Analysis of deterministic chaotic signals

Rochester Institute of Technology

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1-1994

Analysis of deterministic chaotic signals

William Robertson

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ANALYSIS OF DETERMINISTIC CHAOTIC SIGNALS

Approved by:

by

William Robertson

A Thesis Submitted in Partial Fulfillment

of the Requirements for the Degree of

MASTER OF SCIENCE

m

Mechanical Engineering

J.

s.

Torok

Dr. Josef S. Torok -- Thesis Advisor

Hany Ghoneim

Dr. Hany Ghoneim

Teresa A. Wallace

Teresa Wallace

Charles W. Haines

Dr. Charles W. Haines -- Department Chairman

Department of Mechanical Engineering

College of Engineering

Rochester Institute of Technology

ANALYSIS OF DETERMINISTIC CHAOTIC SIGNALS

I, William Robertson, hereby grant permission to the Wallace Memorial Library of the Rochester Institute of Technology, Rochester, NY, to reproduce my thesis in whole or in part providing that no reproduction will be used for commercial use or profit.

ABSTRACT

The prediction of a single observable time series has been achieved with

varying degrees of success. The _quality and duration of the prediction is

dependent on _{many factors,} the two most important _being the reconstruction

technique and the _quantity of data. The goal of this work is to reduce the

computational effort required to achieve _satisfactory predictions. Without new

methods, which are beyond the scope of _{this work, this} requires a reduction in

thesize ofthedata set.

This thesis expands on earlier works _{using the delay} vector space method

and the autocorrelation function for reconstruction and applies this analysis

techniquetoa well knownnon-lineardynamic system._{The embedding delay}and

the sampling rate were varied while _keeping the number of points the same in

order _{to study} the effects of _varying the _sampling rate. The results of this

experimentation show the importance of the _sampling rate and duration of the

sample in the reconstruction and prediction. It is shown that the _sampling

duration may be more important than the number of points. It is apparent from

thischaracteristicthat atime series sampled over alongerduration may contain

Table

Contents

INTRODUCTION vi

ACKNOWLEDGMENTS viii

CHAPTER 1 DYNAMICAL SYSTEMS 1

Periodicsystems...TT .'.'.TT 2

Quasi-periodicsystems 2

Nonlinearsystems 5

Stochasticsystems 7

CHAPTER 2 PHASE SPACE 9

Constructionof phase portrait 9

Phase_trajectory 10

Phase areas 11

Artificial phase space 13

CHAPTER 3 ATTRACTORS 15

Phase space maps 15

Phase spaceflows 16

Normal attractors 17

Pointattractor 17

Periodic(circle) 18

Otherattractors 19

Chaotic_(Strange) attractors 19

CHAPTER4 EMBEDDING DELAY CALCULATION 20

Autocorrelationfunction_(ACF) 20

Mutual information_theory 21

Othermethods 22

Shortfalls 22

CHAPTER 5 HARDWARE& SOFTWARE 30

Hardware 30

Software 30

ACM 32

ARTSPACE.M 33

AUTOCORR.M 35

CORRINT.M 36

EMBED.M 39

LINREG.M 40

LORENZ.M 42

MAIN.M 43

NDIFF.M 46

ROSSLER.M 48

RUNGE.M 49

ULINREGM 51

CHAPTER6 EXPERIMENTAL PROCEDURE 53

Datageneration 53

AbandonmentofRossler 54

Secondarysampling 54

Embeddingdelays 55

Configuration ofMAIN.M 55

RESULTS 56

Casenumber matrix 56

Highest embedding dimensionmatrix 57

Highestattractordimensionestimate matrix 58

Attractor dimensionestimate at De=5matrix 59

ANALYSIS OUTPUT 62

CHAPTER 7 CONCLUSIONS 75

RECOMMENDATIONS 76

Additional reading 77

INTRODUCTION

This thesis is based upon the concept that it is possible to reconstruct the

geometric structure of nonlinear dynamical systems from the measurement of a

single variable, an idea proposed as _early as 1979. This reconstruction then

allows the short term prediction of that variable. The underlying theorem

indicates that reconstruction is possible when an infinite amount of data is

availableforasingle state variable ofthe system.

While the proof of the theorem requires an infinite volume of continuous,

noisefree data, application has shown that a set of 104 to 105 discrete points is

usually sufficient for good reconstruction and prediction capabilities. With the

present level of_computingpowerthat is available, it is notdifficult to reconstruct

systems with two orthreedegrees offreedom with a data set of10,000 points or

more. Itbecomesverytime consumingwhen thedata set orthedimensionofthe

system is larger, with the dimension ultimately playing a larger part in the

computational effort. _Compounding the problems of computational effort is the

restrictions thatafinite data set place on the analysis. The finite data set means

that the choice of reconstruction variable values is no longer arbitrary. This

primarily effects the embedding delay and _sampling rate. _{The embedding delay}

effects the extraction of information from the attractor while the _sampling rate

This work analyzed the _feasibility of _reducing the computational effort _by

reducing the data set size _by the choice of more optimal reconstruction

parameters. It is possible_{to greatly} reducethe data set sizefor low dimensional

mildly nonlinear systems _by careful selection of the data set. _However, in most

cases it is not possible_{to carefully}select thedata set unlessthere is knowledge

ofthesystem already. Since it is desired forthe methods tobe widelyapplicable

to unknown _systems, another method is necessary. The _ability of a _very small

data set, ~100 _points, to give good reconstruction however is the event which

ledtothemore important_discovery, which _{is widely}applicable.

It was discovered from the _geometry of the small _set, that if the data is

sampled over a longer time period a better geometric reconstruction and

prediction was possible. The sample is taken overa _{longer time period,} with the

possibility of _decreasing the actual number of points. This method allows the

systemtovisit more oftheattractor'sspace and _giving a betterstatistical picture

oftheattractor.

The _delay vector space method is applied to the first state variable of the

Lorenzattractor in ordertodemonstratethe effects ofthe _sampling method. The

permutations ofthe embedding delay and _sampling rate are studied to evaluate

their impactontheestimation ofthe fractal dimensionofthesystem which iswell

known. This provides a _{strategy for reducing the} computational effort when

ACKNOWLEDGMENTS

/ would_{like to take the opportunity to thank the following for their patience,}

understanding, and_help whileIpreparedthis thesis:

-Myparents. Thank you _{for putting} me through schoolto this _{level, making}

surethat Ialwayshada placeto sleep, andfoodtoeat.

Dr. Josef_{Torok, my}advisor. Thankyou for the chance to work with one of

thebestandbrightest that Ihave had thegoodfortune to know. Thankyouforall

ofthe_helpand_{especially the}patience as anotherprojectdistractedme.

- Ed

Ziegler, a fellowstudent. Thankyou for_spending the time to teach me

therudiments ofprogramming in the MATLAB environment, andforallhis_helpin

exploringmany different ideasrelatedto this thesis.

Mr. DavidHathaway, myboss at _times, and always a good friend. Thanks

CHAPTER 1

DYNAMICAL SYSTEMS

The_following material is meantto be a briefreview ofdynamicsystems. The

systems are introduced in an order which shows the _{increasing complexity} of

systems which _{leads eventually}tostochastic systems. Chaos is defined'

as:

1. confusion, or confusedmass, offormless matterandinfinite space,

supposedtohaveexistedbefore theordereduniverse.

2. anymixed_mass, withoutdue formor_order;confusion.

3. an_{empty, immeasurable space;}an abyss.

