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

218

D.

C.

Ni

and

C.

H.

Chin

Figure 1 (a) R

-10-

6

Figure

l(b)

R

-10-

2

Figure l(c) R

=1

Figure 1 (d) R

-10

2

Figure lee)

R-10

6

Symmetly Broken

in

Low Dimensional N -Body Chaos

219

Figure 2 (a)

Figure 2 (b)

normailzed

phase

factor

on

the

symmetry

broken

of

original

domains

and

transition

from

stable

doma.ins

to

chaotic doma.ins.

The

compound

influence

of

these

two

parameters

on

the

transitions

is also illustra.ted.

4.1

Normalized

Energy

Level

(

parameter

: a )

Figure

3 (a)

through

(d) show

the

original

domains

of

nnit

circle (i.e., R = 1)

with

normalizpd energy level a = 0.01

in

Figure

3 (a), a = 0.1 in

Figure

3 (b), a = 0.2

in

Figure

3 (c)

and

a = 1

in

Figure

:3

(d)

of

the

function

f =

Z-

l

C\C

2

C;j.

\Ve observe

that

the

original

domain

showing

symmetrical

characteristics

when

the

value

of

normalized

energy

level is small ( a = 0.01 )

as

shown in

Figure

:3

(a). As

the

value of

the

normalized

energy

level increases from 0.1

to

1;

the

origina.l

domain

becomes

gradually

to

a.synl1netrical

fractal

patterns

and

eventually

becomes to chaotic

patterns.

This

observation

also applied

to

the

unit

circles

of

original doma.ins

at

other

function orders.

4.2

Normalized

Phase

factor

(

parameter:

exp(9i

(z)

)

In

the

following, we

examine

4th

order

domains

as

shown in

Figure

1

with

parameter

of

normalized

energy

level a =

OJn

and

normalized

phase

factor

at

88 degrees when

the

original

domain

rema.ins a

stable

domain

and

at

89

degrees when

the

original

domain

beeomes

11

chaotic

domain.

These

two

specific degrees are defined as follows:

g(z) =

1f

X i x (degr-ce/180) (5)

vVe

exmIline

the

original

domains

based

on

the

order

of

iterations

as

shown

in

Figures

"1

(a)

through

(d) for

the

ca.'le

of

88 degrees

and

in

Figures

5 (a)

through

(f) for

the

case

of

89 degrees.

220 D. C Ni

and

C.

H.

Chin

Figure 3 (a) a = 0.

01

Figure 3 (b) a =

0.1

Figure 3 (c) a = 0.2

Figure 3 (d) a = 1

Fig.

3. Symmetry Broken and Chaos induced

by

NonnaIized

Energy

Level

Comparing

with

Figure

l(c),

Figure

4 (a) shows

that

the

low-it.erated

portion

of

fraetals

at

center

of

unit

circle is

rotated

at

88 degrees while

the

bulbs

extended

from

individual

fractal

branches

Jag

an

angle

against

the

cen-

tral

portion

of

fractal

pattern.

This

is

another

symmetry-broken

mechanism

by

the

normalized

phase

factor,

C:Dp([li(Z)).

In

addition,

a

phenomenon

sim-

ilar

to frequency-doubling is

that

the

boundries

of

low-iterated

portion

of

fractals

at

center

is

becoming

a

six-branched

pattern

instead

of

original 3-

branched

pattern.

\Ve call

this

phenomenon

as

branch-ramification.

When

the

iternation

number

increases, we observe spirals formed

to

connect

the

extended

bulbs

back

to

the

original

branches

as

shown

in

Figure

<1

(b) (it-

eranion

number

= 250). As

the

iternation

numbers

are

further

increased

as

Figure

<1

(c)

and

Figure

4 (d),

the

central

portion

of

fractal

rema.ins in

the

same

fractal

boundaries

and

only

the

turns

of

connected

spiral

increases

and

more

extended

bulbs

around

the

inner

boundary

of

the

unit

circle (i.e.,

H = 1)

appear.

