surface fails to

be a manifold because of the branch line, which is a

singularity of a different kind. Points below the

branch line in this representation of the flow (on the

outflow side of the branch line) have two preimages

above the branch line, one in each inflow sheet. This

structure generates positive entropy.

A beautiful theorem of Birman and Williams

justifies the use of the two cartoons shown at the

bottom of Figure 2 to characterize strange attractors

in R

. As preparation for the theorem, Birman and

Williams introduced an important identification for

the nongeneric or atypical points that ‘‘are not

sensitive to initial conditions’’

x  y if jxðtÞyðtÞj !

t !1

0 ½2

That is, two points in a strange attractor are

identified if they have asymptotically the same

future. In practice, this amounts to projecting the

flow down along the stable (

< 0) direction onto a

two-dimensional surface described by the stretching

(

> 0) and the flow (

= 0) directions. This

surface is not a manifold because of lower-

dimensional singularities: splitting points and branch

lines. The two-dimensional surface has many names,

for example, knot holder (because it holds the

periodic orbits that exist in abundance in strange

attractors), braid holders, templates, branched mani-

folds. The flow, restricted to this surface, is called a

semiflow. Under the semiflow, points in the branched

manifold have a unique future but do not have a

unique past. The degree of nonuniqueness is mea-

sured by the topological entropy of the dynamical

system. The Birman–Williams theorem is:

Theorem Assume that a flow 

(i) on R

is dissipative (

> 0, 

= 0, 

< 0 and



þ 

< 0);

(ii) generates a hyperbolic strange attractor (the

eigenvectors of the local Lyapunov exponents



, 

span everywhere on the attractor).

Then the projection [2] maps the strange attractor

SA to a branched manifold BM and the flow 

SA to a semiflow



on BM in R

. The periodic

orbits in SA under 

correspond 1:1 with the

periodic orbits in BM under



with perhaps one or

two specified exceptions. On any finite subset of

orbits the correspondence can be taken via isotopy.

The beauty of this theorem is that it guarantees

that a flow 

that generates a (fractal) strange

attractor SA can be continuously deformed to a new

flow



on a simple two-dimensional structure BM.

During this deformation, periodic orbits are neither

created nor destroyed. The uniqueness theorem for

ODEs is satisfied during the deformation, so orbit

segments do not pass through each other. As a

result, the topological organization of all the

Boundary

layer

Squeeze

Shrink

Stretch

Flow

Branch

line

Squeeze

Flow

xxx

Figure 2 Left: The stretch mechanism is modeled by a two-

dimensional surface with a splitting point singularity. Right: The

squeeze mechanism is modeled by a two-dimensional surface

with a branch line singularity. The Topology of Chaos; R Gilmore

and M Lefranc; Copyright ª 2002, Wiley. This material is used

by permission of John Wiley & Sons, Inc.

Chaos and Attractors 479

unstable periodic orbits in the strange attractor is

the same as the topological organization of all the

unstable periodic orbits in the branched manifold. In

fact, the branched manifold (knot holder) defines

the topological organization of all the unstable

periodic orbits that it supports. Topological organi-

zation is defined by the Gauss linking number and

the relative rotation rates, another braid index.

The significance of this theorem is that strange

attractors can be characterized – in fact classified –

by their branched manifolds. Figure 3 shows a

branched manifold ‘‘for a figure-8 knot’’ as well as

the figure-8 knot itself (dark curve). If a constant

current is sent through a conducting wire tied into

the shape of a figure-8 knot, a discrete countable set

of magnetic field lines will be closed. These closed

field lines can be deformed onto the two-dimen-

sional surface shown in Figure 3. Each of the eight

branches of this branched manifold can be named.

One way to do this specifies the two branch lines

that are joined by the branch in the sense of the flow

(e.g., (a) and () (but not (a)). Every closed field

line can be labeled by a symbol sequence that is

unique up to cyclic permutation. This symbol

sequence provides a symbolic name for the orbit.

For example, (a)()(b)(ba) is a period-4 orbit.

The structure of a branched manifold is determined

in part by a transition matrix T. The matrix element

is 1 if the transition from branch i to branch j is

allowed, 0 otherwise. The transition matrix for the

figure-8 branched manifold is shown in Figure 3.

The Birman–Williams theorem is stronger than its

statement suggests. More systems satisfy the state-

ment of the theorem than do the assumptions of the

theorem. The figure-8 knot, and its attendant

magnetic field, is not dissipative – in fact, it is not

even a dynamical system, yet the closed loops can be

isotoped to the figure-8 knot holder. There are other

ways in which the Birman–Williams theorem is

stronger than its statement suggests.