If these definitions were used in the description of chaotic dynamics it would

seem foolish to have pursued this work. There is a much more complex

relationship which needs to be understood. Chaos is a subset of a much larger

area of_studycalled nonlineardynamics.

It has been discovered" _{by many investigators} that very simple deterministic

systems with a few elements can generate _apparently random behavior. The

randomness in their behavior is afundamental property. It was once thought if

enough data couldbecollected and processed _{that any}system was predictable.

The randomness associated with these deterministic systems is called chaos.

The bulkofthis work is to _develop analysis methods fornonlinearstudies which

mayin thefuture be extendedtochaoticdynamics.

Periodicsystems

Periodicsystems are characterized _bysimple repeated motions at regular

intervals. Thisregularinterval is defined as _{the period, T. From the definition}

oftheperiodfollowstherelationship:

x(t)=_x(t+

T) (1)

It then becomes very simple to predict _any future state of x _{by using}

equation(1). Theperiodicsystemis the simplest ofdynamicsystems.

Quasi-periodic systems

Quasi-periodic systems _may appear much more complex than periodic

systems. Although more complex, the quasi-periodic system _may be

represented as the linear combination of periodic modes. The mathematical

combination isthefinitesummationof periodic modes:

*(')=

2XW

m=]

A system becomes _increasingly complex as the number of excited

SIN(t)

0 2

Figure1._Sin(t)withperiodT=2k

4

t

SIN(4t)

SIN(16t)

0.5

0

sin(16t;

-0.5

,111111

rrn

n

4

t

Figure 3._Sin(16t)

Quasiperiodicfunction

0 2 4 6 8

t

modes,ym(t), have been determined, the summation,x(t), may be evaluated

at _any t. While quasi-periodic systems are more complex than periodic

systems, thedeterminationof predictive equations remains elementary.

For an example _{y-|(t) is sin(t)} as shown in Figure 1, y2(t) is sin(4t) as

shown in Figure2, and_{y3(t) is sin(16t)}as shown in Figure 3. The summation

ofthe three periodic modes results in a much more complex _system, shown

in Figure 4.

Thesummation ofthe three periodic modes yields a simple quasi-periodic

system. As the number of modes _increases, the _complexity of the system

increases until it is not obvious _by inspection that the system is made _up of

multiple modes. Thesystem represented in Figure 4 is_fairly simple, yet when

a small pieceis shown orif therewere more exited_{modes, the} periodic basis

is not evident.

Nonlinearsystems

Nonlinearsystems cannot be represented _by the linear combination of a

finite numberof periodic modes. Instead theequations _governing evolution of

the system contain nonlinear elements or are nonlinear combinations ofthe

state variables. If all the state variables are known and _{measurable, it is} a

simple process to determine the system of equations that govern system

Amore troublesome and _customary case occurs when all state variables

can not either be identified or measured. Mathematicians first hypothesized

that the measurement of a single state variable in a nonlinear system is

sufficient toreconstructthedynamics ofthe system. As a consequence ofthe

system's _{nonlinearity, every} state variable carries all the information

necessaryto reconstructthegeometry. Sinceall the_necessary information is

present in each variable, and the computational tools have been developed

to extract _it, a set of _governing equations can be obtained with some work.

The set of equations contains a single equation which predicts the evolution

ofthe state variable that was _originally sampled. Since in _theory each state

variable contains the _information, it is not important which one is used. In

practice there are small _{(but significant)} variations in the reconstructionfrom

one variabletoanother.

These small variations become significant when the character of non

linear systems is considered. The trajectories within non-linear systems are

highly dependenton initial conditions. Consider then that _any instantaneous

position, basedon thereconstruction, isthe initial conditionfortherest ofthe

evolution. Slight variations in the position at _any given instance result in

trajectories which will _{differ greatly} as the system evolves _giving rise to

considerably different results. This sensitivity to initial conditions provides

techniques, the small errors associated with numerical differentiation and

integrationgive rise to largeerrors in the prediction.

x=100t+ 10t2 +t3/10

x10

Figure5. Non-linear function (cubic).

Stochastic systems

The trajectories in stochastic systems are governed _{primarily by} external

random noise. The intrinsicpropertiesofthesystem are eitherinsignificantor

indiscernible from the stochastic signal's influence. Analytical methods are

not applicable to stochastic systems since the collection of additional data

doesnot contribute additional information.

It is important to test data that appears random _{for underlying periodic,}

quasi-periodic, and nonlinear systems. One must be careful not todismiss a

to remove stochastic noise from a deterministic signal _using standard signal

processing techniques.

a=rand(size(t))

0 20 40 60 80 100

t

CHAPTER 2

PHASE SPACE

The phase space is the geometrical domain required to describe the

instantaneousstate of a system. Eachorthogonal coordinate ofthephase space

corresponds to a system variable. _{Orthoganality} is not a _requirement, but it

eliminates coordinate dependence. _Normally the phase space consists of

familiar quantities such as position, velocity, and higher order derivatives for

each dimension in the occupied space. To _fully describe three dimensional

motion, a minimum of six coordinates _{is necessary} x(t), dx/dt,y(t), dydi, z(t) and

dz dt. _Usually a phase plot uses _only two coordinates hence the designation of

phase planes.

Construction of phase portrait

The phase portrait is constructed in an artificial phase space for use in

system identification. The geometry exhibited _by the system _may be

compared to other systems _{allowing identification} of similar characteristics.

This may be donewithouta prioriknowledgeofthephase space inwhich the

dt2

dx dt

x(t)

Figure 7.A 3-Dnormal phase space createdusingderivatives forcoordinates.

actual phase portraitforthesystem. Lackof_{similarity is due to} the nonlinear

combinations of phases inherent in the _delay coordinates of the artificial

phase space.

Phase

trajectory

Thepath taken on a phase plane is the phase trajectory. Trajectories that

correspond to similar energies pass _closely but do not cross. If the

trajectories crossed attime t the system would be rendered indeterminate _by

the ambiguity of past andfuture states. Figure 8 showsthe phase _trajectory

from a simple harmonic oscillator, the position _x(t) corresponds to a sin

function, while dx/dt corresponds to the cosine derivative. The _trajectory

initial conditions. The trajectories do not cross since _they correspond to

similarenergies.

Phase plot

0.5-dx/dt

-0.5-Figure 8. Phaseplot generatedfrom Figure 1.

Phase areas

Phase areas are bounded _bya set of points in a given area ofthe phase

space. For conservative _systems, the trajectories are area preserving. The

area _{preserving quality} means thatthe area ofthe phase space bounded _by

a fixed set of points is constant for all times past and future. Dissipative

systems have convergenttrajectories that decrease the phase area as time

progresses. Figures 10 and 11 are examples of conservative and dissapative

systems respectively. The energy in the dissapative system tends to zero

1.5

1

0.5

Phase plot

S X

?*

- _ X

S * ~~

. X

/ N

: / , \

/ ' \

i ' \ \

dx/dt _o : > i / i :

-0.5

-1

.-I e;

\ x

* ^.

_^ _ *

s /

~

x ?

X ^

III,.

-1.5 -1 -0.5 0 0.5 1

x(t)

Figure 9. Phaseplot_showingdifferent initialenergies.

Phaseplot

-1.5

-1.5 -1

1.5

-0.5 0 0.5 1

x(t)

Figure10. Conservative system,area_preservingsuchthatA1=A2.

Phaseplot

dx/dt 0

--0.5

Figure 11. Non-conservative_{(dissapative)}system,areadiminishestozero.

