IV. OBSERVATION AND CHARACTERISATION OF CHAOS

In this chapter we analyse the parametervalues for which the map is chaotic. To start with,we compute the power spectrum of MLM for one such p

value, say 0.98, and compare it with that of thelogistic map (see fig. ll). The computation wasperformed using FFT with n = 512. The broad band

spectrum confirms that Xt and ht evolve in an appa­rently chaotic manner for p = 0.98. There are suchuncountable number of parameter values in the range

p@§p<l for which the map is chaotic. In order tostudy the behaviour of the map at these u values indetail, we make use of certain quantitative measureswhich have been successfully used to characterisechaos in nonlinear deterministic systems. We mainlyconcentrate on the Lyapunov exponent (LE) and fractaldimension which are quite general and are applicableto a wide class of problems.

Lyapunov exponents of the map

There are several analytical as well asnumerical works on the LE of the logistic map. Forexample, the dependance of LE of stable and unstableperiodic orbits of the logistic map on the control

°‘ M "A H “7<A>\-5 T

Lou Powca

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-15 ; Le 1 1 I Pl0* i ‘%;;IfNM % ;::1ff %@(B)-s 1 .

L06 POWER

-10 ‘-5 am la H as 1 1 1% 4' o as 1

rasuuzncv

Fig. 11 _The power spectrum for the (A) modulatedand (B) ordinary logistic map for theparameter value u = 0.98.

: 88 :

Taking the product of the Jacobian matrices J(Xi, Xi)at N iterates of the map and letting N*>~4 the ave­rage rate of stretching or compression along the Xand X directions are given by

L1 = in I I A ( 8 (81)N*>m ; i=1 1 ‘- t 1 Nand L = llm ~ N (1) / (82)2 f u A |N~_>m ’* i=1 2

(1) (1)where Al and A2 are simply the diagonal elementsof J (Xi, hi) given by

A1(i)= 4 xi (1 “ 2 xi) (83)and A2(i)= 4“ (1 - 2 >1) (84)As we know L1 and L2 are called Lyapunov numbers

whose logarithm give the two LEs cF'1 and <y—2. Itis easy to see that the LEs of our map are independantof the initial conditions since almost all initialconditions are attracted towards a unique attractor

in the unit square. We now compute a'“1 and 0-2numerically for several values of p in the rangep = 0.84 to p = 1. The method we use has beenfrequently employed in the literature [2o,e] for thenumerical estimation of the LE.

We fix a particular parameter value for the

map. Starting from an initial value (X1, *1), we

: 89 :

iterate the map N times and calculate the Jacobianat all these N iterates. By taking the productof these N matrices, we get a 2x2 matrix, say, M.

If /$1 and /\2 are the eigen values of M, then we.. N (1) /\ .. N 1have A1 - 1:1 A1 and 2 - 1:1 A2( ).

Calculating Al and A2, we can evaluate thequantities (1/N) ln [All and (1/N) ln|/\2|.Repeating this for several N values, we plot thesequantities against N separately. The same isrepeated for other p values and the results are shownin figures 12 and 13. From the behaviour of the

graph, we see that (1/N)1n1/\1| and (1/N)1n |/\2|settle down to almost constant values as N->w,which serve as an approximate estimate of the LE andcan be directly read from the graphs.

A comparison of figures 12 and l3 reveals

some interesting features. As expected,(7-2 isnegative in the regular region and becomes positiveas p crosses over to the chaotic phase. At u=O.96,

a"2 is negative indicating the existence of aperiodic window, where the behaviour is once again

regular. As we increase u, ¢—Q increases steadily

0i5.

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0 \\\\\\\¥ ~» _{ %_' %Q66A " 0 W 0 J _ jl§4_e ,07 ~~~s ea ~ — 0 _ _+ _ V W 0- i _ A500 1000 N 1500 2000Fig. 12 An approximate numerical estimation of the

Lyapunov exponent of the MLM for the X-degreeof freedom. Note that it is negative for allvalues of p,

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Lyapunov exponent for the A-degree offreedom which is same as that of thelogistic map. As p increases to gm,and above, the LE cross over froma negative to positive value reaching

: 92 :

and reaches a value¢-Jln 2 corresponding to u=l.(There are several periodic windows in between with

negative<y-2 values which cannot be seen in ourgraph). The behaviour of 5*1 is quite different.lt is always negative irrespective of whether p isin the regular region or chaotic region. It has asmall negative value in the regular region and reachesits maximum when the system just crosses over to thechaotic state. With further increase in p, itsvalue is reduced. The reason why 6“1 is alwaysnegative may be that the X-direction is always foldedunder the mapping, as we see below.

