Skiadas C.H., Dimotikalis I. (editors) Chaotic Systems: Theory and Applications

78

C.

T.

J.

Dodson

By

inspection

, we

can

see

that

the

entropy

gradient

components

are

each in

one variable

only

and

in

particular

the

first

component

has

solution

so

the

mean

increases

exponentially

with

time.

Such curves

represent

typical

trajectories

for processes

subordinate

to

gamma

distributions;

the

processes

become

increasingly disordered as K,

-+

1.

The

entropy

gradient

curves cor-

respond

to

systems

with

ex

te

rnal

input

-

the

mean

increases as disorder in-

creases.

Th

e

asymptote

is K, =

1,

the

exponential

case

of

maximum

disorder.

Conversely,

the

reverse

direction

of

the

curves

corresponds

to

evolution from

total

disord

er

to

other

states

(clustered

for K, < 1,

and

smoothed

out,

'more

crystal-like', for

K, > 1) while

the

mean

is allowed

to

r

ed

uce-som

ew

hat

like

the

situation

after

the

Big

Bang,

see

Dodson

[4].

3

Constrained

degeneration

of

order

Lucarini

[ll] effectively

illustrated

the

dege

neration

of

order

in his

perturba-

tions

of

the

simple 3D cubic

crystal

lattic

es (SC,

BCC,

FCC)

by

an

increas-

ing

spatial

Gaussian

noise. Physically,

the

perturbing

spatial

noise

intensity

corresponds

somewhat

to

a

lattice

temperature

in

th

e

structural

symmetry

degeneration

.

With

rather

moderate

levels of noise,

quit

e quickly

the

three

tessellations

became

indistinguishable.

In

the

presence

of

intense

noise

they

all converged

to

the

3D

Poisson-Voronoi tessellations, for which

exact

ana-

lytic

results

are

known

[6].

Moreover, in all cases

the

gamma

distribution

was

an

excellent

model

for

the

observ

ed

probability

density

functions of all

metric

and

topological

prop

ert

ies. See also

Jarai-Szabo

and

Neda

[9]

for some

analytic

approximations

using

gamma

distributions

for two

and

three

dimen-

sional Poisson-Voronoi cell size

statistics.

Lucarini

provided

plots

showing

the

evolution of

the

mean

and

standard

deviation

of

th

ese

properti

es as

they

converge

asymptotically

towards

the

Poisson-Voronoi case,

illustrating

the

degeneration

of

crystallinity

from K, ~

00

to

lower values.

Of

course,

the

constraint

of

remaining

tessellations,

albeit

highly disor-

dered

ones, precludes convergence

down

to

the

maximum

entropy

limit

K, =

1.

In

fact

the

limiting

valu

es

are

K,

;::,;;

16 for

number

of vertices

and

the

same

for

number

of edges

and

K,

;::,;;

22

for

the

number

of

faces;

actually

these

are

dis-

crete

random

variables

and

the

gamma

is

not

appropriate.

However, for

the

positive r

ea

l

random

variables,

polyhedron

volume in

the

limit

has

K,

;::,;;

5.6

and

polygon

face

area

K.

;::,;;

16.

Lucarini

[10]

had

report

ed similar findings for

the

2D case

of

perturbations

of

the

three

regular

tessellations

of

the

plane:

square,

hexagonal

and

triangular.

There

also

the

gamma

distribution

gave a

good

fit for

the

distributions

during

the

degeneration

of

the

regular

tessella-

tions

to

the

2D Poisson-Voronoi case;

the

limiting

values were K,

;::,;;

16 for

the

pe

rimeter

of

polygons

and

K,

;::,;;

3.7 for areas.

On

the Entropy Flows to Disorder 79

We

can

give

another

perspective

on

the

level

of

constraint

persistent

in

the

limits for

thes

e disordered tessellations: for infinite

random

matri-

ces

the

best

fitting

gamma

distributions

for eigenvalue spacings

have",

=

2.420, 4.247, 9.606 respe

ct

ively for orthogonal,

unitary

and

symplectic Gaus-

sian ensembles

[2].

Thes

e values in a sense

indicat

e

the

increasing

statistical

constraints

of

algebraic relations as

the

ensembles change

through

orthogo-

nality,

unit

ar

ity

and

symplectivity.

In

the

context

of

analytic

number

theory,

gamma

distributions

give ap-

proximat

e

distributions

for

the

r

epo

rted

data

on spacings between zeros

of

the

Riemann

zeta

fun

ct

ion

[12].

