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

68

L.

I.

Dobrescu, M. Neamlu,

and

D.

Opri~'

2

The

mathematical

model

with

discrete

time

and

consumer

sentiment

Let

Yt = y(t),

tEN

be

the

income

at

time

step

t

and

let:

1.

the

investment

function

It

=

I(t),

t = 2

..

N,

be

given by:

It

=

V(Yt-1

-

Yt-2)

-

W(Yt-1

-

Yt_2)3,

v > 0, W

:;0,

0;

2.

the

consumption

function

C

t

= C(t), t = 2

..

N,

be

given by:

C

t

= a +

Yt-I

(b

+

CSt-I),

a:;o,

0, b > 0, c

:;0,

0,

where

St

=

S(t),

t = 2

..

N,

is

the

sentiment

function

given by:

1

St=

,

EE[O,I];

1 +

Eexp(Yt-1

- Yt)

3.

the

saving

function

E

t

=

E(t),

t = 2

..

N,

be

given by:

E

t

=

d(Yt-2

-

Yt-I)

+

mSt-l,

d:;o,

0, m

:;0,

0.

The

mathematical

model

is

described

by

the

relation:

(1)

(2)

(3)

(4)

(5)

From

(1), (2), (3), (4)

and

(5)

the

mathematical

model

with

discrete

time

and

consumer

sentiment

is given by:

Yt

= a +

dYt-2

+

(b

-

d)Yt-1

+

V(Yt-1

-

Yt-2)

-

W(Yt-1

-

Yt_2):l+

CYt-1 + m _ N

+ ( )' t - 2.. .

1 +

Eexp

Yt-2

-

Yf-I

(6)

The

parameters

from

(6)

have

the

following economic

interpretations.

The

pa-

rameter

a

represents

the

autonomous

expenditures.

The

parameter

d is

the

control,

dE

[0,1]

and

it

characterizes

a

part

of

the

difference

between

the

incomes

obtained

at

two

time

steps

t - 2

and

t - 1, which is

used

for

consumption

or

saving

in

the

time

step

t.

The

parameter

c, c E

[0,

1], is

the

trend

towards

consumption.

The

pa-

rameter

m,

m E [0,1]' is

the

trend

towards

the

saving.

The

parameter

b,

b E

(0,1),

represents

the

consumer's

reaction

against

the

increase

or

decrease

of

his income.

When

the

income

(strongly)

decreases,

the

consumer

becomes

pessimistic

and

con-

sumes

° < b < 1

of

his income.

When

the

income

(strongly)

increases,

the

consumer

becomes

optimistic

and

consumes

b < b + c < 1

of

hi" income.

Note

that

Souleles

([7]) finds,

in

fact,

that

higher

consumer

confidence is

correlated

with

less

saving

and

increases

in

relation

to

expected

future

resources.

The

parameters

v

and

w,

11

> 0,

11J

:;0,

°

describe

the

investment

function.

If

W = °

the

investment

function

is

linear.

The

parameter

E,

E E

[0,

1],

describes

a family

of

the

sentiment

functions.

For

different values

of

the

model

parameters,

we

obtain

the

following classical

models:

1.

for a = 0, b = 1 - s, d = 0, m = 0, E = 0, s E

(0,1)

from

(6) we

obtain

the

Hick-Samuelson

model

([6]):

Yt =

(1

+ v -

S)Yt-1

-

VYt-2,

v>

O,S E

(0,1);

Hick Samuelson Keynes Dynamic Economic Model 69

2. for v = w = 0, c = 0, rn = 0, (6) gives us

the

Keynes

model

([6]):

Yt

=

d(YI-2

-

Yt-l)

+ a +

bYt-l;

3. for v = w = 0, c =

1,

(6) leads

to

the

model from

[2]:

CYt-l

+ rn

Yt

=

d(Yt-2

-

YI-l)

+ a +

bYt-l

+ (

.)'

tEN;

. 1 +

exp

Yt-2

-

Yt-l

4.

for a = O,b =.5 -

I,d

=

O,Tn

=

O,W

= 1 +

V,c

= 0, from (6) we

get

the

Puu

model ([7]):

YI

= V(YI-1 -

Yt-2)

-

(1

+

V)(YI-l

- YI_2)3 -

(1

-

S)Yt-l.