\Ve claim tha.t

this

set

of

parameters,

namely, a = 0.01

and

degTee

"'.c

88, defines a st.able

domain.

vVhen

the

rotahon

angle increases from 88 degrees

to

89 degrees, as

shown

in

Figure

5, we observe

that

the

branch-ramification

as

shown

in

Figure

5 (b)

and

·Figure 5 (c)

are

not

stopping

at

particular

iteration,

these

newly growing

branches

further

extend

and

conned

to

the

angle-lagged

exteneded

bulbs.

As

Symmetry Broken

in

[,ott'

Dimensional N-Body Chaos

221

Figure 4 (a) iteration = 50

Figure 4 (b) iteration = 250

Figure 4 (c) iteration = 1000

Figure 4 (d) iteration = 2000

Fig.

4.

Stable

Domains

with

different

iteration

at

88

degrees

th

e

iteration

number

is

equal

to

100 shown in Figure 5

Cd),

we observe

the

spirals, as shown in

Figur

e 4,

appear

and

compet

e for connecting

to

the

extended bulbs.

As

the

it

e

ration

number

increases

to

200, shown in Figure

5

(8),

the

branch-ramification

and

bulb-connecting compe

tetion

are

further

developed a

nd

the

original domains eventually becomes a chaotic domain as

shown in Figure 5 (f).

This

indicates

that

all points in

the

original domain C

1

:::;

R ) will eventually diverge

at

a given

numb

er of iteration.

As

the

normalized energy level increases,

the

chaotic transitions occur

a.t

lower

rotatio

n angles

of

the

norm

alized

phas

e factor, crp(g(z)) . Table

1 illustrates

the

eases for a = 0.1, a = 0.2,

and

a = 0.3. As normalized

energy level increases

to

a c

rita

l value,

the

origianal domains become chaotic

domains

without

an

y

rotation

applied as shown in

Fi

gure

3.

Table

1.

C

haotic

n·an

sition in

Parametric

Space

222

D.

C.

Ni

and

C.

H.

Chill

Figure 5 (a) iteration

50

Figure 5 (b) iteration =

70

Figure 5 (c) iteration = 80

Figure 5 (d) iteration = 100

Figure

5 (e) iteration =

200

Figure 5 (f) iteration = 1000

Fig.

5.

Chaotic

Domains

with

different

ih:ration

at

89

degrees

5

Remark

In

this

paper

we examined

the

transiton

to

chaotic

status

of

the

original

domains

of

fUllction f = hez) TIdca:p(g;(z))[(a;

-z)/(1-

aiz)]}.

The

nor-

malilled energy level

and

the

normalized

phase

factor

arc

two

parmnters

in

the

parametric

space

of

this

function induce

symmtry

broken in

the

fractal

patterns.

Accompanying with

symmetry

broken, we observed

the

trasitions

from

stable

domains

to

chaotic domains

under

the

iufiunee

of

combined ef-

Symmetry Broken

in

Low Dimensional N-Body Chaos 223

feet

of

these

two

parameters.

Branch-ramification

and

bulb-connecting com-

petion

are

the

observed

phenomena

during

the

transitions.

We will

further

investigate

mathematical

and

physical implications

of

these

computational

observations.

References

l.Benoit

B.

Mandelbrot,

The

Fractal

Geometry

of

Natur

e, New York,

W.H.

Free-

man

and

Company, 19S3.

2.K. Falconer,

Fractal

Geometry:

Mathematical

Foundations

and

Applications,

New York,

John

Wiley & Sons, 1990.

3.J. Milnor,

Dynamics

in

On

e

Complex

Variable, Vieweg, 2000.

4.A.

Fathi

and

J.

-C. Yoccoz (Eds.),

Dynamical

Systems: Michael

Herman

Memorial

Volume,

Cambridge

University

Press, Feb. 2006.

5.David

C. Ni

and

C. H.

Chin,

Z-

l

C

1

C2C3C

4

System

and

Application,

TIENCS

workshop,

Singapore

,

August

1-5, 2006.

6.David C. L. Ni

and

C. H.

Chin,

A Novel

Approach

for Designing

Fractal

Anten-

nas

, Proc. of

Intern

a

tional

Conference

on

Wireless

Information

Networks

and

Systems,

pp.