Artificial phase space ^

"~'

A phase space is considered artificial when the coordinates of the space

do not havethe usual temporal significance. For the purposes of_embedding

sampled _data, an artificial phase space is generated _{by creating} phase

coordinates based on a set of_{time delays xn}yielding:

y(n)=

[x(?),

r,),x(t+_r,),...,x(t+

r((M))]

Although eachtime _{delay may be} at least in principle picked arbitrarily'".IV

it iseasierto picka single value ofx and use multiples oftfor the _embedding

coordinates. Thisyields artificial phase space vectors oftheform:

"'

y(n)=

[x(t),x(t

(c/-l)r)]

Thissubstitution for the usual phase coordinates is acceptable"

sincethe

time lagged coordinate elements of_y(n) are nonlinear combinations of the

local time derivatives.

X(t+2T₎

X(t+T₎

X(t)

Figure12. A 3-Dartificial phasespace created_usingdelayparametersfor thecoordinates.

IV

R. Mane, Dynamical SystemsandTurbulence, Lecture Notes in Mathematics, 898, 230

(1981) v

CHAPTER 3

ATTRACTORS .

An attractor is a phase space _{entity that} attracts phase trajectories _lying

within its basin of attraction. It is a set or sub-space on which analytic or

experimental data accumulates. The phase_{trajectory asymptotically} approaches

theattractor as t-oo. Transientsarethe resuit of an initialconditionthat iswithin

the basin of attraction, but that is not on the attractor. The attractor _may be a

map, a set ofdiscretepoints, or a_flow, a continuous sub-space region.

Phase space maps

Maps are the result of data from discrete systems. In discrete systems

eachfuture set of coordinates is mapped _directlyfrom the _preceding set. The

system exists_only atdiscretestates, the set of which is predetermined _bythe

initial conditions. The Henon attractor is an example of a discrete mapping.

The Henonattractorisgenerated _bythe_following set ofiterative relations:

|x1(W

l-14x1():|

1x2(

Henon attractor

0.4

0.3

0.2

0.1

y o

-0.1

-0.2

-0.3

-0.4

"*****.

:-V*:^-""'*x, ""*" *

-V* V*:..

_

's. \

:* tx.

.

1

1 -\

I

-.'

/

y

__rf *

-1.5 -0.5 0

x

0.5 1.5

Figure 13. Henon attractor,an example of a mapping. Two initial transient points _maybe

seen at_[0,0]and [1,0].

Initial values are selectedforxtfn), after which _any number of points _may

becalculated. The choice ofinitial conditions depends on the interest in the

system. _{If the only interest is in} the _geometry of the system the initial

conditions are unimportant as _long as _they are within its basin of attraction.

The _trajectory will approach the attractor given enough iterations for

transients todisappear. If afuture or exact state ofthe system is of _interest,

then the current state ofthe system mustbe known.

Phasespace flows

Flows are the result of continuous systems. In continuous _{systems, the}

the system can be _{determined for any} time and set of initial conditions. The

Rossler attractor is an example of a flow mapping. The Rossler attractor is

constructed _{using thefollowing}set ofdifferentialequations:

'i=

-(y +z)

y=_x+ ay

z-b+xz+cz

(6)

Typically for the Rossler attractor _{a=0.2, b=0.2,} and c=-5.7, although these

are not _{the only} possible values. As with the phase space _{maps, if the only}

interest _{is the geometry} of _{the system, the} choice of initial conditions is

unimportant _assuming that _they lie within the basin of attraction. If a future

state of the system at a given time is of _interest, then the initial state

becomes important.

Normal attractors

Normal attractors are not sensitive to initial conditions. Small changes in

theinitial conditions havelittleeffect onthe_trajectoryast-*.

Pointattractor

The _trajectory of the point attractor is invariant with time after the

transients in the system have vanished. The choice of initial conditions

must place the _trajectory within the attractors basin of attraction. The

basin of attraction is the set of initial states that will converge to the

placed in anywherewithin the bowl itwill come torest in thecenter _every

time. The center is the point attractor and the volume of the bowl is the

basin ofattraction. This isshown in Figure 1 1 ofChapter2.

Rosslerattractor

-10 -10

Figure 14. Rossler attractor, an example of a flow. Integrated with a time _step of 0.1

secondsfromaninitial condition of[1 1 1].

Periodic _(circle)

The _trajectory of the periodic attractor forms a circle. After the

transients havevanished the state ofthe system continues torepeat with

period T such that x(t)=x(t+T). The trajectories from initial conditions

within thecircle spiral _{outward, asymptoticallyapproaching}the circle. The

trajectoriesfrom initial conditions which lieoutsidethe circle spiral inward

OtherAttractors

As with dynamical systems there are _{corresponding increasingly}

complex attractors. The torus is the attractor analogous to the

quasi-periodic system. It has greater _complexity than the circle while

maintaining linearity. The complexity of quasi-periodic attractors

increasesuntil nonlinearelementsarepresent.

Chaotic

(Strange)

The trajectories of strange attractors are sensitive to variation or

uncertainty in the initial conditions. Variation in the initial conditions should

be thought of as a different initial state, whereas _uncertainty could be

uncertainty in measurement or computational error. Trajectories within the

strange attractor are _{exponentially divergent,} that is points which are _very

close _initially will have greatly different evolutions. _{Therefore the uncertainty}

associated with _determining the state of a system has direct _bearing on the

CHAPTER

EMBEDDING DELAY CALCULATION

The first efforts in the analysis _{involved estimating} either the _embedding

delay or _{the embedding dimension. Early} methods were prone to errors from

noisy data or an improper choice of time delay. Subsequent methods have

offered some promise _{in correcting} these shortfalls, but no method has been

showntowork wellfor everysystem.

Autocorrelation function

(ACF)

The first attempt to determine an optimal _{embedding delay} used the

autocorrelation function. Although the autocorrelation function is suitable for

largeand small data sets, theembeddingdelaytis usually selected_using an

autocorrelationfunction if_N(jat is less than 10,000. Theapplicationto smaller

data sets is possible because the autocorrelation process picks a smaller x

than would mutual information theory. The autocorrelation is performed to

determine the first local minimum or decorrelation time. The decorrelation

value. The_{embedding delay}is then selected suchthat tis 1/10to1/20ofthe

first local minimum1"

or zero. This is an _arbitrary choice_{that may} work well in

practice. The _sampling rate should correspond to _{approximately} the same

value as t. For dynamical time series _{resulting from mappings,} such as the

Henon set, the first local minimum of the autocorrelation occurs at the first

point. With no other intermediate points _{available, the first} point is selected.

The first zero orlocal minimum ofthe autocorrelation wasthought to ensure

linearly independent coordinates. In application to higher order systems the

ACF can yield incorrect results7"

due to correlations between other

coordinates and _{may lead to} a collapse of the attractor in one or more

coordinates.

Mutual information

theory

Mutual _{information, /, theory} is usually applied to larger data sets. The

mutual information _theory approach looks at the general dependence oftwo

variables, overcoming the autocorrelation's limitation to _measuring linear

dependence. For the creation of phase _{portraits, the first local} minimum of/

is used. The mutual information _{theory generally} picks x _{significantly larger}

than autocorrelation would and is therefore not suitable for use with small

Vl

Abarbanel, H.D., Brown, R., Kadtke, J.B., Prediction in chaotic nonlinear systems: Methodsfortimeseries withbroadbandFourier spectra, Physical ReviewA, 41, 1782 (1990) v"

Buzug, Th., Pfister, G., Optimal _delay time and _{embedding dimension for delay-time} coordinates_byanalysis oftheglobal static andlocaldynamicalbehaviorof strange _attractors,

data sets. Fraser and _Swinney applied mutual information theory71"

to the

Roux and Rossler equations in 1986. The _{embedding delay} x _may be

selected _usingmutual information_theoryfor

Ndat>10,000. ₍₇₎

The _theory works well for low _embedding dimensions but becomes

impractical above an _embedding dimension of four. In addition to the

embedding dimension limitation the algorithm to calculate the mutual

information is recursive and not _easily coded _{using MATLAB,} so it is not

employed inthiswork.