3

The

dynamic

behavior

of

the

model

(6)

Using

the

method

from

Kusnetsov

([5]),

we

will analyze

the

system

(6), considering

the

parameter

a as

bifurcation

parameter.

The

associated

map

of

(6) is F :

]R2

--+

]R2

given by:

FG)

=

(a

+

(b

- d +

v)y

+

(d

-

v):

-

w(y

- z)3 +

l+E~!~;(:'-Y»)

(7)

Using

the

methods

from

[3],

[4]

and

[6],

the

map

(7)

has

the

following properties:

Proposition

1.

(i).

If

(1

+

c)(I-

b)

- c >

0,

then,

for

the

map

(7), the fixed

point

with

the

positive

components

is

Eo

(Yo, zo), where:

Zo

=

Yo

and

l+c

Tn

PI =

(1

+ c)

(1

-

b)

- c ,P2 =

(1

+ c)

(1

-

b)

- c;

(ii).

The

Jacobi

matrix

of

the

map

F

in

Eo

is given by:

A =

(all

a

12

)

1 ° '

where:

all

=

P3

a

-

P4

+

P5,

a12

=

-P3

a

+ P4,

P:l

=

(1~~)2pl,P4

= d -

11

-

(1~1J:)2

-

(1~~)2P2,P5

= b +

(I~E);

(iii).

The

characteristic

equation

of

matrix

A is given by:

A2

-

allA

-

a12

=

0;

(iv).

If

the

model

parameters

d,

v,

c,

b,

c,

rn

satisfy

the following inequality:

(1

+ d -

v)(l

+ c)((1 +

c)(l

-

b)

-

c)

- rnc(1 -

b)

>

0,

(8)

(9)

(10)

(11)

then,

for

equation

(1

OJ,

the roots have the

modulus

less

than

1,

if

and

only

if

a E

(al,a2),

where

2P4

-

P5

- 1 1 +

P4

al

=

a2

=--

2P3'

P3

(12)

70

L.

I.

Dobrescu,

M.

Neam/u,

and

D. Opriij

(v).

If

the

model

parameters

d,

v,

c,

b,

c,

Tn

satisfy

the

inequality

(11)

and

a =

al

then,

one

of

equation

(10)'s

roots

'is

-1, while

the

other

one

has

the

modulus

less

than

1;

(vi)

If

the

model

parameteTs

d, v, c,

b,

c,

Tn

satisfy

the

inequality

(11)

and

a =

a2

then,

the

equation

(10)

has

the

TOots

fJoI(a)

=

fJo(a),fJo2(a)

=

/i(a),

wheTe

IfJo(a)

I = 1.

Using

[6]

and

Proposition

1,

with

respect

to

parameter

a,

the

asymptotic

stability

conditions

of

the

fixed

point,

the

conditions

for

the

existence

of

the

flip

and

Neimark-

Sacker

bifurcations

are

presented

in:

Proposition

2.

(i).

If

(1

+ c)(1 -

b)

- c >

0,

the

inequality

(11) holds

and

2P4-

Ps

- 1 > 0,

then

fOT

a E

(aI,

(2)

the

fixed

point

Eo

is

asymptotically

stable.

If

(1

+ c)(1 -

b)

- c > 0,

the

ineq'uality (11) holds

and

2P4

-

P5

- 1 < 0,

then

for

a E

(0,

(2)

the

fixed

point

Eo

is

asymptotically

stable;

(ii).

If

(1

+ c)(1 -

b)

- c >

0,

the

inequality

(11)

holds

and

2P4

-

P5

- 1 > 0,

then

a =

al

is

a flip bifuTcation

and

a =

a2

is

a

NeimaTk-SackeT

bifuTcation;

(iii).

If

(1

+ c)(1 -

b)

- c >

0,

the

inequality

(11) holds

and

2P4

-

P5

- 1 <

0,

then

a =

a2

is

a

Neimark-SackeT

bifuTcation,

4

The

normal

form

for flip

and

Neimark-Sacker

bifurcations

In

this

section, we describe

the

normal

form in

the

neighborhood

of

the

fixed

point

Eo,

for

the

cases a =

al

and

a = a2.

We consider

the

transformation:

UI

= Y -

Yo,

(13)

where

Yo,

Zo

are

given by (8).