157-160, (Eds. S.O.

Mohammand

et

al.), Barcelona, Spain,

July

2007.

7.David C. Ni

and

C. H.

Chin,

Broadband

Fractal

Antennas

,

Proc.

of

Int

e

rnational

Symposium

on

Antennas

and

Propagation

, Taipei, Taiwan,

pp.

6S

(Abstract),

October

27-30, 200S

S.C.

H.

Chin,Hypertranscendental

Geom

e

try

,

Preprint

NCTU-IMS-OS01, Taiwan,

R.O.C.

9.David C. Ni

and

C. H.

Chin,

Herman

Ring

Classification

on

Function

f =

h(z)

Il{exp(gi(z))

[(ai -

z)/(1

- a;z)]} ,

accepted

for

publication

in

the

5th

International

Conference 2009 -

Dynamical

Syst

e

ms

and

Applications, Con-

stantz

a,

Romania

,

June

15-1S, 2009

224

On

Markov

Processes:

A

Survey

of

the

Transition

Probabilities

and

Markov

Property

Gabriel

V.

Orman

Department

of

Mathematical

Analysis

and

Probability

"Transilvania"

University

of

Bnl~ov

500091

Brasov,

Romania

(e-mail:

ogabriel@unitbv.ro

)

Abstract:

Some

problems

concerning

the

Markov

processes

are

discussed

shortly.

Among

other

things,

it

is

presented

an

e

xt

e

nded

Markov

property.

Also,

two

variants

of

the

strong

Markov

property,

as

they

are

synthetized

by

Kiyosi Ito,

are

discussed.

Keywords:

random

walk

,

transition

probabilities,

transition

operators,

Markov

prop-

erty.

2000

MS

Classification:

60H25, 60H30, 60J1O,

60J27,

60J60.

1

Markov

processes

Generally

speaking

,

it

can

be

considered

a

probl

em

where

compu

t

ation

is

split

among

se

veral

processors

,

operating

and

transmitting

data

to

one

anoth

er

asynchronously.

Such

algorithms

are

only

being

to

come

into

promin

ence,

due

to

both

the

developm

e

nts

of

decentralized

processing

and

applications

wh

ere e

ach

of

several

locations

migth

control

or

adjusted

local variable

but

th

e

criterion

of

concern

is global.

For

example

a

current

de

centralized

application

is

in

Q-learning

where

the

component

update

at

any

time

depends

on

the

state

of

a

Markov

process.

The

usual

clas

s

of

Markov

processes

which

we involv

many

tim

es

has

some

re

strictions

that

it

does

not

cover

many

int

ere

sting

processes.

This

is

the

reason

for

which

we

try

often

to

obtain

some

extensions

of

this

notion.

Some

research

es

in

such

a

dir

e

ction

are

due,

among

others,

to

Kiyosi Ito.

In

this

cont

e

xt

we

shall

refer

here

to

Markov

proc

esses

and

their

impact

for

some

practical

problems

(see for

example

[4], [5],

[6]).

1.1

Transition

probabilities

Let S

be

a

state

space

and

it

is

supposed

that

a

particle

moves in

S.

Also,

it

is

supposed

that

the

particl

e

starting

at

x at

the

present

moment

will

mov

e

into

A C S

with

proba-

bility

pt(x,A)

after

time

t

"irresp

e

ctively

of

its

past

motion";

that

is

to

say

this

motion

is

considered

to

have a

Markovian

character.

The

tran

s

ition

probabilities

of

this

motion

are {Pt(x,

A)h

,x,A

and

it

is c

onsidered

that

the

tim

e

parameter

mov

es

in

T = [0, +00).

The

stat

e

space

S is

consid

ered

to

be

a compact

Hausdorff

space with a countable

open base so

that

it

is

homeomorphic

with

a

compact

separ

a

ble

metric

space

by

the

Urysohn's

metri

z

ation

theorem.

The

a-field

genera

ted

by

the

op

en

space

(the

topological

a-fieldon

S)

is

denoted

by

K(S).

Therefore

, a

Bor

el

set

is a

set

in

K(S).