Th

e best fit

gamma

distributions

to

spac-

ings between

the

first two million zeros of

the

Riemann

zeta

function

has

"':::::;

5.6

[2].

For

the

SARS disease epidemic

outbreak

[3,14]

the

gamma

di

str

ibution

gave a good

model

for

the

infection process

with",

:::::;

3,

cf.

[5].

4

Bivariate

gamma

processes

consider

the

bivariate case

of

a

pair

of

coupled

gamm

a processes.

The

McKay

family

can

be

thought

of

as giving

the

probability

density

for

the

two

random

variables X

and

Y = X + Z where X

and

Z

both

have

gamma

distributions

.

This

smooth

bivariate family M

of

density functions is defined

in

the

positive

octant

of

random

variables 0 < x < y <

(Xl

with

parameters

CYi, C,

CY2

E

~+

and

prob

ability

density

functions

cCCXl+CX2)XCXl-i(y _

x)cx

2

- i

e

-c

y

m(x,

y)

=

r(CYdr(CY2)

(6)

The

margin

al

density

functions,

of

X

and

Yare:

x>o

(7)

c(CX1

+CX2)y(CX

1 +cx2

)-

le-

c

y

my(y)

=

r(CYi

+

CY2)

y>

o.

(8)

Note

that

we

cannot

have

both

marginal

distributions

exponential.

The

covariance

and

correlation coefficient of X

and

Yar

e:

CYi

{;!£;i

0"12 = 2""

and

p(X, Y) = .

C

CYi

+

CY2

Unlike

other

bivariate

gamma

families,

the

McKay

information

geometry

is

surprisingly

tractable

and

there

are

a

number

of

app

li

cat

ions discussed in

Arwini

and

Dodson

[2].

In fact

the

parameters

CYi, C,

CY2

E

~+

are

natural

coordinates

for

the

3-dimensional manifold M

of

this

family.

The

Riemannian

80

C.

T.

1.

Dodson

information

metric

is

(9)

It

is difficult

to

present

graphics

of

curves

in

the

McKay

3-manifold

M,

but

it

has

an

interesting

2-dimensional

submanifold

M1 C

M:

001 = 1.

The

density

functions

are

of

form:

(10)

defined

on

0 < x < y < =

with

parameters

c,002

E

jR+.

The

correlation

coefficient

and

marginal

functions

of

X

and

Yare

given by:

(11)

(12)

(13)

In fact 002 =

l~t,

which

in

applications

would give a

measure

of

the

vari-

ability

not

due

to

the

correlation.

The

matrix

of

metric

components

[gij 1

on

M1

is

(14)

Some geodesics

emanating

from

(002,

c) =

(1,1)

E M1

are

shown

on

the

left

of

Figure

3.

The

density

functions

can

be

presented

also

in

terms

of

the

positive

parameters

(OO1,CJ12,OO2)

where

CJ12

is

the

covariance

of

X

and

Y

(15)

x>O

(16)

y >

O.

(17)

On the Entropy Flows

to

Disorder

81

c

Fig.

3.

Geodesics passing through

(c

,

(2)

=

(1,1)

(left)

and

a

contour

plot

of

the

en

tropy

wdh

some

int

egral gradi

ent

CU'f'1J

C8

(right),

in

the

McKay

submanifold

1111

which has a1 = 1.

The entropy function is

On

the

right of Figure 3 is a con

tour

plot

of the en

tropy

showing

its

grad

ient

field

and

some integral

cur

ves.

It

may be helpful

to

express

the

entropy

in

terms

of

a2

and

p, which gives

The McKay

entropy

in M1

with

respect

to

a2, P is shown in Figure 4 (left)

with

the

gradient

flow

on a cont

our

plot (right)

tog

et

her

w

ith

th

e a

ppro

xim

at

e

locus curve of maximum entropy.

The

two

thinner

curves show the loci of

a 1 = 1 (

upp

er

curve)

and

a1

+

a2

= 1 (lower curve), which correspond,

respectively,

to

Poisson

random

processes for

th

e X

and

Y var

iab

les.

82

C.

T.

1. Dodson

a2

Fig.

4.

Surface representation

of

the

Shannon

entropy function 8

m

f01'

the

s'U.bmani-

fold

!viI

of

the M

cK

ay family 'with respect to

p(L7'ameter'

a2 and correlation coefficient

p (left), and contour plot (right). Superimposed also are gradient flow arrows and

the thick

curve is the

loc

'us

of

ma

xim

um

entmpy.