Othermethods

Other methods _{employ different techniques} such as

pseudocycle1" based

time delays and the first minimum logarithm of a generalized correlation

integral."

All the methods worktoa greater orlesser_degree, butnone _supply

information aboutan optimal_{embedding dimension.}

Shortfalls

The largest shortfall with all methods is the failure to pick the best

embedding delay. All ofthe methods in general pickthedelay, x, arbitrarily.

In _theory, with an infinite supplyof noisefree_data, thechoice of_delaymay in

Vl"

Fraser, A.M., Swinney, H.L., Independentcoordinatesforstrange attractors frommutual information,PhysicalReviewA, 33, 1134(1986)

Destexte, A., etal, Physics LettersA, 132, 101 (1988)

factbealmost arbitrary. If the _delaypicked is to short, the attractor _{lies along}

the diagonal of the artificial space and its characteristics are difficult to

distinguish. This problem is even more evident in low dimensional systems

where noise ispresent. If the _delay picked is too _longthen the points are not

strongly correlated due to the _sensitivity of the _trajectory as explained in

Chapter 1. In practice all methods give a _delay which _{may be} used to

qualitatively reproduce the system. The effects of the _delay selection on

reconstruction are_{illustrated below using}2000 pointsofLorenz-xdata.

Phase plot oftime series 150

Artificial phase, delay0

Figure 16. Artificialphase plot witha_delayof zero causesthecomplete collapse ontothe hyper-diagonalwiththeconsequentlossof information.

Artificial phase, delay4

Figure 17. The delayof_4, l^o"1

Artificial phase, delay8

Figure 18.Delayof _8,1/10 the decorrelation value,gives a reasonable expansion. Notice astheloopsexpand_{away from}the_{diagonal, they}contract_{along it.}

Artificial _{phase, delay} 16

Figure 19. Thecontinued expansion inone direction isassociated_bya contractionin the otherdirectionat

1/5*

Artificial phase, delay40

Figure 20. At a _delay of _40, one half the decorrelation value, the embedding shows symmetrical_foldingandinformation isbeinglost. Thereconstructionhas lostmost ofits geometrical_similaritytothephase plot.

Artificial phase, delay80

EMBEDDING

For problems with N large enough, dA(N,d) will become independent of d

when theattractor _{is properly}embedded in Rd- _{Formal analytical} results'" dictate

the embedding dimension d>2dA+l to ensure that all of the geometric

information about the attractor is exposed in the _embedding space. _Usually in

practice however, _d>d/^ is sufficient. _{Operationally, d^} is increased until _d/\

becomes a constant. The first value of _dg forwhich _d/\becomes constant then

becomestheembedding dimension thatis used.

One of the first methods""

was suggested _by Packard et al. in 1980. The

method consisted of embedding data into a phase space _{using the time} series

and its successive derivatives for coordinates. The state vectors in the

d^-dimensional embeddingspace are given _by

('.)=

dt

</''-'#,)

dta

(8)

xi Takens, F., Dynamical Systems and Turbulence, Lecture Notes in

Mathematics, 898, 366, (1981)

The derivative based method worked, but was _{computationally intensive, very}

sensitiveto noiseand not _necessarilyoptimal.

Takens published a _{very promising} theorem in 1980 which is the basis for

most analysistoday. His method employs time _delay coordinates which _{may be}

chosen _arbitrarily at least in theory. The attractor is reconstructed from a scalar

timeseries_{^(t^)}, where_keKrjand

K0={*GN0;A:<iVdat}. ₍₉₎

\ is the observed _quantity and _N<jat is the number of data points. The state

vectorsforadE-dimensional_embedding space are given_by

<*.)

ft.)

t)

(10)

j{ts +

<dE

-1))_

where_ts=sTaand seSnfor

S0 ={sc-N0-s<N^-(r/Ta)(dE-l)} (11)

withsamplingtimeTaand _delaytimex.

Stricttheorem requirements necessitate an infinite number of noisefree data

points with infinite resolution. The theorem also requires dE>2n+1 (where n is

the dimension oftheflow in original space) for there to be a_{topological mapping}

fromtheoriginal phase spaceto theembedding space. In application, an infinite

methodis _{computationally} intensiveanddoes notindicate an optimal _embedding

dimension.

Broomhead and _King used singular value decomposition in 1986,XI"

Using

SVD on the _trajectory matrix _they were able to obtain the number of non zero

singularvalues_{corresponding to}an optimal _embedding dimension.

Several others"'

have mentioned methods based on the requirement of n

initial conditions_{to uniquely} solvea set of n _{ordinary differential} equations. The

method determines the minimum number of conditions _{necessary to} set _{up the}

attractortrajectory.

All of the methods above are capable of _embedding the time series into a

phase space. Some ofthe methods yield an optimal _{dimension for embeddings,}

butail are computationally intensive.

xiii Broomhead, D.S., King, J._P., Physica _{D, 20,} 271 ₍₁₉₈₆₎ xiv Bumeliene, S., et_{al., Liet Fiz.} Rinkinys, 28, 569 ₍₁₉₈₈₎

CHAPTER

HARDWARE & SOFTWARE

Hardware

The_twentysix cases which ranto a normal termination required atotal of

123 billion _floating point operations _(flops) foran average of4.7 billion flops

per case. Two computers were used to perform _{the analyses,} a _Gateway

486/25 DX and a _Gateway486/66 DX2. The 486/25 performed calculations

at an average rate of ~77 kflops/s while the 486/66 performed the

calculations at an average rate of -189 kflops/s. The combination of

computers used over290 hoursof_{processing time to}run the26cases.

Software

The first version ofMATLAB was written at the _Universityof New Mexico

and Stanford Universityin the late 1 970's. MATLABwas intended tobe used

for courses in matrix _theory, linear algebra, and numerical analysis. The

subroutine packages for matrix _{manipulation, but} wanted a tool that did not

require the user to write programs in Fortran. The programming syntax still

has the look and feel of Fortran but does not require _{any knowledge} of

Fortran or_programming ingeneral.

Theprogram has now returned to its origins with the publication of

an inexpensive Student Edition ofMATLAB. There are also other versions of

MATLAB which are available on a _variety of platforms. The software which

has been developed here is meant to be run _using a Pro-MATLAB version.

The Student Edition has all the _{necessary functions}to perform the _analysis,

butwas never meantto handle the large datasets.

Many different programs were written to accomplish this work. Some of

the original programs no longer exist or are not _necessary, while new

programshave beenwrittenand others have been adapted. All ofthe current

programs run _properly under MATLAB v4.2b for Windows. _They have been

adapted to utilize some of the graphical abilities of _Windows, but _by no

means do _they accomplish this fully. The Windows version is a major

improvement over the DOS based Student versions _{capabilities,} although I

have not had time toexplore all ofthe newfeatures. Due to changes in the

MATLAB software some of the programs _may not run on older versions of

MATLAB due tothe elimination of certain commands. I do not know howthe

Most of the programs are documented to an extent. A briefstatement is

included to give the basic purpose of each function or program. Within each

function or _{program, in the code,} comments _{have been included specifically}

where important user configurable parameters are. _{Additionally,} there are

description below foreach moduletoprovide greaterinsight.