It is apparent from Figure 3 that the figure-8

branched manifold can be built up Lego

fashion

from the two basic building blocks shown in

Figure 2. This is more generally true. Every

branched manifold can be built up, Lego

fashion,

from the stretch (with a splitting point singularity)

and the squeeze (with a branch line singularity)

building blocks, subject to the following two

conditions:

1. outputs flow to inputs and

2. there are no free ends.

The figure-8 branched manifold is built up from

four stretch and four squeeze building blocks. As a

result, there are eight branches and four branch

lines.

Two often-studied strange attractors are shown in

Figures 4 and 5. Figure 4 shows the details of the

Ro¨ ssler dynamical system. A similar spectrum of

features is shown in Figure 5 for the Lorenz equations.

The knot holder in Figure 5e is obtained from the

caricature in Figure 5d by twisting the right-hand lobe

by  radians.

Branched manifolds can be used to characterize

all three-dimensional strange attractors. Branched

manifolds that classify the strange attractors gener-

ated by four familiar sets of equations (for some

control parameter values) are shown in Figure 6.

The sets of equations, and one set of parameter

values that generate strange attractors, are presented

in Table 1.

The beauty of this topological classification of

strange attractors is that it is apparent, just by

inspection, that there is no smooth change of

variables that will map any of these systems to any

of the others for the parameter values shown.

Branched manifolds can be described algebrai-

cally. In Figure 7 we provide the algebraic

aα

αβ

αa

bβ

βα

βb

aααβαa

bββαβb

0000 0

0011 0

0000 1

011

000

100

1100 0000

0000 1100

0011 0000

0000 0011

Figure 3 Figure-8 knot (dark curve) and the figure-8 branched

manifold. Transition matrix for the eight branches of the figure-8

branched manifold is also shown. Flow direction is shown by

arrows. The Topology of Chaos; R Gilmore and M Lefranc;

John Wiley & Sons, Inc.

480 Chaos and Attractors

description of two branched manifolds. Figure 7a

shows the branched manifold that describes experi-

mental data generated by many physical systems.

The mechanism is a simple stretch-and-fold defor-

mation with zero global torsion that generates a

typical Smale horseshoe. There are two branches.

The diagonal elements of the matrix identify the

local torsion of the flow through the corresponding

branch, measured in units of . Branch 0 has no

local torsion, and branch 1 shows a half-twist and

has local torsion þ1. The off-diagonal matrix

elements are twice the linking number of the

period-1 orbits in the corresponding pair of branches.

Since the period-1 orbits in these two branches do not

link, the off-diagonal matrix elements are 0. The

period-1 orbits in the branches labeled 1 and 2 in

Figure 7b have linking number þ1, so the off-diagonal

matrix elements are T(1, 2) = T(2, 1) = 2 þ1. The

array identifies the order (above, below) that the two

branches are joined at the branch line, the smaller the

value, the closer to the viewer. These two pieces of

information, four integers in Figure 7a and eight in

= x + ay

= b + z(x – c)

(a) (b) (c)

–4

–2

= –y – z

z(t )

x(t )

(d)

(e)

Figure 4 The Ro

ssler dynamical system. (a) Ro

ssler equations. (b) Time series z(t) and x(t) generated by these equations, and

(c) projection of the strange attractor onto the x–y plane. (d) Caricature of the flow and (e) knot holder derived directly from the

caricature. Control parameter values (a, b,c) = (2:0, 4:0, 0:398): The Topology of Chaos; R Gilmore and M Lefranc; Copyright ª 2002,

Wiley. This material is used by permission of John Wiley & Sons, Inc.

= Rx – y – xz

= –bz + xy

= –σx + σy

(a) (b)

(e) (d)

(c)

z(t )

x(t

)

–10

–20

Figure 5 (a) Lorenz equations. (b) Time series x(t) and z(t) generated by these equations, and (c) projection of the strange attractor

onto the x–y plane. (d) Caricature of the flow and (e) knot holder derived directly from the caricature by rotating the right-hand lobe by 

radians. Control parameter values (R, , b) = (26:0, 10:0, 8=3): The Topology of Chaos; R Gilmore and M Lefranc; Copyright ª 2002,

Wiley. This material is used by permission of John Wiley & Sons, Inc.

Chaos and Attractors 481

Figure 7b, serve to determine the topological organi-

zation of all the unstable periodic orbits in any

strange attractor with either branched manifold.