Now, LE is a measure of the exponential

separation of nearby points in phase space and henceis proportional to the rate of loss of informationregarding the state of the system [5]. Also, thedegree of chaos in a system can be measured in termsof this rate of loss of information. The rate of lossof information contained in an infinitesimal unit cell

of phase space of the logistic map as well as the

MLM is same and is proportional to (TF2. But notethat in the case of the logistic map, this intervalis l-dimensional of length, say, E whereas in MLMthis much amount of information is lost from a square

: 93 :

of side £_. In this sense, the'modulated' logisticmap is less chaotic compared to the logistic map.It is interesting to note that this result is inagreement with the observation by Tomita [64], whohas studied the same model with p¢-l, eventhough ina different context. He used the system as anexample of unilateral chaotic modulation to show thatthe degree of chaos can be reduced by an appropriatemodulation or coupling.

The strange attractors of the map

In this section we show the existence of

strange attractors for MLM for many values of theparameter ( in fact uncountable in number, as weshall see below). In order to decide whether anattractor is periodic or chaotic, one must look atthe spectrum of LEs characterising the attractor.

We have seen that, of the two LEs 6-1 and 5-'2 ofthe map, q—1 is always negative independent of thevalue of p. This indicates that when<r'2 > 0,the attractor can possibly be "strange" due to thestretching and folding of the trajectories on theunit square. We now choose one such value, say,p = 0.895 and show the corresponding attractor infig. l4a by plotting 8000 points after the initial5000 points were discarded. Taking a small region

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Fig. 14 A plot of the attractor of MLM and itsself similar structure for p = 0.895.

: 95 :

of the attractori indicated by a box in the figureand enlarging it we get fig. l4b and repeat thisprocess once again to get fig. l4c. From the figures,it becomes clear that the attractor has a selfsimilar structure, which is in fact very much similarto that of the Henon attractor [26]. We can alsogive a simple reason to support this numerical evidencFrom the Cantor set structure of the A,-values in the

interval [0,1] for F2 > 0, it directly follows thatany two arbitrarily close points on the attractorwill have another point in between (for sufficientlylarge number of iterations) making the attractor ‘selfsimilar‘. But it is found that the self similarstructure becomes less pronounced within the limits ofnumerical precision for larger values of the parameterThe reason for this can be attributed to the observed

variation in the values of <7-'1 and Q"‘2 with p, shownin fig. 16 below.

We now present in fig. 15 the numerical plotof the attractor for six different values of p. Forp just beyond pm, the attractor consists of severaldisjoint sets and finally becomes a single piece forsufficiently large value of p. This structurevariation necessarily reflects the band merging in thelogistic map. Now, the values of p for which

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: 97 :

(y-2 > O are uncountable in number and form a setof positive measure on the parameter axis [65,57].So, in principle, the map can be said to generateuncountable number of strange attractors in theunit square apart from the infinite number ofperiodic cycles already shown. Here, we want tomake one point clear. It is seen from figure l6below that for sufficiently large values p, thesum of LEs is larger than zero implying that the areaelements grow on the average. But we call theresulting fractal set a ‘strange attractor‘ becauseit satisfies the other properties of an attractor.For example, it is a bounded region to which almostall other initial conditions in the unit squareget attracted asymptotically. Moreover, it forms aninvariant set whose every part is eventually visitedby the iterates. It should be noted that a basicfactor responsible for the existence of these‘attractors’ with growing area elements is the con­finement of the mapping to the unit square. This isdiscussed in more detail below.

Fractal and information dimensions

One important question regarding a strangeattractor is its dimension. Eventhough there are avariety of different definitions of dimension, the

: 98 :

most relevant ones are i) the fractal dimension,depending only on the metric properties and ii) theinformation dimension which depends on metric as

well as probabilistic properties. The fractaldimension of a set is given by

1.Do = 1'“ (as)£—>a log (1/€.)

where, if the set in question is a bounded subsetof a m—dimensional Euclidean space Rm, then N (€ )is the minimum number of m-dimensional cubes of side

G} needed to cover the set. The fractal dimensionis only a measure of the ‘strangeness’ of an attractorand says nothing about the dynamics on it. Fromfigure 15 it can be easily seen that some regions ofthe attractor are more frequently visited by thetrajectories than others. So, in order to understandthe dynamics on a chaotic attractor, one must alsotake into account the distribution or density ofpoints on the attractor. This is more preciselydiscussed in terms of the information dimension and

is given by

D1 = 2:0 I‘ Us _ (awlog (l/ €. )N(€)where I(€)= 2 Pi log (l/Pi) (87)i=1

and Pi is the probability contained within the ith

cube. It can be easily shown that Do )»D1, where

= 99: --6‘/Q08“if 11> »--.'-,1 '1 . 9: *2 J,,'i>" -..-' 3 -W ,-' \ J < __.

*1‘. [I-. P

the equality sign holds if the attractor is uniform.