A Survey

of

the Transition Probabilities and Markov Property 225

The

mean

value

m = M(/-I) = JR x

/-I

(

dx)

is

used

for

the

center

and

the

scattering

degre

e

of

a

one-dimensional

probability

measure

/-I

having

the

second

ord

er

moment

finite;

and

the

varianc

e

of

/-I

is

the

following

On

the

other

hand

,

from

the

Tchebichev's

inequality

we

hav

e (for

any

t > 0)

1

/-I(m -

tu,

m +

tu)

<

-2

- t

so

that

several

properties

of

I-dimensional

probability

measures

can

be

de

riv

ed.

But

in

the

case

when

the

considered

probability

measur

e h

as

no

finite

second

order

moment,

u

becomes

useless.

In

s

uch

a

case

one

ca

n

introduc

e the

central

value

and

the

dispersion

that

will

play

similar

roles

as

m

and

u for

general

I-dimension

al

probability

measur

es.

Note

1.

We

remind

that

J.L.

Doob defined the

central

valu

e,

= ,(/-I) as being

the

real

number,

which

ve

rifyes

the

following

relation

r

arctg(x

-,)

/-I(dx)) =

O.

JR

Here,

th

e

existe

nce

and

the

uniqueness

of

1 follow

from

the

fact

that

arctg(

x -

I)

de-

creases

strictly

and

contin

uous

ly

from

~

to -

~,

for

x fixed

and

1

moves

from

-00

to

2 2

+00.

Th

en,

the

dispersion

<5

is

defined

as

follows

Regardin

g

to

the

transition

probabiliti

es

{Pt(x,

A)}

tET,XES,AEK (S) , it is s

upposed

that

they

satis

fy

the

follo

wing

conditions:

(1)

for t

and

A fixed,

a)

the

transition

probabilities

are

Bore

l

measurabl

e

in

x;

b)

Pt(x,

A)

is a

probability

measur

e in A;

(2)

p

o(x,

A)

= 6

x

(A)

(i.e.

the

6-measure

concentrated

at

x);

(3)

Pt(x,·)

---'we

ak

Pt(xo

, ·)

as

x

->

Xo

for

any

t fixed;

[

that

is

to

say

x

l!...rICioJ

f(y)pt(

x,dy)

=./

f(y)pt(

xo,

dy)

for all

continuous

functions

f

on

Sj;

(4)

Pt(x,

U(x))

---.

1

as

t 1 0, for

any

neighborhood

U(x)

Of

x;

(5)

the

Chapman-Kolmogorov

eq

uation

holds:

Ps+t(x,A)

=

}SPt(x,

dy)ps(y,

A).

226

G.

V.

Orman

In

a

similar

manner

the

transition

operators

can

be

defined.

It

is

consider

ed

that

C =

C(S)

is

th

e

spac

e

of

all

cont

inuous

function

s

(it

is a

separabl

e

Banach

space

with

a

supremum

norm).

The

following

operators,

denoted

by

Pt

,

(ptf)(x)

=

is

Pt(x,

dy)f(y),

f E C

are

called

transition

operators.

Now

th

e

conditions

for

the

t

ransition

probabilities

can

be

adapte

d

to

the

transition

operators.

We

do

not

insist

here

on

this

case (for

more

details

see

[4],

[

6],

[2]).

Let

us

observe

that

the

conditions

(1) - (5)

above

are

satisfied

for

Brownian

transition

probabilities.

One

can

define

1 _

(,,

_

x)2

pt(x

,dy) =

=e

2,2

dy

tv

27r

inR

pt(oo

,

A)

=

o=A.

Definition

1.

A

Markov

process is a

system

of

stochastic processes

{Xt(w),t

E

T,w

E

(O,K,Pa)}aES.

[For

each

a E

S,

{Xt}tES

is a

stochastic

proce

ss

defined

on

the

probability

space

(0

, K , Pal

l.

The

tran

s

ition

probabilities

of

a

Markov

proc

ess

are

{pet, a,

Bn.

Let

us

de

note

by

{Ht}

the

transition

semigroup

and

let

R

",

be

the

resolvent

operator

of

{Ht}.

