Calculation of the fractal dimension via the correlation integral

Chaos. Solimnr .4 FractaL Vol. 1. No. 3. pp. 273 --SO. 1991 Printedin Great Brimin

!396@0779/9153.00+.0 Rrgamon PIUS plc

Calculation of the Fractal Dimension via the Correlation Integral PETER STELTER* Oesterleyst-

5. D-3000

Hanoover

1, FRG

and THOMAS PFINGSTEN Bergkammstrasse 2, D-3000

H-over

91. FRG

(Received 2 I May 199 1)

Ahstrnct

- There are numerous methods for calculating the fractal dimension of a discrete chaotic time series.

If these methods are based on P so called ‘box counting” algorithm,

they are extremely CPU time consuming

and sensitive due IO changes of the box size and the number of sampled data points. it is possible to calculate the correlation correhrion

inregml.

The

To avoid these problems

dimension as the slope in a double logarithmic

of the number of independent pairs of the state coordinates on the chaotic attractor. has been applied to a regular and

computerperfornumce

diagram of the

correlation dimension is a lower bound of the fractal dimension and is an estimation In this paper the method

chaotic time series of Y cwtilever which is excited by friction forces. Tho

for culculuting the correlation integrid has been incrfzz%cd. by

connected

pointer arrays.

1. INTRODUCTION If one obtain a chaotic time series from an unknown physical system, then there exists the problem of distinguishing

between the noise from the environment

or measurement

part of the signal. Another task is to obtain some information attractor

in order to develop an appropriate

solved by calculating to the simulated

the comlotion

mathematical

dimension of the chaotic

model of the system. These problems can be

integral of the discrete time series. This method has been applied

data of the Lorenz equations

time series of a cantilever,

device and the deterministic

of the (fractal)

with a defined noise level and to experimental

obtained

which has been excited with dry friction forces. The data has been sampled

with an especially developed laser vibrometer.

2. PROBLEM STATEMENT

AND MEASUREMENT

DEVICE

Dry friction is a complex problem, because it is influenced by many parameters,

like normal force, relative

velocity, roughtness of the surface, the material of the specimen and many others. But in every case dry friction leads to nonlinear equations of motion. In order to study the behaviour of continuous structures under the influence of dry friction forces an experimental laser vibrometer

set up has been constructed,

see Figure 1. The

has been especially developed in order to measure directly the phase plots of a point of

the continuous structure

with a high precision. The resolution

this device is able to reveal the nonlinear character

is theoretically

of the motion. 273

up to Au = 40 nm. Only

P. STELTERand T.

274

PFMGSTE3

marble plate

laser vibrometer

Figure 1: Test set-up with the cantilever

3. CONSTRUCTION The result of the experiment

OF A PSEUDO

with the laser vibrometcr

the deflection of the tip of the cantilcvcr

and the laser vibrometer

w(t) := w(4,t).

STATE

SPACE

is a discrete scalar time series, in our case it is ‘I‘o make further investigations

to generate new state variables. This can bc done by numerical diffcrcntiation Due to the unavoidable noise in the signal this leads often to unacceptable obtained

by applying the time d&y

method

mcntioncd

by Packard

it is necessary

or integrating

results. Better

IS]. Pseudo

state

processes.

results can be

vectors

zj(t)

has

been picked from the time scrics w(t) in the following manner, cf. Eq. (1). 1)7’..)],

k = l(1) rnB,

j = l(1)

N - -& - (mB - 1)z.

Here N is the number of sampled data points of the discrete time series, mg is the embedding Ts is the time delay or separation

points. The procedure

time, to is the starting

can be visualized by means of Figure 2. The embedding

time delay Ts arc of great influence on the following operations. rules to obtain appropriate

dimension,

time and At is the time step between two data Therefore

dimension rno and the

it is important

to have some

values for rnB and Ts:

1. The embedding dimension rno has to be so large, that the trajectories state space. This condition

can be proved by calculating

do not intersect in the pseudo

the so called intersection

probabiky,

cf.

131. But this procedure requires a great deal of CPU time, therefore it is a better practice to increase the embedding dimension stepwise, until the correlation

dimension

becomes constant.

2. The delay time Ts can be in principle of any size. Good results have been obtained, pseudo state vectors are lrneorly independent. function

C,,,

This condition

Eq. (2), of the time series. An alternative

calculate the mutual information

when the

can be proved with the cooariance

way for determining

the time delay, is to

cf. (11, but this method needs also large CPU time.