In order to compute these quantities, wemade use of the familiar box counting algorithm[18]. The unit square was partitioned into boxesof side 5,. The number of boxes N which containedat least one point of the attractor as well as the

number of points ni contained in each box were thencounted. The probability Pi is given by nilnwhere n is the total number of points on the attractor.Calculating N(€) and I (€) for various values of £and plotting ln N (€) and I (€) separately against

ln (1/(), Do and D1 were obtained as the asymptoticslopes respectively. Our results are presented in

table ll and in all cases Do > D1 as is required.

We know that the dimension of a set depends

on its structure or distribution of points in it.For example, the dimension of a Cantor set dependsvery much on the way it is constructed [15]. Strangeattractors can exhibit a wide variety of shapes andthe complexity of these shapes will be related insome way to the relative amounts of stretching andcompression which in turn depends on the LEs. To seethis possible dependence of the dimension on the LEs,

we plot the numerically calculated values of<r-1 and5-2 as a function of the parameter p. in fig. 16.

é

parameter values

Table II Fractal and information dimensions of the

strange attractors of IflLJ4 for various

P D1

éiiti

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0.898

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0.915

0.92

0.930.94

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1.04571

1.09677

1.20536

1.4400

1.06666

1.23188

1.30435

1.33928

1.58819

0 -——___“ *__

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1.02325

1.05572

1.1500

1.4000

1.05520

1.19492

1.2500

1.32121

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Fig. 16 Lyapunov exponents qrl and 0-2 Of MLMfor various values of p. This ie enlyto show the variation of the pos}t1vevalues of 0-2 and the cqrrespondmgvalues of ¢-—l with g.

: 102 :

We have shown only a few positive values of <T“2

and the corresponding values of 0-1. In between,there are infinite number of p values with<rb< 0of which only one corresponding to the period 3

window is given. Prom the variation ofG"1 and 0'2,it becomes clear why the strange attractor becomesmore and more stretched out as p increases­

Now, it is well known that a direct

estimation of the dimension D1 of the strangeattractor can be made in terms of the LEs makinguse of the Kaplan-Yorke conjecture [66,18].However, it is found that this is not possible for .the MLM. We now show that, this is due to a typicalproperty of noninvertible maps bounded on an interval.As a first step, note from figure 16 that for larger

values of p,, Q-1 + (F2 > O which implies that thestretching is globally predominant. For anN-dimensional flow or the related (N-1)-dimensionalinvertible map to be chaotic, the largest LE should

be > O while E ¢——i < 0 [6,6'7], implying that thephase space volume must contract globally. The mainpoint is that for such systems, there is a welldefined condition in terms of the LEs which determines

: 103 :

the existence of a strange attractor. Now, fornoninvertible maps on an interval, this linkbetween the volume contraction and the exponentialdivergence of nearby trajectories is broken [6].For such systems, a bounded chaotic motion can occur

even if § 6“i > O as is evident from our map. Thereason for this depends crucially on the fundamentalproperty of transformation, namely, the noninverti­bility. While the sensitive dependance on initialconditions stretches any small initial displacementby an average stretching factor resulting in anexponential increase in the displacement, the non­invertibility helps the mapping to remain bounded inthe interval. To see how this happens in our map,note that points lying on a line parallel to theX-axis are mapped on to another line parallel to it.Moreover, for every point on the first half of theinitial line, there is a symmetric point on the secondhalf, both being mapped on to a single point, implyingcompression along the X-direction. Now, it can beeasily seen that half of the interval with A in therange O< )~<l/2 is stretched along the oppositedirection by an amount which increases with the valueof p. So, when p is sufficiently large, the wholeplane should be folded back into the unit square about

: 104 :

a line parallel to the X-axis, in order to remainbounded in it. The whole process of confinementis analogous to that in the quadratic map whichhas been discussed in detail by Berge et.al [4].In fact, they show that noninvertibility is essen­tial for a mapping of R into R to be capable ofproducing chaos. It is then evident that thisfundamental property does play a role along withthe sensitive dependance on initial conditions inthe formation of strange attractors in MLM. So,it is not surprising that a direct relationshipconnecting the fractal dimension and the LEs, suchas the one conjectured by Kaplan and Yorke, doesnot exist for our map. This is also in agreementwith the fact that the conjecture has been shown tohold only for those maps for which the phase space

volume contracts after each iteration (ggri <0) andthe contraction is, in fact, uniform everywhere[68, 6].

Finally, for a more complete physicalcharacterisation of strange attractors we requirethe knowledge of the generalised fractal dimensionsDq [17]. They can also be used to obtain a globalscaling measure of the strange attractor. The fact

: 105 :

that Do and D1 are greater than one indicatesthat MLM may be useful in studying certainhigher dimensional chaotic phenomena.