The periodic orbits are identified by a repeating

symbol sequence of least period p, which is unique

up to cyclic permutation. The symbol sequence

consists of a string of integers, sequentially identify-

ing the branches through which the orbit passes. For

a branched manifold with two branches, there are

two symbols. The number of orbits of period

p, N(p), obeys the recursion relation

pNðpÞ¼2



kp=2

1¼kjp

kNðkÞ½3

Table 2 shows the number of orbits of period

p  20 for the branched manifolds with two and

three branches shown in Figure 7. The number of

orbits of period p grows exponentially with p, and

the limit h

= lim

p !1

log (N(p))=p defines the topo-

logical entropy h

for the branched manifold. The

limits are ln 2 and ln 3 for the branched manifolds

with two and three branches, respectively. The

linking numbers of orbits up to period 5 in the

Smale horseshoe branched manifold are shown in

Table 3, which identifies each of the orbits by its

symbol sequence (e.g., 00111).

Table 1 Four sets of equations that generate strange attractors

Dynamical

system ODEs

Parameter

values

x = y  z

ssler

y = x þ ay (a, b, c) = (2:0, 4:0, 0:398)

z = b þ z(x  c)

Duffing

x = y

y = y  x

þ x

þA sin(!t)

(, A, !) = (0:4, 0:4, 1:0)

van der Pol

x = by þ (c  dy

)x (b, c, d, A, !) =

y = x þ A sin(!t)(0:7, 1:0, 10:0, 0:25, =2)

x = x þ y

Lorenz

y = Rx  y  xz (R, , b) = (26:0, 10:0, 8=3)

z = bz þ xy

(a) (b)

′

–1

Figure 6 Branched manifolds for four standard sets of

equations: (a) Ro

ssler equations, (b) periodically driven Duffing

equations, (c) periodically driven van der Pol equations, and

(d) Lorenz equations. The Topology of Chaos; R Gilmore and

permission of John Wiley & Sons, Inc.

0–1

000

120

220

–2 –10

012

(a) (b)

Figure 7 Branched manifolds are described algebraically. The

diagonal matrix elements describe the twist of each branch.

The off-diagonal matrix elements are twice the linking number of

the period-1 orbits in each of the two branches. The array

describes the order in which the branches are connected at the

branch line. (a) Smale horseshoe branched manifold. (b) Beginning

of a ‘‘gateau roule

’ ’ (jelly roll) branched manifold.

Table 2 Number of orbits of period p on the branched manifolds

with two and three branches, shown in Figure 7. The integers

(p) are constructed by replacing 2

by 3

in eqn [3]

Period

Two

branches

Three

branches Period

Two

branches

Three

branches

(p) N

(p) pN

(p) N

(p)

1 2 3 11 186 16 104

2 1 3 12 335 44 220

3 2 8 13 630 122 640

4 3 18 14 1 161 341 484

5 6 48 15 2 182 956 576

6 9 116 16 4 080 2 690 010

7 18 312 17 7 710 7 596 480

8 30 810 18 14 532 21 522 228

9 56 2184 19 27 954 61 171 656

10 99 5880 20 52 377 174 336 264

482 Chaos and Attractors

Tables of linking numbers have been used

successfully to identify mechanisms that nature uses

to generate chaotic data. This analysis procedure is

called topological analysis. Segments of data are

identified that closely approximate unstable periodic

orbits existing in the strange attractor. These data

segments are then embedded in R

. Each orbit is

given a trial identification (symbol sequence). Their

pairwise linking numbers are computed either by

counting signed crossings or using the time-

parametrized data segments and estimating the

integers numerically using the Gauss linking integral

LinkðA; BÞ

4

ðt

Þr

ðt

Þr

ðt

Þj

ðt

Þdr

ðt

This table of experimental integers is compared with

the table of linking numbers for orbits with the same

symbolic name on a trial branched manifold. This

procedure serves to identify the branched manifold

and refine the symbolic identifications of the

experimental orbits, if necessary. The procedure is

vastly overdetermined. For example, the linking

numbers of only three low-period orbits serve to

identify the four pieces of information required to

specify a branched manifold with two branches.

Since six or more surrogate periodic orbits can

typically be extracted from experimental data,

providing



= 15 or more linking numbers, this

topological analysis procedure has built-in self-

consistency checks, unlike analysis procedures

based on geometric and dynamical tools.

Basis Sets of Orbits

A branched manifold determines the topological

organization of all the periodic orbits that it

supports. Whenever a low-dimensional strange

attractor is subjected to topological analysis, it is

always the case that fewer periodic orbits are

present and identified than are allowed by the

branched manifold that classifies it. This is the case

for strange attractors generated by experimental

data as well as strange attractors generated by

ODEs. The full spectrum occurs only in the

hyperbolic limit, which has never been seen.