ACM

The ACM function performs an autocorrelation on the sampled data and

returnsthe decorrelation index foruse _{in selecting} the _{embedding delay. The}

autocorrelationforcontinuousdata looks like the following:

AC(T)

This form however isnot_veryusefulforadiscrete timeseries. Instead the

following discretized form is used, where x is an index ratherthan a time as

above:

^C(r)=

-^-|>.-*.+r

0<r<Ar

(13) 1=1

ACM is a _shorter, faster, slightly less precise version ofAUTOCORR.M.

To reduce the computational effort, ACM only processes the first 1000

points. Experiments with data sets of different sizes showed that the

reduction in size to 1000 points did not _{significantly} affect the decorrelation

index that the function returns. ACM uses the Hankel matrix _function,

0,

isa simple mattertochange MAIN.M so thatAUTOCORR.M iscalled instead

sincebothfunctionsuse thesame arguments.

% ACM

% MATLABfunctionthatperforms anautocorrelationontheinputvector

% The indexofthefirstzero orlocal minimumisreturned. ACMisa shorter

% versionof_{AUTOCORR.M,usingonly}thefirst 1000pointstoreducethe

% computationaleffort. Reductionto 1000pointsalsoallowsuseofHANKEL.M

% whichspeedscalculation.

%

% function decorr= ac(a)

% Copyright_(c) 1992-_{1994WilliamRobertson}

function decorr= ac(a)

% Takefirst 1000pointsforautocorrelation

if_max(size(a))> _{1000; a=a(l:1000);}end

ac=a*hankel(a); x=max(size(a)):-_1:1

;

ac=ac./x;

plot(ac); title('AutocorrelationPlot');xlabel('Index');

% Findfirstzero orlocalminimum

N=max(size(ac));

fori=l:N

ifac(i)<=0

decorr=i;

disp(['Autocorrelationzero at_{lag',int2str(i),'.'])} return

elseif_{(ac(i)-ac(i+l))}<0 decorr=i; disp(['

Autocorrelationlocalminimum at_{lag',int2str(i),}' _'])

return

end

end

ARTSPACE.M

TheARTSPACE.M program was written to create artificial phase spaces

embedding delay on the embedded time series. ARTSPACE.M was used to

generate Figures 16-21 in Chapter 4. The program calls NDIFF.M to

calculate the first time derivative of the series. This is used to generate a

phase plot so that subsequent artificial phase plots _using different

embeddingdelays may becomparedtothe real phase plot.

% ARTSPACEM

% MATLABprogramthatcreates artificial phase spacesbasedonTakens'_delay

% vectors. Onephase spaceiscreatedforeach_delaybelowthedecorrelation % index. Thisprogramcallsthefunctions NDIFF.MandACM.

% Copyright _(c) 1992-_{1 994 William Robertson}

clc

clear all

flops(O);

tic;

% loadthedata froma.matfile, alternativelythedatacouldbecalculated

load lorenzx

x(:,l)=time; x(.,2)=value;

cleartimevalue; N=max(size(x));

disp(['Read',int2str(max(size(x))),'

pointsfromtime_series.'])

% the_{following loop} reducesthenumberof pointsto2000, it isnot _necessaryto do

this

ifN>2000

disp('Usinglast2,000points oftime_series') y(l:2000,:)=x(N-1999:N,:);

end

disp('Calculatingfirstphasefortime_series.')

dt=abs(y(l,l)-y(2,l));%assumesa uniformtime_step

x=y(:,2);

clear_y

[x,dx]=ndiff(2,4,x,dt); N=max(size(x));

figure;plot(x,dx,'g'); %phaseplotfrom derivativecalculation

title('Phaseplot');

xlabel('x'); ylabel('dx');

[rc]=size(x); ifr>c; x=x';end

decorr=ac(x);

% generates all artificial phases from _delay 1 to decorrelation _delay this _may cause

problemsif

% decorr is _large, second argument in for _loop is increment, increase to reduce

number of plots

for i=l:l:decorr

y=x(l:N-i);z=x(l+i:N);

figure;plot(y,z,'g'); title(['

Artificialphase,_{delay',num2str(i)])}

xlabel('x(i),);ylabel('x(i+j)');

end

disp(['Elapsedtime',num2str(toc),'_seconds.']) disp(['Floatingpoint operations=

', num2str(flops),'.']) disp('Normalterminationof_ARTSPACE')

AUTOCORR.M

The AUTOCORR.M function performs an autocorrelation on the input

vectorand returnsthe decorrelation index. Unlike ACMwhich is based upon

it, AUTOCORR.M uses the entire _{input vector, ultimately yielding} a better

estimate ofthedecorrelation index. Thiscauses a highercomputational effort

due to the larger size. The larger size further requires the use of a less

efficientalgorithm, increasingthecomputational effort.

% AUTOCORR.M

% MATLAB Functionthatperformsan autocorrelation ontheinputvector

% Thedecorrelationindexisreturned_indicatingthefirst localminimum or zero of

% theautocorrelationfunction.

%

% functiondecorr=

autocorr(a)

% Copyright_(c) 1992-_{1 994 William Robertson}

function decorr=

autocorr(a)

disp('Calculatingautocorrelationfunction')

a=a(:);

N=max(size(a));

acorr=[l _1];

acorr( 1)=sum(a.*a)/N;

% becauseofthe_possibilityof averylargetimeseriestheHANKEL functioncannot

beused,

% it requiresN squared elements and storagerequirementsbecome prohibitive. The

loopmethod

% employed below is however _very inefficient in MATLAB, so with a better

estimate, comesthe

% price of greatertime

forj=0:N

i=l:N-j;

acorr(2)=sum(a(i).*a(i+j))/(N+l-j); % checkfor firstzero

ifacorr(2)<=0

decorr=j; disp(['

Autocorrelationzeroat_{lag ',int2str(j)])}

return

% checkfor first localminimum

elseif_{(acorr(l)-acorr(2))}<0

decorr=j; disp(['

Autocorrelationlocalminimum at_{lag ',int2str(j)])}

return

end

acorr(l)=acorr(2);

end

CORRINT.M

The CORRINT.M function calculates the correlation integral for use in

estimating the limit ofthe attractor dimension. As the _{embedding dimension}

increases the estimate ofthe attractor dimension _{asymptotically} approaches

the fractal dimension of the attractor. The method was first proposed _by

Grassberger and Procaccia after which it is named. The

Grassberger-Procaccia algorithm hastheform:

where _0(x) is the Heaviside function. The operation of the Heaviside

evaluation isgiven_by thefollowing:

. f0 r <S

<*-*)-{,

d5)

Theiler modified the algorithm to reduce the nearest neighbor effects on the

correlation integral which _may cause errors in the estimation ofthe attractor

dimension. A_{term, W,}was introduced, changing theformto:

ThetermW iscalculated _bythefollowing:

r=

()"+l

where N is the number of points in the time series, and m _{is the embedding}

dimension. W must be rounded up to the next integer value since it is to be

used inthe summation as anindex.