The two

thinner

C1lr'Ves

show the

loci

of

al

= 1 (upper curve) and

al

+

a2

= 1 (lower curve), which correspond,

respectively, to Poisson

random processes

for

the X and Y variables.

Qualitatively,

what

we

may

see in Figure 4 is

that

for correlated ran-

dom variables

X

and

Y s

ubordinate

to

the

bivariate

gamma

dens

ity

(10),

the

maximum entropy locus is roughly hyperbolic.

The

maximum

entropy

curve is

rather

insensitive

to

the

correlation coefficient p when (X2 > 1

and

the

difference Y - X is dispersed more evenly

than

Poisson.

When

Y - X is

actually Poisson

random

,

with

0;2

= 1,

the

critical value is

at

p ~ 0.355. How-

ever,

as

0;2

reduces

fnrth

er- -corresponding

to

clustering

of

Y - X

values

-

~so

the

locus

turns

rapidly

to

increasing correlation.

If

we

take

the

situation

of

a bivariate

gamma

process

with

constant

marginal

mean

values,

then

the

McKay probability density has co

nst

ant

correlation coe

ffici

ent

p; in

this

case

the

gradient

flow

li

es along lines of

constant

0;2·

Dodson

[5]

gives

an

application

to

an

epidemic model in which

th

e la-

tency

and

infectivity for individuals in a

population

are jointly

distributed

properties controlled by a bivariate

gamma

distribution.

5

Weibull

processes

Like

the

gamma

family,

the

Wei bull family of distributions contains

the

ex-

ponential

distribution

as a

sp

ecial cas

e;

it

ha

s wide application in models for

reliability

and

lifetime

statistics

for

random

variable t >

O.

The

probability

density function

can

be presented in

terms

of positive

parameters

~,f3

f3

( )

/3-

1

w :

jR+

---;

jR

: t

h>

~

f .

e

~

(t

/

O

{3

.

(21)

On

the Entropy Flows to Disorder

83

Fig

.

5.

Surface repl-csentaJion

of

the Sha.nnon

cntmpy

ftmction

Sw

for

the Weibnll

family

(left).

and

conto'Wr

plot

(T'ight) with

gmdient

flow

and

integml

carves.

In

applications

of

(21), reliability

R(t)

is

the

probability

of

survival

to

time

t

and

it

is

the

failure

rate

Z (t)

at

time

t

through

i

·

X

(t) (

1)

(I

>

_'t!i.:(J

r.'

1JJ

" j

R(t)=w(t)dt=e

(f~)

and

Z(t)

=_

..

=[3

-.

t . R(

t)

.

~

The

INeihull

mean,

Htandard

deviation

and

entropy

are

1

/1",

= C

T(l

+ - )

<,

.

(1

(22)

(24)

(25)

(26)

In

(25) 1 is

the

Euler

constant.

approximately

0.577.

In

cFt.'le

3 = * for

("

) I

positive integer

n,

then

the

coefficient

of

variation

is !!3L =

-(":~')i

--

1,

and

/1

-

11.'

n.

we

sec

that

tl

= 1 reduces (21)

to

the

exponential

density

with

Itw

=

CJ

w

=

~

and

unit

eoefIicient of variation.

Figure

5 shows

11

surface

plot

of

the

Weibull

entropy

Sw

and

a

corresponding

contour

plot

,\lith

gradient

flow

and

some

integral

curves.

There

is a resemblanee

to

the

gamma

distribution

entropy

in

Figure

2 a.nd from (26)

again

here

we

have

/-lw(t)

=

/-l'/u((J)e

t

,

but

in

the

\Veibull case

the

asymptotic

curve

has

f3

= r

~

0.577, which corresponds

to

it

process

with

coefficient

of

variation

~

4.65

compared

with

unity

for

the

exponential

distribution

and

Poisson

randomness.

84

C.

T.

1.

Dodson

Acknowledgement

The

author

is

grateful

to

V.

Lucarini

for discussions

concerning

the

interpre-

tation

of

his

simulations.

References

1.S-I.

Amari

and

H.

Nagaoka.

Methods

of

Information

Geometry.

American

Math-

ematical

Society,

Oxford

University

Press,

Oxford,

2000.

2.Khadiga

Arwini

and

C.

T.

J.

Dodson.

Information

Geometry: Near

Randomness

and

Near

Independence. Springer-Verlag, New York, Berlin, 2008.

3.T.

Britton

and

D.

Lindenstrand.

Epidemic

modelling:

aspects

where

stochasticity

matters.

http://arxiv.org/abs/0812.3505,

pages

1~16,

2008.