With

respect

to

(13),

the

map

(7)

is

G:]R2

--->

]R2,

where

and

gl(UI,1L2) =

--1

r +

(b

- d + V)UI + (d -

V)U2

-W(UI

- U2)3+

+c

CUI

+ r

+-----,------,-

I+Cexp(U2-

UI)

g2(UI,

U2)

=

uI,

T =

cYo

+

Tn

The

map

(14)

has

0(0,0)

as fixed point.

We consider:

(14)

Hick Samuelson Keynes Dynamic Economic Model

71

We develop

the

function

G(u),u

=

(U1

,U2)T in

the

Taylor series

until

the

third

order

and

obtain:

1 1

G(u)

=

Au

+

"2B(u

,

u)

+ "2C(u,

u,

u)

+ 0(luI

4

)

where

A is given by (9)

and

B(u

,

u)

=

(B1(u

,

u)

, of,

C(u,

u ,

u)

=

(C

1

(u

,

u,

u),

of

(15)

For a =

aI,

given by (12)

with

the

condition

(v)

from

Proposition

1,

we have:

Proposition

3.

(i).

The

eigenvector

q E ]R2,

given

by

Aq

=

-q

has

the

compo-

nents:

q1

=

1,

q2

=

-l.

(16)

(ii)

The

eigenvector

h E ]R2,

given

by

hT

A =

_hT

has

the

components

:

1

a12

hI

=

---

,h

2

=

--

--.

1 +

a12

1 +

a12

The

relation

< q, h

>=

1 holds.

(iii).

The

normal

form

of

the

map

(7)

on

the

center

manifold

in

0(0,0)

is

given

by:

1

7)

--->

-7) + 6v7)3 + 0(7)4),

where v =

-1+1

(130

-

31

21

+

31

12

-103)

+ 1 3

(1

20

-

2111

+

102)2.

a12

-

all

-a1

2

The

proof

results from

stra

ight calculus using

the

formula (6):

v

=<

r,

C(q

, q,

q)

>

+3B(q

,

(I

-

A)-l

B(q,

q)),

1 =

(~ ~)

.

For a = a2, given by (12)

with

the

condition

(vi) from

Proposition

1, we have:

Proposition

4.

(i).

The

eigenvector

q E

([2,

given

by

Aq

=

J-i1q,

, where

J-i1

is the

eigenvalue

of

the

matrix

A ,

has

the

components

:

q1

=

1,

q2

=

J-i2

=

Ill·

(ii)

The

eigenvector

h E

([2,

given

by

hT

A =

J-i2hT,

where

J-i2

=

Ill'

has

the

components:

1

J-i1a12

hI

= 2 ' h2 =

2·

1 +

J-i1

a12

1 +

J-i1

a12

The

relation < q, Ii

>=

1 holds.

72

L.

1.

Dobrescu,

M.

Neamfu, and D.

Opri~

Using (15)

and

(16) we have:

We

denoted

by:

Bl(q,

q)

= 1

20

+

2111/12

+

102/1~,

Bl

(q,

q) = 1

20

+

111

(/11

+

/12)

+ 1

02

/11/12,

Bl

(q, q) = 1

20

+

2111/11

+

102/1i.

920

= VIB1(q,q),911 = VIB1(q,q),902 =

VI

B1

(q,q),

h~o

=

(1

-

VI

- Vl)920,

hil

=

(1

-

VI

- Vl)911,

h62

=

(1-

VI

-

Vl)902,

h~o

= -(Vl/12

+Vl/11)B

1

(q,q),hil

= -(Vl/12

+Vl/11)B

1

(q,q),

h62

= -(Vl/12 + Vl/11)B

1

(q, q),

(17)

2

-1

(h~o)

-1

(hil) 2

-1

(h62)

w20

=

(/11

I -

A)

h~o'

Wl1

=

(I

-

A)

hil'

W02

=

(/12

-

A)

h62'

(18)

where

I =

(~

~),

A is given

by

(9)

and

r20

= B

1

(q,w2o),rll

= B

1

(q,Wl1),

Co

= C

1

(q,

q,

q) =

130

+

(/11

+

2/12)1

21

+

/12(2/11

+

/12)112

+

/11/1~103,

921

=

V2(r20

+ 2rl1 + Co). (19)

Using

the

normal

form for

the

Neimark-Sacker

bifurcation

of

the

dynamic

sys-

tems

with

discrete

time

([6])

and

(17), (18), (19),

we

obtain:

Proposition

5.