The function of the correlation integral is to sum all of the inter-point

distances, S, greaterthanor equal torforeach r. CORRINT.M evaluatesthe

correlation integral at each r defined _by the parameters _{rmin, rmax,} and

numbin. The parameter rmin determines the lower _scaling region and rmax

determines the upperscaling region. Theparameter numbin determines how

many points _including rmin and rmax there will be. The choice of rmax and

set too large part or all ofthe _{linear scaling} region will be lost. _Conversely if

rmin is set _{too small,} statistical aberrationsatthe lower points _may affectthe

dimensionestimate.

% CORRINT.M

%

% MATLAB functionthatcalculatescorrelationintegral function for_plotting

% _{log C(N,r)}vs. _logr.Thecorrelationintegral isusedtoapproximatethe

% dimensionof a non-linear or chaotictimeseries.

%

% function_{[r,CNr]=corrint(xn,W)}

% Copyright_(c) 1992-_{1 994 WilliamRobertson}

function_{[r,CNr]=corrint(xn,W)}

ifnargin==l;W=0; end;

[R,C]=size(xn);

ifC>R;xn=xn';[R,C]=size(xn);end;

% The_followingtwolinesare usedtosetthe_scalingregionfortheinterpoint

% distancecalculations.The lowerand upperlimitsof5%and_{60% respectively}

Vo haveworkedwell,butthereisno analyticalsignificance. Theexpression

% inparenthesisisa quick(&_dirty)waytoapproximatethehypersphereradius

Vo oftheembeddedtimeseries.Itismuchfasterthan_calculatingtheEuclidean or

% taxicabdiameter.

rmin=0.05*sqrt((max(xn(:,l))A2)*C);

rmax=0.60*sqrt((max(xn(:,l))A2)*C);

% the_followingsetthenumber of_evenlyspaced radiiforthecorrelationintegral

% basedupon rminand rmax

numbin=25;

r=linspace(rmin,rmax,numbin);

xn2=xn;

CNr=zeros(size(numbin)); fori=l:R-l;

b=xn2(l,:);xn2(l, :)=[]; xn2(R,:)=b; d=sqrt(sum(((xn-xn2).*(xn-xn2))'));

fork=l:numbin;C(k)=sum(d<=

r(k));end

CNr=CNr+C;

end

r=log(r);

CNr=log(CNr/RA2); 0,

EMBED.M

EMBED. M is a MATLAB function that maps the sampled time series into

an n-dimensional space. The _embedding is performed _using a delay,

associated with _{the decorrelation index. The function} checks to make sure

there are enough points after embedding. If the number of points after

embedding would be less than half of the number of points in the original

time series, then the function terminates. Theembedding is accomplished _by

indexing a column vector _by the _delay and _{inserting it sequentially into}

columns oftheartificial space array.

% EMBED.M

%

% MATLAB functionthatmapsx vectorintondimensions using_delayspecifiedat

% thefunctioncalling. Ifthevectoristoosmallforthenumber ofdimensions

% an errorisgenerated andthefunctionterminates.

%

% function[xn]=embed(x,n,delay)

% Copyright_(c) 1992-_{1 994 William Robertson}

function[xn]=embed(x,n,delay)

% Calculatenumber of pointsfor_embedding

[N,c]=size(x);

ifc>N;x=x';N=c;end

npts=N-(n-l)*delay;

ifnpts<N/2

fprintf('Insufficient number of points for _embedding in %3.0f

dimensions._',n)

return

end

% Embeddata intocolumn coordinate vectors forj=l:n

xn=[xnx(0-l)*delay+l:N-(n-j)*delay)];

end

disp(fNumberof embedded points=',num2str(npts)])

LINREG.M

LINREG.M is a MATLAB function that computes the coefficients for a

linear function in the least squares sense on a preselected range of the

correlation integral results. The correlation _{integral, (CORRINT.M),} yields

log(r) vs. _log(CNr) data. The lower region of this _{data, corresponding to}

smaller inter-point _distances, has a slope which scales with the attractor

dimension.

The range selection is made in the line that says xsub=x(1:6). The

contents of x are predetermined _by CORRINT.M. The coefficients of the fit

are calculated without regard for the qualityoffit. It is assumed thatthe user

has had some experience with the time series and has set _{up CORRINT.M}

properly. If this is not the case then ULINREG.M should be substituted for

LINREG.M in MAIN.M.

The leastsquaresformulation fora linearfitstarts withthefollowing:

(18)

Syis thesum ofthesquares ofthe residuals. This functioncan be minimized

with respect to

a,-by taking the appropriate partial derivatives. The most

common formofthis is usuallyshown as:

a\ =

7 Vs

5>H2>.) ₍₁₉₎

0,

Orthe resultofthe processcanbeexpressed inmatrixformas:

[[^]]w=^rm

Close inspection ofthefunction's _coding reveals the _simplicity of_calculating

the linear leastsquares _usingthe matrix method.

% LINREG.M

%

% MATLAB functionthatcomputesleast squareslinear fitto theregion ofdata

% preselected_bytheuser,usuallythefirst fewpointsofthecorrelation % integral. The_slopefa]and_intercept[b]are returnedto the_callingprogram. % Linearequation[y]=a[x]+b.xsand ys are usedtodrawalinethrough the

% fitted datapoints. SimilartoULINREG.Mbutnotinteractive.

%

% function_{[xs,ys,a,b]=linreg(x,y)}

% Copyright_(c) 1992-_{1 994 WilliamRobertson}

function_{[xs,ys,a,b]=linreg(x,y)}

x=x(:);

y=yO);

figure

plot(x,y,'+g')

title('Correlationintegralplot')

xsub=x(l:6);

ysub=y(l:6);

xsub=[xsub,ones(size(xsub))] ;

A=xsub'*xsub; B=xsub'*ysub;

ifdet(A)~=0;a=A\B;end

b=a(2); % intercept

a=a(l); %slope

xs=[xsub( 1);xsub(max(size(xsub)))];

ys=a*xs+b;

clg

plot(x,y,'+g',xs,ys,'~r')

LORENZ.M

dt

=_-ax+

ay

dy

dt

=-xz+ bx-_y

dz dt =_xy-cz VO % % % % % % % % % % % %

LORENZ.M calculates the time derivatives which define the Lorenz

attractor. Thevelocities are calculatedfrom position_{using thefollowing} set of

coupled differentialequations:

lax

=_-ax+

ay

(21)

where _{typically a=10, b=8/3,} and c=28 although the coefficients _{may be}

different.

LORENZ.M is called _by an integrator such as RUNGE.M for evaluation.

The data generated _by LORENZ.M was used for most ofthe work on this

thesis. A total of 300,001 points were calculated _using a _{time step} of 0.01

seconds and the initial position [1 1 1]. The Lorenz attractor has a fractal

dimensionof2.067.

LORENZM

ThisMATLABprogram calculatesthederivativesoftheLorenzattractor.

Thederivativesare used_bya numericalintegrationroutine such

asRunge-Kutta.

Lorenzattractorset ofdifferentialequations dx/dt=-ax+_ay

dy/dt=-xz+bx

-y

dz/dt= xy- _cz

[xdot]=lorenz(t,x) tin input isa_dummyargumentfor integrator

function_{[xdot]=lorenz(t,x)}

a=10; b=8/3;

c=28;

dx=-a*x(0+_a*x(2);

dy=-x(l)*x(3)+_c*x(l)-x(2); dz=x(l)*x(2)-b*x(3); xdot=[dx_dydz];

MAIN.M

The MAIN.M routine providesthe basiccontrol structurefor the collection

of programs and functions that perform the analysisfor the time series. The

MAIN.M program also has some configurable parameters such as the

embedding delay and the zfactorwhich determines the _{secondary sampling}

ofthe time series. A_diary file is opened and all screen outputis also written

to a text file. The text file can be saved as a record of how the data was

processed and whatthevarious parameters were setto.