The orbits that are present are organized exactly

as in the hyperbolic limit – that is, as determined by

the underlying branched manifold. As control para-

meters change, the strange attractor undergoes

perestroikas. New orbits are created and/or old

orbits are annihilated in direct or inverse period-

doubling and saddle–node bifurcations. The orbits

that are present are always organized as determined

by the branched manifold. Orbits are not created or

annihilated independently of each other. Rather,

there is a partial order (‘‘forcing order’’) involved in

orbit creation and annihilation. This partial order is

poorly understood for general branched manifolds.

It is much better understood for the two-branch

Smale horseshoe branched manifold.

The forcing diagram for this branched manifold

is shown in Figure 8 for orbits up to period 8. It is

typically the case that the existence of one orbit in

a strange attractor forces the presence of a

spectrum of additional orbits. Forcing is transitive,

so if orbit A forces orbit B(A ) B) and B forces C,

then A forces C:ifA ) B and B ) C then A ) C.

For this reason, it is sufficient to show only the

first-order forcing in this figure. The orbits shown

are labeled by their period and the order in which

they are created in a particular highly dissipative

limit of the dynamics: the logistic map (U-sequence

order in Figure 8). For example, 5

describes the

second (pair) of period-5 orbits created in the

Table 3 Linking numbers of orbits to period 5 in the Smale horseshoe branched manifold with zero global torsion

012

000000000000000

100111211222211

0101122322443322

01101223433555533

00101232433554433

0111 0 2 344544887744

0011 0 1 233434555544

0001 0 1 233443555544

01111 0 2 4558558109955

01101 0 2 4558551088855

00111 0 2 354755986755

00101 0 2 354755987655

00011 0 1 233444555545

00001 0 1 233444555554

Chaos and Attractors 483

logistic map in the transition from simple, non-

chaotic behavior to fully chaotic (hyperbolic)

behavior.

The orbits in the forcing diagram are organized

according to their one-dimensional entropy

(horizontal axis, U-sequence order) and their two-

dimensional entropy (vertical axis). Nonchaotic

(‘‘laminar’’) behavior occurs at the lower left of

this figure, where both entropies are zero. Fully

chaotic behavior occurs at the upper right, where

both entropies are ln 2. As control parameters

change, a dynamical system that can exhibit chaos

generated by a stretch-and-fold mechanism follows a

path in the forcing diagram from the lower left to

the upper right. Each such path is a ‘‘route to

chaos.’’ The Smale horseshoe mechanism exhibits

many different routes to chaos: each follows a

different path in the forcing diagram.

The state of a strange attractor at any stage in its

route to chaos can be specified by a ‘‘basis set of

orbits.’’ This is a set of orbits whose presence forces

the existence of all other orbits that can concur-

rently be found in the attractor, up to any finite

period. The basis set of orbits can be constructed

algorithmically. The algorithm is as follows:

1. Write down all the orbits that are present in

order of increasing two-dimensional entropy

from left to right.

2. For orbits with the same two-dimensional entropy,

order by increasing one-dimensional entropy.

3. Remove the ‘‘highest’’ (rightmost) orbit from this

list, together with all the orbits that it forces.

This is the first basis orbit.

4. Of the orbits remaining, again remove the right-

most and all the orbits that it forces. This is the

second basis orbit.

5. Continue until all orbits have been removed.

For any finite period, the above algorithm

terminates because there is only a finite number of

orbits. For example, if the orbit 5

is present as well

as all orbits with lower one-dimensional entropy,

the basis set is 8

R,7

F,8

. As control

parameters change, a strange attractor undergoes

perestroikas that are quantitatively determined by

changes in the basis sets of orbits.

0.70

0.65

0.60

0.55

0.50

0.45

0.40

0.35

Other

finite order

Period

doubled

Well

ordered

Entropy

Forcing of horseshoe

orbits to period 8

U-sequence order

(a)

(b)

2/5 1/3 1/4 1/5 1/6

1/2 1/7 1/81/61/52/7

1/43/8 1/33/7 2/5

Wo :

QOD :

ln 2

Braids

Figure 8 (a) Forcing diagram for orbits up to period 8 in the Smale horseshoe branched manifold. (b) The sequence (‘‘universal

order’’) in which orbits are created in the highly dissipative limit, which is the logistic map. The Topology of Chaos; R Gilmore and

M Lefranc; Copyright ª 2002, Wiley. This material is used by permission of John Wiley & Sons, Inc.

484 Chaos and Attractors