(i). The solution

of

the

system

(6)

in

the neighborhood

of

the fixed

point

(Yo,

zo) E

lR

n

is given by:

- 1 2 2 2 _ 1

2-2

Yt

=

Yo

+

/12

X

t

+

/11

X

t

+

'2W20Xt

+ W

l1

XtXt +

'2

W02Xt

,

- 1 1 2 1 _ 1

1-2

Yt-l

=

Zt

=

Zo

+

Xt

+

Xt

+

'2W20Xt

+ W

ll

XtXt + '2W02Xt,

where

Xt

is the solution

of

the following equation:

1 2 _ 1

-2

1

2-

Xt+l =

/1Xt

+ '2920Xt +

911

X

t

X

/,

+

'2902·7;t

+ '2921XtXt.

(ii) A eomplex variable

transformation

exists so that the equation (20) becomes:

Wt+l

=

/11Wt

+ C

1

W;Wt

+ O(IWtI4),

where

Cl=920911(tL2-3-2tLIl+

Igl11

2

+

19201

2

+921

2(/1i -

/11)(/12

-

1)

(1

- /1Il 2(/1i -

/12)

2'

(ii'i).

Iflo

=

Re(e-

W

C

1

)

< 0, where B = arg(/1Il,then

in

the neighborhood

of

the

fixed

point

(Yo,

zo) ther'e is a stable

limit

cycle.

Hick Samuelson Keynes Dynamic Economic Model 73

5

The

Lyapunov

exponents

If

a1

> °

then

for a E (0,

ad

or

a E

(a2

'

1)

th

e

system

(6)

has

a

complex

behavior

and

it

can

be

esta

blished

by

computing

the

Lyapunov

exponents.

We will use

the

algorithm

QR

from

[6]

and

the

determination

of

th

e

Lyapunov

exponents

can

be

obtained

by

solving

the

following

system:

3 cYt + m

Yt+1

= a +

(b

- d + V)Yt + (d - V)Zt - W(Yt - Zt) + ( )

1 +

exp

Zt

-

Yt

Zt+

1 =

Yt

Xt+1

=

arctan

( cos

Xt

. )

111

cos

Xt

-

h2

sm

Xt

At+1

=

At

+ In(I(/11 -

tanxt+d

COSXt

COSXt+1

-

h2

sinxt

cOSXt+11)

/-It+1

=

/-It

+

In(1

(/11 -

tan

Xt+1

+

1)

sin

Xt

cos

Xt+1

+

h2

cos

Xt

cos xt+11),

with

f

_

811(

)-b

d+

3 (

)2+

c

+

c

(c

+m+CYt)ex

P

(

Zt

- Yt}

11

- - Yt,

Zt

- - v - W

Yt

-

Zt

2 '

8y

(1

+ c exp(Zt - Yt))

f

8h(

) d 3 (

)2

c(

m+cYt)exp(

Zt

- Yt)

12 = - Yt,

Zt

= - v + W

Yt

-

Zt

- 2 .

8z

(l+cexp(Zt

- Yt))

The

Lyapunov

exponents

are:

.

At

.

/-It

L1 =

hm

-,L

2

=

lUll

-

t~

oo

t

t~oo

t

(20)

If

one

of

the

two

exponents

is positive,

the

system

has

a

cha

otic

be

havior

.

6

Numerical

simulation

The

numerical

simulation

is

done

using

a

Maple

13

program.

We

consider

different

values for

the

parameters

which

are

used

in

the

r

ea

l

economic

processes.

The

bifurcation

parameter

is

a.

F

or

a = 250, v = 0.1, W =

0,

C = 0.1, b = 0.45, c = 1, m = 0.5

and

the

con

trol

parameter

d = ° we

obtain

in

Figl

th

e e

volution

of

the

income

in

the

time

domain

(t,yd,

in

Fig

2

the

evolution

of

the

income

in

the

phase

space

(Yt

- 1,Yt).

In

both

these

cases,

the

system

is

not

c

haotic

.

Fig.

1.

(t, Yt), for d = ° Fig.

2.

(Yt-1 , Yt), for d = 0

74

L.

I.

Dobrescu,

M.

Neamju, and

D.

Opri}

For

the

control

parameter

d = 0.6

Fig3

displays

the

evolution

of

the

income

in

the

time

domain

(t,

yd,

Fig

4

the

evolution

of

the

income

in

the

phase

space

(Yt-l,

Yt)·

The

Lyapunov

exponent

is

positive

and

the

system

has

a

chaotic

behavior.