% MAIN.M

%

% Maincontrolroutineforsystemidentificationprograms_using

% MATLAB.

% Copyright_(c) 1992-1994 William Robertson

%

% Asingletimeseries_{(t, F(t))}isreadfromadata fileor created % _usingaMATLABfunction.

%

% Anautocorrelationisperformed onthedatatodetermine % thedecorrelationtimeforusein_embeddingthedata.

%

% Thedimensionofthesystemisestimated_byembeddingthedata in

% higher dimensionalspaces until convergance ofthecorrelation % functionslopesisobtained. (Slopeof_lnC(r)vs._ln(r)converges

%

% ThefileDATA.MATwas createdbeforetheexecution ofMAIN.M. Inthe % file DATA.MATtwovectors werecreated, TIMEandVALUEsuchthatindexed % pairs represent a state.

%

clc

clear all

flops(O);

diarysession.txt tic;

disp('Timeseries analysis program_byWilliam_Robertson.')

disp([date])

tl=clock;disp([num2str(tl(4)),':',num2str(tl(5)),':',num2str(tl(6))])

disp('Loadingtime_series')

disp('Data_beingloaded from_DATA.MAT.')

load data

dt=time(2)-time(l); N=max(size(value)); disp(['Read',int2str(N),'

pointsfromtime_series.'])

% z# selects_everyzthpoint

ifN>5000

z=l;

disp(['Using5,000points oftime series,every',num2str(z),'point(s).']) i=l:5000;

j=i*z;

x(i)=value(N-j);

end

clearvalue;

disp('Calculatingfirstphasefortimeseries.') % SeeNDIFF.Mforexplanationof_callingparameters

[x,dx]=ndiff(2,4,x,dt);

plot(x,dx,'g')

title('Phaseplot oftimeseries')

xlabel('x(t)') ylabel('dx/dt')

drawnow

% Anautocorrelationisperformedtodeterminethedecorrelationtime

% whichisusedindeterminingthedelayincrementassociated withtheembedding

% process.

%

disp('Calculatingautocorrelationfunction.') decorr=ac(x);

disp(['Elapsedtime',num2str(toc),'

seconds'])

%

% Embedthe timeseriesinto _successivelyhigher dimensions. The

embed-% _ing stopswhenthe_{embedding dimension}nisgreaterthan2a+l,where % aisan approximation oftheattractordimension.

m=l; a=0; while_m<=(2*a+l); m=m+l; disp(' ') disp([num2str(m),'-Dimensional embedding.']) delay=ceil(decorr/2);

disp(['Usingembeddingdelayof_{',int2str(delay), '.'])} [xm]=embed(x,m,delay);

% W is Theiler modification to Grassberger-Procaccia algorithm, set W=l to

eliminate effect

W=ceil(decorr*(2/max(size(x)))A(2/m))+l;

dispensingW factorof_{',int2str(W),'.'])} [r,CNr]=corrint(xm,W);

disp('Finishedcorrelationintegral.') [rsub,CNrsub,a,b]=linreg(r,CNr); disp('Finished_LINREG.') R=[Rr]; CNR=[CNRCNr]; rs=[rs_rsub]; CNrs=[CNrsCNrsub]; A=[A a]; tl=clock;disp([num2str(tl(4)),':',num2str(tl(5)),':',num2str(tl(6))]) disp(['Elapsedtime ',num2str(toc),' seconds.']) end

disp('Embeddingprocedurecomplete.')

% Showresults ofembeddings.

figure;plot(R,CNR,'+g',rs,CNrs,'--r') text(.80,.15+.05*(m-l),'De Da','sc')

xlabel('ln(r)')

ylabel('ln(CNr)')

for i=2:m

disp([' \int2str(i),'

\num2str(A(i-l))]) end

drawnow

disp('

')

tl=clock;disp([num2str(tl(4)),':',num2str(tl(5)),':',num2str(tl(6))])

disp(['Elapsedtime',num2str(toc),'_seconds.'])

disp( ['Floatingpoint operations=

', num2str(flops),'.']) disp('Normalterminationof_MAIN.M')

diaryoff

NDIFF.M

NDIFF.M is a numerical differentiation function that was _originallywritten

to provide real phase plots. It can perform _{backward, centered,} and forward

differences on equi-spaced data. Forward and backward differences are

performed _{using 2} or3 pointformulas, whilethe centered difference can be

performed _{using 2,} 4, or6 points. NDIFF.M returns two _vectors, a subset of

the original time series and the time derivatives that correspond to the

subset.Theusageof multiple pointdifferentiationformulas meansthatnot all

the derivatives can be evaluated, thereby reducing the useable number of

points. The two point backward difference formula is based upon the

followingfinite divideddifference:

f(x,)

f(X,)~f(X-iKoh

h

Thethree pointbackwarddifference is:

/(x>3/(x,)-4/(x,)+/(x3)+o/;2

^

f(x)=f^-f^-(W

2/7

The fourpoint centered _{difference is based upon;}

(24)

/(r|)

/M-8/(x.,)

+

8/(xJ-/(xrt)

12/7 v ;

Thesix point centered difference is based upon:

f

)

+_9/(^)~

45/(,.,)+

4S/(x,)

-9/(x,+2)+

/(x,.3)

+Qh& ₍₂₆₎ 60//

The twopointforward difference is based upon

w^u/MiW.^ ₍₂₇₎

Thethreepointforward difference isbasedupon:

/(x)=-3/(x,)+4/(x,+1)-/(x,+2)_o;;2

^

% NDIFF.M

%

% Thisprogram uses_backward,centered, orforward divided differencetech

no niquesto_numericallyestimatederivatives, dx/dt. The inputargumentsare: %

% type -onecharacter_{string giving}typeofdivided differencetoperform % npts -numberofdatapointstousein_calculatingthedivided difference % x -vectorwithx values

% dt -dtvalue %

% Examplesof valid combinations % dx=ndiff(l,2,x,dt) dx=ndiff(l,3,x,dt)

% dx=ndiff(2,2,x,dt) dx=ndiff(2,4,x,dt) dx=ndiff(2,6,x,dt)

% dx=ndiff(3,2,x,dt) dx=ndiff(3,3,x,dt)

%

% function[xs,dxs]=ndiff(type,npts,x,dt)

function_{[xs,dx]=ndiff(type,npts,x,dt)}

np=max(size(x));

iftype==l % backward differenc

ifnpts==2

dx=(x(2:np)-x(l:np-l))/dt; xs=x(2:np);

elseifnpts==3

dx=(x(3:np)-4*x(2:np-l)+3*x(l:np-2))/(2*dt); "

-xs=x(3:np);

end

end

iftype=2 %centereddifference

ifnpts=2

dx=(x(3:np)-x(l_{:np-2))/(2*dt);}

xs=x(2:np-l);

elseif npts==4

dx=(-x(5:np)+8*x(4:np-l)-8*x(2:np-3)+x(l:np-4))/(12*dt); xs=x(3:np-2);

elseif npts==6

dx=+x(7:np)-9*x(6:np-l)+45*x(5:np-2)-45*x(3:np-4)+9*x(2:np-5)-x(l:np-6);

dx=dx/(60*dt);

xs=x(4:np-3);

end

end

iftype=3 % forward difference

ifnpts=2

dx=(x(2:np)-x(l:np-l))/dt;

xs=x(l:np-l);

elseif npts==3

dx=(-x(3:np)+4*x(2:np-_1)-3*x(1

:np-2))/(2*dt);

xs=x(l:np-2);

end

end

ROSSLER.M

ROSSLER.M is a function much like LORENZ.M. It is called _by an

[dx dt cfy dl di -y. -= x+ay dt

=_b+xz

(29) % % % % % % % % % % % % % %

where _{typicallya=0.2, b=0.2,} andc=4.6.