4.C.

T.

J.

Dodson.

Quantifying

galactic

clustering

and

departures

from

random-

ness

of

the

inter-galactic

void

probablity

function

using

information

geometry.

http://arxiv.org/abs/astro-ph/0608511,

2006.

5.C.

T.

J.

Dodson.

Information

geometry

and

entropy

in

a

stochastic

epidemic

rate

process.

http://arxiv

.

org/abs/0903.

2997, 2009.

6.S.R.

Finch.

Mathematical

Constants.

Cambridge

University

Press,

Cambridge,

2003.

7.A. Gray.

Modem

Differential

Geometry

of

Curves

and

Surfaces 2

nd

Edition.

CRC

Press,

Boca

Raton,

1998.

8.T-Y.

Hwang

and

C-Y. Hu.

On

a

characterization

of

the

gamma

distribution:

The

independence

of

the

sample

mean

and

the

sample

coefficient

of

variation.

Annals

Inst.

Statist.

Math.,

51(4):749~753,

1999.

9.F.

Jarai-Szabo

and

Z.

Neda.

On

the

size

distribution

of

poisson

voronoi

cells.

http://arxiv.org/abs/cond-mat/0406116v2,

2008.

1O.Valerio

Lucarini.

From

symmetry

breaking

to

poisson

point

processes in

2d

voronoi

tessellations:

the

generic

nature

of

hexagons.

1.

Statist.

Phys.,

130:1047~1062,

2008.

I1.Valerio

Lucarini.

Three-dimensional

random

voronoi tessellations:

From

cubic

crystal

lattices

to

poisson

point

processes.

1.

Statist.

Phys.,

page

In

press,

2008.

12.A. Odlyzko.

Tables

of

zeros

of

the

riemann

zeta

function

http://www.dtc.umn.edu:80/-od1yzko/zeta_tab1es/index.htm1.

2009.

13.A.

Papoulis.

Probability,

Random

Variables

and

Stochastic Processes.

McGraw-

Hill, New York.

14.WHO.

Cumulative

number

of

reported

probable

cases

of

severe

acute

respiratory

syndrome

(sars).

http://www.who.int/csr/sars/country/en/,

2003.

85

Linear Communication Channel Based on Chaos

Synchronization

Victor Grigoras 1 and Carmen Grigoras

2

Faculty

of

Electronics, Telecommunications and Information Technology,

'Gh. Asachi' Technical University, Iasi, Romania

2 Faculty

of

Medical Bioengineering, University

of

Medicine and Pharmacy

"Gr. T. Popa",

Ia~i,

Romania

Romanian Academy -

Ia~i

Branch, Institute

of

Computer Science, Romania

(e-mail: grigoras@etc.tuiasi.ro )

Abstract: This paper presents the possibility

of

designing a linear communication

channel by modulating chaotic analog systems. After presenting the general setup,

conditions for correct demodulation and linear dynamic input-output behavior are

demonstrated. For a linear dynamic relation between the modulating and demodulated

signals, channel equalization is used to achieve wider bandwidth transmission. The

presented case studies, regarding the Lorenz and Chen systems, highlight the

applicability

of

the proposed method for high speed digital communication. The

overall performance

of

the resulting communication system is analyzed in terms

of

speed, security and occupied frequency bandwidth. The concluding remarks point

towards some directions

in

further research.

Keywords: Chaos synchronization, channel equalization, Lorenz system, Chen

system.

1.

Introduction

The present contribution aims at characterizing an analog modulation

method to build a wide-band communication channel. The proposed approach

is

based on chaos synchronization between the emitter and receiver ends

in

order to achieve a supplementary level

of

security under the standard digital

encryption layer. The general synchronization setup is based on the system

partitioning method, first introduced by Carroll and Pecora in [1]. Considering

some reasonable restrictions for the chaotic system,

we

develop a method

for the characterization

of

the input-output relation

of

the proposed

communication channel, namely between the modulating signal applied at the

emitter end and the demodulated signal obtained at the output

of

the receiver.

The linear dynamic system that models the studied relation shows the

frequency limitations for the modulating signal. In order to increase the

transmitted signal bandwidth, a feed forward equalizer is proposed and its

efficiency studied. Case studies regarding third order chaotic systems

of

the

Lorenz

[2]

and Chen

[3]

types aim at proving the feasibility

of

the proposed

method. The Lorenz system, chosen for implementation advantages

[4]

and

for synchronization simplicity [5-6], proved

to

be a good prototype for the

proposed method, allowing easy synchronization and demodulation. In order

to apply the general setup to the Chen system, some coefficient modifications

had to be made

to

allow the proposed demodulation scheme.