..

'

,

Fig.

3.

(t,

Yt),

for d = 0.6

Fig.

4.

(Yt-l,

Yt),

for d = 0.6

For

different

values

of

the

parameters,

we

can

obtain

a

Neimark-Sacker

bifurca-

tion

point

or

a flip

bifurcation

point.

7

Conclusion

Hick

Samuelson

Keynes

dynamic

model

with

discrete

time

using

investment,

con-

sumption,

sentiment

and

saving

functions

is

studied.

The

behavior

of

the

dynamic

system

in

the

fixed

point's

neighborhood

for

the

associated

map

is

analyzed.

We

establish

asymptotic

stability

conditions

for

the

flip

and

Neimark-Sacker

bifurca-

tions.

The

QR

method

is

used

for

determining

the

Lyapunov

exponents

and

they

allows

us

to

decide

whether

the

system

has

a

complex

behavior.

Using

a

program

in

Maple

13, we

display

the

evolution

of

the

income

in

the

time

domain

and

the

phase

space.

References

l.C.

Carroll, J. Fuhrer,

and

D.

Wilcox. Does consumer sentiment forecast household spend-

ing?

If

so, why?

American

Economic

Review, S4:1397-140S, 1994.

2.L.1. Dobrescu,

and

D. Opris. Neimark-Sacker bifurcation

for

the

discrete-delay Kaldor

model.

Chaos, Solitons

fj

Fractals, doi:1O.1016/j.chaos.2007.10.044, 2007.

3.L.I. Dobrescu,

and

D.

Opris. Neimark-Sacker bifurcation for

the

discrete-delay Kaldor-

Kalecki model.

Chaos, Solitons

fj

Fractals, doi:1O.1016/j.chaos.200S.09.022, 2008.

4.M. Doms,

and

N.

Morin. Consumer sentiment,

the

economy,

and

the

news media. Federal

Reserve

Bank

of

San

Francisco, Working

Paper

2004-09.

5.Y.A. Kuznetsov. Elements of Applied Bifurcation Theory.

Applied

Mathematical Sciences

112, Springer-Verlag, New York, 1995.

6.M.

Neamtu

and

D. Opris. Economical games. Disctrete economical dynamics. Applica-

tions. Mirton,

200S

(in Romanian).

7.T.

Puu

and

1.

Sushko. A business cycle model with cubic nonlinearity. Chaos, Solitons

and Fractals, 19:597-612, 2004.

S.N.

Souleles. Expectations, heterogeneous forecast errors, and consumption: Micro evi-

dence from

the

Michigan Consumer Sentiment Survey. Journal

of

Money, Credit and

Banking,

36:39-72, 2004.

9.F.H. Westerhoff. Consumer sentiment

and

business cycles: A Neimark-Sacker bifurcation

scenario.

Applied

Economics

Letters, 14:1-5, 2007.

On

the

Entropy

Flows

to

Disorder

School

of

Mathematics

University

of

Manchester

M13

9PL,

UK

c.

T.

J.

Dodson

(e-mail:

ctdodson@manchester.ac

. uk)

75

Abstract:

Gamma

distributions,

which

contain

the

exponential

as a

special

case,

have

a

distinguished

place

in

the

representation

of

near-Poisson

randomness

for

sta-

tistical

processes; typically,

they

represent

distributions

of

spacings

between

events

or

voids

among

objects.

Here

we

look

at

the

properties

of

the

Shannon

entropy

function

and

calculate

its

corresponding

flow curves,

relating

them

to

examples

of

constrained

degeneration

from

ordered

processes. We

consider

univariate

and

bivariate

gamma,

and

Wei

bull

distributions

since

these

also

include

exponential

distributions.

Keywords:

Shannon

entropy,

integral

curves,

gamma

distribution,

bivariate

gamma,

McKay

distribution,

Wei

bull

distribution,

randomness,

information

ge-

ometry.

MSC

classes: 82-08; 15A52.

1

Introduction

The

smooth

family

of

gamma

probability

density

functions

is given

by

f

.

[0

00)

----)

[0

00)

. x

----)

e-";2 x

K

-

1

(~r

.,

,.

r(K)

iL,

K >

O.