ROSSLER.M

MATLAB functionthatcalculatesthederivativesoftheRosslerattractor.

The derivativesare used_bya numericalintegrationroutine such

asRUNGE.M.

[xdot]=rossler(t,x)

Copyright_(c) 1992-1994William Robertson

Rosslerattractor dx/dt=-(x+_z)

dy/dt=_x+

ay

dz/dt=_b+xz- _cz

[xdot]=rossler(t,x) tin input isa_dummyargumentfor integrator function[xdot]=rossler(t,x) a=0.2; b=0.2; c=4.6; dx= -x(2) -x(3); dy=x(l)+_a*x(2);

dz=b+ x(l).*x(3)-c*x(3);

xdot=[dx_dydz];

RUNGE.M

RUNGE.M is the MATLAB implementation ofa classical 4th order Runge

Kutta integration routine. A set ofparameters, /c,are calculated based on the

selected _by the dt argument in the _{calling function.} The next value of the

function iscalculated by:

jvi =y,+

-(k]+2k2+2L+k4) dt (30)

wherethek,are calculated by:

A*,y.)

, ,. dt

, dt (31) % % % % % % % % % % % % % % % k^f(t,+dt,y, +k3dt)

A fixed time _step is used_{to simplify the} analysis. RUNGE.M was used to

integrate both the Lorenz and Rossler differential equations. A higher order

integrator could be _used, but it is _probably not worth the additional

computational effort. The integration tends to smooth out _any noise in the

data where as the differentiation function will exaggerate noise and cause

furtherpropagation.

RUNGE.M

MATLAB functionthatimplementsthe4thorderRunge-Kuttaroutinefor

numericalintegration. Performsintegrationon polynomials passedfrom

a main control program. Returnstheintegrated functionsto themain program.

F ti dt tf yO t y

-functionfileusedtoevaluatethefunction

-initialtime

-integration_stepsize -finaltime

-initialvalue vector of state variables

-returnedtimevector

-returnedsolution,one column per state variable

% Copyright_(c) 1992-1994WilliamRobertson

function_{[t,y]=runge(F,ti,dt,tf,yO)}

t=(ti:dt:tf)';

y=zeros(length(t),length(yO));

y(i,:)=yo;

fori=l:l:length(t)-l

kl=feval(F,t(i),y(i,:));

k2=feval(F,t(i)+dt/2,y(i,:)+dt*kl/2); k3=feval(F,t(i)+dt/2,y(i,:)+dt*k2/2); k4=feval(F,t(i)+dt,y(i,:)+dt*k3);

y(i+l,:)=y(i,:)+dt*(kl+2*k2+2*k3+k4)/6;

end

ULINREG.M

ULINREG.M is a MATLABfunction thatcomputesthe linear least squares

fitto data from thecorrelation integral. The data from the correlation integral

is plotted and the user _{interactively} selects the region of data to curve fit.

After the range is selected, the slope is calculated. The figure is re-plotted

showing theslope andthe best fit line. The usercanthenaccept or rejectthe

selection. If the selection isrejected, the useris promptedtore-select.

% ULINREG.M

%

% MATLAB functioncomputesleastsquareslinearfitto theregion ofdata

% selected_graphicallybytheuser. Theuser_mayacceptor rejecttheselection

% beforeproceeding. The slope[a]and_interceptfb] are returnedto the_calling

% program,xs and ys are subsets representthebest fit line fortheselection.

% Linearequation[y]=a[x]+b.

%

% function[xs,ys,a,b]=ulinreg(x,y)

% Copyright_(c) 1992-_{1 994 William Robertson}

function[xs,ys,a,b]=ulinreg(x,y) x=x(:);

button=0;

% Selectregionofdatatocurvefit.

whilebutton~=2

figure

plot(x,y,'+g')

title('Correlationintegral_plot')

xlabel('Picktwopointsin linearregion of_graph')

[xloc,yloc,button]=ginput(2); index=find(x>=

min(xloc)&x<=

max(xloc));

xsub=x(index);

ysub=y(index);

xsub=[xsub,ones(size(xsub))];

A=xsub'*xsub; B=xsub'*ysub;

if det(A)~=0;a=A\B;end

b=a(2); % intercept a=a(l); %slope

xs=[xsub(l);xsub(max(size(xsub)))];

ys=a*xs+b;

clg

plot(x,y,'+g',xs,ys,'~r')

title(['

Attractordimensionapproximatedtobe_{',num2str(a)])}

xlabel('Leftbuttonto_tryagain,Rightbuttonto_continue')

[xloc,yloc,button]=ginput(1),

CHAPTER

EXPERIMENTAL

The original plan was to perform the _battery of experiments on both Rossler

and Lorenz data. This would have doubled the run time of the computers and

therewere some_{inherent difficulties in using the Rossler}attractor.

Data generation

The Rosslerequations were _{integrated using} the function RUNGE.M for

60 seconds with a dt=0.01 seconds from an initial condition of [0 0 0]. The

initial integrationwas to remove the transients caused _by an initial condition

within the basin of attraction, but not on the attractor. A second integration

was performedusing a larger time stepof0.05 seconds toprovidea_working

datasetof60,001 points_spanningatotal of3000seconds.

The Lorenz data was similarly generated _{using RUNGE.M from} an initial

condition of[111],with dt=0.01 seconds. Inspection ofthe equations show a

point attractor for the initial condition of [0 0 0]. The Lorenz data also

spanned 3000 seconds, but had to be integrated with a smaller _{time step,}

To speed _{up the programs,} the data for each attractor was written to a

.MAT file. This allowed MATLAB to read the data from the file rather than

recreate it for each case. Although the investigations used _{very large data}

sets, thismethod yieldssignificanttimesavings evenforsmall data sets.

AbandonmentofRossler

The Rossler attractor was_only used _briefly in this investigation due to its

weak non-linearity. The fractal dimension of the Rossler attractor is,only

2.027, a result oftheweak non-linearity. Thecorrelation integral estimate of

the attractor dimension from this analysis should approach the fractal

dimension asymptotically from below. For the Rossler attractor this requires

that the estimation be 99% of the actual dimension in order _{to accurately}

indicate the need for three coordinates to describe the attractor. Most

combinations of embedding delay and _{secondary sampling} yielded

convergencetoadimension estimatejust lessthantwo.

Secondary

To further reduce the effects of _{any transients,} all points used for the

analysis were taken from the end of each data set. Seven secondary

sampling schemes were used, every point, every second _{point, every third}

point, every fifth point, every tenth point, every

25th

point, and _every

point. Even at_every 50th

point,this leftmore than 50,000 pointsfortransients

tosettle out in the Lorenz data.

Embedding

Four different embedding delay fractions were used at each _secondary

sampling rate. _{The embedding delay} fractions were _one, one _half, one _fifth,

and one tenth of eitherthefirst local minimum or zero. In two cases this led

to non-convergence, and in one _{case, the duplication} of results. In the 28

cases which were run, 26 resulted in normal termination. The duplication

caused _by the _{embedding delay} selection resulted in there _being