86

V.

Grigoras and

C.

Grigoras

The next section concentrates on the demonstration

of

the proposed method

in the general case. The error dynamics method is used, leading to the

conclusion that the input - output relation

of

the proposed channel is linear.

Simulation results, confirming the theoretical solution, are presented in the

third section, for two case studies.

2. General Results

The communication channel based on chaos synchronization is built around

a nonlinear chaotic emitter described by the general state equations:

x'=f(x)

where x denotes the N-dimensional state vector and f :

lR

N

~

lRN

is the

nonlinear state transition function.

In

order to achieve chaos synchronization, using the emitter subsystem

approach, we presume that the above state equations can be partitioned in the

form:

j

XI

' =

./;

I

(XI)

+

./;?

(XR)

,

-()

lXI]

./;1:

lR

~

R

'/;

2:

lR

N

-

I

~

lR

;

xR

=f

21(XI)+f22 xR ; X=

;f

.m

mN-

I'f .

mN

-I

mN

-I

X

R

21 .

""

~

""

,

22

'

""

~

""

y=XI

If

we choose to transmit the first state variable,

Xl,

the synchronizing

receiver results in the form:

X

R

'

=i

22

(XR

)+f

2

1(Y)

where the

';'

signals and functions denote the receiver correspondents

of

the

emitter ones.

Presuming the receiver has been correctly designed, in the case

of

parameter matching

((

21

(-) = f21

(-);£22

(-)

= f

22

(-

p,

the error dynamics

IS

globally asymptotically convergent

to

zero:

E'=f

22

(xR)-f2

2

(x

R

) ;

E(t)

~

0

where the error E

(t

) = x

R

(t)

- x

R

(t)

denotes the difference between the emitter

and receiver state vectors.

To achieve transmission

of

the useful information signal,

met),

a direct

modulation approach

is

used at the emitter end, by modifying the first state

equation:

XI'='/;I (XI)+./;2

(xR)+m(t)

At the receiver end, demodulation is implemented by appending the

previous receiver state equations with a similar one, without modulation:

XI'=~I(XI)+~

2

(XR)

The demodulated signal is algebraically obtained giving the receiver output

equation:

Linear Communication Channel Based

on

Chaos Synchronization 87

The dynamic time evolution

of

the demodulated signal is governed by the

resulting nonlinear differential equation, obtained by subtracting the previous

state equations:

m

'(t)

=

.1;1

(Xl)

-

~

1 (j\) +

i12

(XR)

-

112

(XR)

+

m(t)

As we presumed the error dynamics to be globally asymptotically stable, in

the case

of

parameter matching

(~1

(.)

=

.I;

1

(.);

~2

(.)

=

i12

(.)), the second

difference converges to zero

i12(XR)-

112(X

R

)-70

After the synchronizing transient, the demodulated signal differential

equation

is

simplified to the form:

m'(t)

=

111

(X1)-

111

(i\)+m(t)

Assuming the average value

of

the modulating signal is zero,

we

can

decompose the algebraic nonlinear function,

ill

(.)

in a power series around

this value, resulting:

.1;1

(X)

= f

~

d"!:1

(O)·x"

n=O

n. dx

If

the nonlinear function,!II(.), is smooth enough and the modulating signal

small,

we

may neglect the higher order terms, to obtain:

m

'(t)

=

d.l;

I

(0)·

met) + met); met) = XI

(t)

-

iJt)

dx

The resulting linear dynamic system, characterizing the proposed

communication channel based on chaos modulation,

is

stable

if

the structural

constant,

a,

is negative:

a=d.l;l(O)<O

dx

In the complex frequency domain, the input-output characterization

of

proposed communication channel is given by the transfer function:

H(s)

=

M(s)

=_1_

M(s)

s+a

The obtained result shows that the modulating signal cannot be perfectly

recovered unless the maximum frequency in its spectrum

<is

(much) less than

the structural constant,

a.

Aiming our results at applications in higher

frequency transmission,

we

propose a single order equalizer connected in a

feed forward topology. The real zero

of

the equalizer must compensate the

channel pole, while the pole

of

the equalizer

is

chosen at a much larger

frequency,

at,

to limit the overall pass band

of

the communication system.

HE(s) =

s+a

l/mc'

s+

1

The overall communication system

is

linear, characterized by the transfer

function:

Skiadas C.H., Dimotikalis I. (editors) Chaotic Systems: Theory and Applications

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