(1)

Here

iL

is

the

mean,

and

the

standard

deviation

17,

given

by

K

(;;:)2,

is

proportional

to

the

mean.

Hence

the

coefficient

of

variation

fo

is

unity

in

the

case

that

(1)

reduces

to

the

exponential

distribution.

Thus,

K = 1

corresponds

to

an

underlying

Poisson

random

process

complementary

to

the

exponential

distribution.

When

K < 1

the

random

variable

X

represents

spacings

between

events

that

are

more

clustered

than

for a

Poisson

process

and

when

K > 1

the

spacings

X

are

more

uniformly

distributed

than

for

Poisson.

The

case

when

iL

= n is a

positive

integer

and

K = 2 gives

the

Chi-

Squared

distribution

with

n - 1 degrees

of

freedom;

this

is

the

distribution

of

~

for

variances

8

2

of

samples

of

size n

taken

from a

Gaussian

population

(Je

with

variance

17~.

The

gamma

distribution

has

a

conveniently

tractable

information

geome-

try

[1,2],

and

the

Riemannian

metric

in

the

2-dimensional

manifold

of

gamma

distributions

(1)

is

(2)

76

C.

T.

1.

Dodson

So

the

coordinates

(p"

K,)

yield an

orthogonal

basis

of

tangent

vectors, which

is useful in calculations because

then

the

arc

length

function is

simply

The

system

of

geodesic

equat

ions is difficult

to

solve analytically

but

nu-

merical solutions using

the

Mathematica

programs

of

Gray

[7]

were

obtained

in

[2].

Figur

e 1 shows a

spray

of some

maximally

extend

ed

geodesics

emanat-

ing from

the

point

(p"

K,)

=

(1,1).

Geodesic curves have

tangent

vectors

that

are

parall

el along

them

and

yield

minimal

information

arc

length

distances

in

the

gamma

2-manifold.

2.00

1.75

K,

1.50

1.25

1.00

0.75

0.50

0.25

0.5 1.0

1.5

2.0 2.5

3.0

/1

Fig.

1.

Some

examples

of

maximally

extended geodesics passing through

(/1,

K,)

=

(1,1)

in

the

gamma

2-manifold.

We

note

the

following

important

uniqueness

property:

Theorem

1

(Hwang

and

Hu

(8)). For independent positive random vari-

ables

with a common probability density function

f,

having independence

of

the sample mean and the sample coefficient

of

variation is equivalent to f

being the

gamma

distribution.

This

property

is one

of

the

main

reasons for

the

larg

e

number

of

applications

of

gamma

distributions:

many

near-random

natural

processes have

standa

rd

deviation

approximately

proportional

to

the

mean

[2].

On

the Entropy Flows

to

Disorder 77

2

Gamma

entropy

flows

Papoulis

[13]

Chapter

15

gives

an

account

of

the role

of

the

Shannon

e

ntropy

function in probability theory,

stocha

st

ic processes

and

coding theory.

The

entropy

of (1) is shown in Figure 2 using

Sf

= -1

00

f lo

gf

dx

:

lR

2

+

-+

lR

(tt,

r;,)

~

r;,

-- log

(~)

+

log(r

(r;,))

-

(r;,-1)1,b(r;,)

(~3)

with

gradient

(

~,

_

(r;,

-1)

(r;,1,b/(r;,)

-1)

) .

J.i

r;,

(4)

where

1,b

=

r;:

is

the

digamma

function.

At

fixed r;"

the

e

ntrop

y increases

like log

J.i.

At

fixed m

ea

n

J.i,

the

maxim

um

entro

py is given

by

r;,

= 1,

th

e

exponential

distribution

case of maximal disorder or chaos.

Fig.

2.

Shannon

entmpy

function

Sf

for the

gamma

family

as a sm:face with respect

to

mean

fJ,

and",

(left) and as a conto'

ur

plot with entropy gradient flow and

int

egral

c

urv

es (right),

The

a

symp

tot

e

is",

=

1,

th

e expone

ntial

case of

maximum

disorder.

Fi

gure 2

on

the

right shows entropy as a contour plot

with

supe

rim-

posed also some examples

of

integral curves of

the

entropy

gradient

flow

fie

ld, namely c

urv

es c

sat

is

fy

ing

c:

[0,(0)

-+

]R2:

c(t) =

V'Sf

l(c(t))

.

(5)

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

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