215

Calculationof the fractaldimension

Figure 2:

Construction

of a pseudo state space with an embedding dimension for mn = 3 and Ts = At

lim ;- / [w(t) - m J’ dt T-m 0

The vectors are linearly independent,

when the covariance function vanishes. In our code the delay

time Ts has been chosen as the first zero crossing of the covariancc function shows a embedded time series and the corresponding 4. CORRELATION It is obvious and has been mentioned systems takes place on chaotic physical system,

it is necessary

and the fmcml dimension of the attractor. Gmssbeqer

obtain

an appropriate

an information

about

of dissipative

mathematical

integral C(r)

heap

model for the

the noise level of the signal

This can be done with the correlation

and Prucaccio in 1983, 121. Th e correlation

space is defined in Q.

that the long term bchaviour

cf. 141. A sampled discrete time series is a structured

of points in the pseudo state space. In order to construct underlying

covariance function.

INTEGRAL

by several authors,

&ractors,

(Ts = ~0). Figure 3

integral introduced

by

of a time series in a pseudo state

(3).

Here N is the number of the sampled data points, H( * ) is the HEAVYSIDE unit step function, r is the radius of a hypersphere

in the pseudo state space, and Z[J are the state vectors, which has been defined

in E&. (1). When the correlation

integral C(r)

is plotted versus the radius of the hypersphere

rithmic diagram three regions with different slope m will show up.

in a (double) loga

276

P. .%EIXER and T. m-INGSTEN

Figure

3:

1. For

C(r)

Measured

small

-

and embedded

values

of P, (r

time

<

series

PO), area

with

I, see

rms holds, which means that

the corresponding

Figure

a pure

4 and

random

covariance

in the

noise

function

presence

fills up the

of noise

whole

the

pseudo

relation

state

space

(m = r7QJ). 2.

For

medium

slope

(C(r)

values

of r,

rDc ). The

-

Amension

correlation

(rg

<

t-A), area II, there

r <

constant

slope

DC of the chaotic

attractor

m zz

The

correlation

is a mcnsurc 3.

For large

values

cmbcdding the radius In order

The

noise

parameters

noise,

with

Figure Eq.

rA. The

This

radius

to find a connection

of the random

lit within

area

means,

III,

the chaotic standard

regime

showed,

that

s has

a “true”

added

chaos.

‘I’hc results linear

with

value

has been carried out.

deterministic

s depends

large

attractor

and a characteristic

equations

to the signal.

deviation

it

GAussian

are shown

on the radius

in

rO, cf.

(5)

Eq.

in order With

to obtain

been

hence

the hypcrsphcrc

of the chaotic

Lorenz

the standard

s(ro) \Vith

lit within

well known

in order

deviation

Dr (DC < D,),

to zero (m = 0), at sufficient

of the attractor

of the “diameter”

the

for the

attractor.

rg of the “edge” in the C(r)-plot

using

constant

(.[I

dimension

will converge

all points

the radius

with

’

of the fractal

the slope

that

a region

is an estimation

)I

log (TX/f,)

on the corresponding

rA is an estimation

a simulation

slmulatlons

mt(-.oo

coordinates

between

process

a well defined

4. Many

state

of r, (r > rA),

dimensions.

log lC(rz)IC(Ti

lim

exists

C(r)-diagram,

(m = DC)

DC is a lower bound

dimension of the active

normally

in the logarithmic

(5)

it is possible

to characterize

the dcscribcd

measured.

6 the correlation bc seen,

test

la’or more

that

“edge”

an equivalent

the measurement

noise

set-up,

1 a regular

detailed

integral the

to calculate

N 0.4%0.

has

set

Figure

information been

at the

plotted

radius

versus

r,, is not

standard

or the noise

about

(5) deviation

from

and a chaotic

the measurement the radius full dcvelopcd,

from

a given

“edge”

radius

r0

the environment. time scrics device

of the cantilever

see (61, 171 and

of the hypersphere. because

the signal

F’rom the which

has been

[S]. In Figure picture are

can

obtained

~&ulntion

Figure

4:

of the frnctaldimension

Influence of noise on the correlation

277

integral for the parameters:

Q = 16.0,

p = 40.0,/3= 4.0,rno = 4 and N = 15000 (a) without noise s = OR, (b) noise level with s = 0.8% and (c) noise level with s = 2.3o/o

w,(lO) = ;;

5 \qJ

5 array

Figure

5:

w,

Structure

array

4 w,

of the pointer arrays for example (w(l)

w(2) = 11, w(3) = 12, w(4) = 10, w(5) = 10)

= 10,

P. mrna

218

by the laser vibrometu

and T. PFMGS~ZN

are nearly noiseless. A DC, mB-plot

converges with increasing

embedding

dimensions

shows that the slope in the medium region

to constant

DC = 5.1 in the chaotic cme. This means, that an appropriate four

state coordinates

and simulated

values: DC = 2.3 in the regular case and mathematical

modei must have more than

in the regular and more than ten in the chaotic case. A comparison from measured

data of the cantilever

show, that the regular motion can be described

by a model with

two mode shapes, cf. 181.

5. PROGRAM The input of the FORTRAN77

DESCRIFITON

program is a scalar time series w(t) of integer numbers. The program has

been devided into three parts: 1. Calculation

of the time delay in accordance

2. Determination 3. Calculation

of a connected

of the correlation

to the statements

pointer array by a recursive procedure, integral according

to Eq. (3).

A major part of the CPU time is used for the calculation

of the norm of the difference vectors Ilzi - z,[l.

In order to reduce the CPU time, it is of great advantage

to make an estimation

bccausc the vectors in the pseudo state space fits the inequality

11% That

in section 3,

ZjI(

2

(=*i

-

for the vcktor norm,

(Eq. 6).

111)

means, if the diflcrcncc of two mcasurcd data points of the time series is larger then the radius r of

the hypcrsphere,

then the norm cannot lie within the radius P. Due to this property, a limitation

number of calculated

difference vectors is possible.

In order to avoid the calculation

of the

of all state vectors,

first the values of the time series has been divided first into classes. This procedure has been realised by two connected array

w,:

pointer arrays wl and ws, see Figure 5.

contains

wz: contains

array

the connection

of a particular

value to a data point,

a reference between a data value to the pointer of a further data point with the same

value. The development operations

of the pointer array has been realised by re-write operations

is a sequential

list of all measured

only. The result of this

points with the same value. With

the described

linked

pointer arrays one has to take into account only the state vectors which fits Eq. 6. This saves computer power especially at small values of the hypersphere. Furthermore

the program computes

double calculating. CYBER

the difIerence

vectors only for one time, while Eq. (3) requires a

This feature saves again 50% CPU-time.

990 main frame computer.

The calculation

has been carried out on a

For 15 000 data points and a maximum radius of 25 % of the largest

data value the program needs about 1000 CPU seconds for calculating

the correlation

integral.

279

Calculation of the fractd dimension

2.4’

2.6 ’

I

2.6

3.0

I

3.2

I

mB

Figure

O: Correlation

integral

sions (a) regular

for measured

motion,

data from the cantilever

(b) chaotic

6. CONCLUDING It has been shown in this paper, estimation equations deviation

of the random by adding

plot changes.

Further

investigation

integral

plots

a correct robust,

noise depends

with measured

is an estimation

mathematical effcctivc,

noise

model.

linearly

Calculating which

integral

This property data.

on the radius

data of a cantilever

of the correlation

and fast procedure,

signal.

to the simulated

different

embedding

dimen-

REMARKS

of the correlation

noise level in a measured

a Gaussian

of the random

integral

that calculation

with

(N = 1~000)

motion

dimension,

~0, at which

which

has been implemented

show, the slope

the constant

dimension

tool to obtain

has been shown

Our experience

show, that

the correlation

is an effective

gives

an

for the Lorenz

that

the standard

in the correlation

slope in the correlation

a hint

at the dimension

via the correlation

in a FORTRAN77

code.

integral

of is a

P. STELTER~~T.PFING~~N

280

It is worth noting, that the fmcta.l dimension coordinates

for the long term behaviour

is only a lower bound for a sufficient

of the system.

number of state

Which means, that the original state space can

have a larger dimension than the fractal dimension. Acknowledgements This reasearch has been carried out at the technical under the supervision

of Professor

Volkswagen foundation

Dr.-big.

under the contract

university of Harmover, West Germany

Karl Popp. The work has been financed by the No. I/63177.

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