Now pet, a,

B),

Ht

and

R",

can

be

expressed

in

terms

of

the

process

as

follows.

Theorem

1.

Let

f

be

a

fun

ction

in

C(S).

Then

10.

p(t,a,B)

=

Pa(X

t

E

B).

20. For

EaO

=

.f~

·Pa(dw) one has

Htf(a)

=

Ea(f(Xt)).

30. R

",

f (a) =

Ea

Uo=

e-"'t

f(Xt)dt).

Proof.

One

can

observe

that

1 °

an

d 2

0

are

immediately.

Now, for 30,

it

is

consid

er

ed

the

following

equality:

R",f(a)

= 1= e- ",t

Hd(a)dt

= 1=

e-",t

Ea(f(Ht))dt

.

But

f

(X

t (w)) is

right

continuous

in

t for w fixed

and

mea

surab

le

in

w for t fixed

and

therefore

it

is me

asurable

in

the

pair

(t, w).

Thus,

the

Fubini's

th

e

orem

can

be

used

and

one

gets

that

is

just

30..

Definition

2.

The operator e

t

:

°

--t

° defined as follows

Uhw)(s) =

w(s

+ t)

for

every s E

T,

is called

th

e "shift operator".

Evidently

,

(the

semigroup property).

Notation.

For

e a

a-field

on

0 ,

the

spac

e

of

all

bounded

e-m

eas

urable

functions

will

be

denoted

by

B(O,

e),

or

simple

B(e).

A Survey

of

the Transition Probabilities

and

Markov Property 227

2

Markov

property

We

shall

develop

below

some

aspects

concerning

the

Markov

property

and

in

a

vision

of

Kiyosi Ito.

Firstly

the

classical

and

the

extend

ed

Markov

prop

e

rtie

s

are

present

ed

.

Then

the

strong

Ma

rkov

property

will

be

considered.

The

Markov property is e

xpressed

in

the

theorem

below.

Theorem

2.

Let

be

given r E

K.

Th

e following

is

tru

e

that

is to say

Note

2.

Th

e following

notation

can

be

used

Proof.

It

will

be

suffice

to

show

that

(1)

for r E K a

nd

DE

K

t

.

On

e

can

distinguish

three

cases.

[I.]

Let

us

consider

rand

D as follows:

and

with

o

:S

Sl

<

S2

< ... <

Sn

o

:S

t 1 <

t2

< ... <

tm

:S

t

and

B

i

,

Aj

E

K(S).

Now

it

will

be

obs

e

rved

that

the

both

sides

in

(1)

are

expressed

as

int

e

gr

als

on

srn+n

in

terms

of

transition

probabilities.

Thus,

one

can

see

that

they

are

equal.

[II.]

Let

now

be

r

as

in

th

e case

[I.]

and

let

us

denot

e

by

D a

general

member

of

Kt.

For

r fixed

the

family

'D

of

all

D's

satisfyin

g (1) is a

Dynkin

class.

If

M is

the

family

of

all

M's

in

the

case

[I.]

th

en,

this

family

is

multiplicative

and

Me

'D.

In

this

way

it

follows

'D(M) c

'D

=

K(M)

= K t

and

one

can

co

nclude

that,

for r

in

the

case [I.]

and

for D g

ene

ral

in K

t

,

th

e

equality

(1)

hold

s.

[III.]

(General

case.)

This

case

can

be

obtained

in

a

same

manner

from

[

II.]

by

fixing

an

arb

i

tra

ry

D E K

t

.

It

will

be

obtained

that

Pa(r)

is

Bor

el

me

a

surable

in

a for

any

r E

K.

Corollary

1.

Ea(G

0 e

t

,

D)

E

a(F

· (G 0

ell

)

Ea(G

0 etlKtl

Ea(EXt

(G),

D)

for

G E

B(K),

DE

Kt,

Ea(F.

EXt

(G))

for

G E

B(K),

FE

B(K

t),

EXt

(G)

(a.s.)(P

a

)

for

G E B

(K)

.

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

Подождите немного. Документ загружается.