Dow Jones Index time series exhibit irregular or *fractal* fluctuations on all time scales from days, months to years. The apparently irregular (nonlinear) fluctuations are selfsimilar as exhibited in inverse power law form for power spectra of temporal (spatial) fluctuations. Inverse power law form for power spectra of *fractal* fluctuations is generic to all spacially extended dynamical systems in nature and is identified as *self-organized criticality*. Selfsimilarity implies long-range space-time correlations or non-local connections. It is important to quantify the total pattern of *fractal* fluctuations for predictability studies, e.g., weather and climate prediction, stock market trends, etc. The author has developed a general systems theory for universal quantification of the observed *self-organized criticality* in dynamical systems. The model predictions are as follows. (1) The power spectra of *fractal* fluctuations follow the universal and unique inverse power law form of the statistical normal distribution. (2) The non-local connections or long-range correlations exhibited by the *fractal* fluctuations are signatures of quantum-like chaos in macroscopic spatially extended dynamical systems. (3) The apparently irregular geometry of the *fractal* fluctuations forms the component parts of a unified whole precise geometrical pattern of the logarithmic spiral with quasiperiodic *Penrose* tiling pattern for the internal structure. Conventional power spectral analyses will resolve the logarithmic spiral pattern as an eddy continuum with progressive increase in eddy phase angle. (4) The eddy continuum exhibit dominant periodicities which are functions of the golden mean (**@** 1.618) and the numerical value *T* of the primary perturbation time period (days, months years). The numerical values of the first *13* dominant cycles for unit time scale (days, months or years) for the primary perturbation time period, i.e., *T* *= 1*, are *2.2*, *3.6*, *5.8*, *9.5*, *15.3*, *24.8*, *40.1*, *64.9*, *105.0*, *170.0*, *275.0*, *445.0*, *720.0* . These dominant periodicities are robust cycles inherent to dynamical systems and are independent of all other factors except the numerical value of the primary perturbation cycle time period *T*. Examples of persistent primary perturbation cycles in nature are, the daily cycle of solar heating of atmospheric flows, the day- to- day stock market trading cycle, etc . (5) Periodicities up to about *3.6* time scale units (days, months or years) contribute up to 50% of the total variance.

Continuous periodogram power spectral analyses of normalised daily, monthly and annual Dow Jones Index for the past 100-years show that the power spectra follow the universal inverse power law form of the statistical normal distribution in agreement with model prediction. The fractal fluctuations of Dow Jones Index therefore exhibit *self-organized criticality* which is a signature of quantum-like chaos on all time scales from days to years. All the data sets exhibit periodicities close to model predicted values of about 2.2, 3.6, etc., time units in days, months, years respectively for data sets of daily, monthly and annual values. Also 50% contribution to total variance is contributed by periodicities up to the model predicted value of about *3.6* time units in days, months, years. The apparently noisy, irregular *fractal* fluctuations of dynamical systems such as the Dow Jones Index in the present study, contribute to the formation of robust selfsimilar space-time geometrical structures which indicate the surprising resilience of market economy over a period of 100 years.

Irregular (nonlinear) fluctuations on all scales of space and time are generic to dynamical systems in nature such as fluid flows, atmospheric weather patterns, heart beat patterns, stock market fluctuations, etc. Mandelbrot (1977) coined the name *fractal* for the non-Euclidean geometry of such fluctuations which have fractional dimension, for example, the rise and susequent fall with time of the Dow Jones Index traces a zig-zag line in a two-dimensional plane and therefore has a *fractal* dimension greater than one but less than two. Mathematical models of dynamical systems are nonlinear and finite precision computer realisations exhibit sensitive dependence on initial conditions resulting in chaotic solutions, identified as *deterministic chaos*. *Nonlinear dynamics and chaos* is now (since 1980s) an area of intensive research in all branches of science (Gleick, 1987). The *fractal* fluctuations exhibit scale invariance or selfsimilarity manifested as the widely documented (Bak, Tang, Wiesenfeld, 1988; Bak and Chen, 1989; 1991; Schroeder, 1991; Stanley, 1995; Buchanan,1997) inverse power law form for power spectra of space-time fluctuations identified as *self-organized criticality* by Bak *et al*. (1987). The power law is a distinctive experimental signature seen in a wide variety of complex systems. In economy it goes by the name *fat tails*, in physics it is referred to as *critical fluctuations*, in computer science and biology it is *the edge of chaos*, and in demographics it is called *Zip's law* (Newman, 2000). Power-law scaling is not new to economics. The power law distribution of wealth discovered by Vilfredo Pareto (1848-1923) in the 19^{th} century (Eatwell, Milgate and Newman, 1991) predates any power laws in physics (Farmer, 1999). One of the oldest scaling laws in geophysics is the *Omori law* (Omori, 1895). It describes the temporal distribution of the number of aftershocks which occur after a larger earthquake (i.e., mainshock) by a scaling relationship.The other basic empirical seismological law, the *Gutenberg-Richter law* (Gutenberg and Richter, 1944) is also a scaling relationship, and relates intensity to its probability of occurrence (Hooge *et. al*., 1994). Time series analyses of global market economy also exhibits power-law behaviour ( Bak *et al*., 1992; Mantegna and Stanley, 1995; Sornette *et al*., 1995; Feigenbaum and Freund, 1995; Chen, 1996a,b; Stanley *et al*., 1996; Feigenbaum and Freund, 1997a,b; Gopikrishnan *et al*., 1999; Plerou *et al*., 1999; Stanley *et al*., 2000; Feigenbaum, 2001a,b) with possible multifractal structure ( Farmer, 1999 ) and has suggested an analogy to fluid turbulence (Ghasgaie *et al*., 1996; Arneodo *et al*., 1998). The stock market can be viewed as a self-organizing cooperative system presenting power law distributions, large events in possible co-existence with synchronized behaviour (Sornette *et al*., 1995). Sornette *et al*. (1995) also conclude that the observed power law represents structures similar to '*Elliott waves*' of technical analysis first introduced in the 1930s. It describes the time series of a stock price as made of different *waves*, these *waves* are in relation to each other through the *Fibonacci* series. Sornette *et al*. (1995) speculate that '*Elliott waves*' could be a signature of an underlying critical structure of the stock market. Incidentally the *Fibonacci* series represent a *fractal* tree-like branching network of selfsimilar structures (Stewart, 1992). The commonly found shapes in nature are the helix and the dodecahedron (Stoddart, 1988; Muller and Beugholt,1996) which are signatures of selfsimilarity underlying *Fibonacci* numbers. The general systems theory presented in this paper shows (Section 2) that *Fibonacci* series underlies fractal fluctuations on all space-time scales. Chen (1996b) has identified color chaos and persistent cycles with characteristic period of around three to four years in time series analyses of *Standard and Poor* stock price indices. * * Historically, basic similarity in the branching (fractal) form underlying the individual leaf and the tree as a whole was identified more than three centuries ago in botany (Arber,1950). The branching (bifurcating) structure of roots, shoots, veins on leaves of plants, etc., have similarity in form to branched lighting strokes, tributaries of rivers, physiological networks of blood vessels, nerves and ducts in lungs, heart, liver, kidney, brain ,etc. (Freeman, 1987; 1989; Goldberger *et al*., 1990; Jean, 1994; ). Such seemingly complex network structure is again associated with *Fibonacci* numbers seen in the exquisitely ordered beautiful patterns in flowers and arrangement of leaves in the plant kingdom (Jean, 1994; Stewart, 1995). The identification of physical mechanism for the spontaneous generation of mathematically precise, robust spatial pattern formation in plants will have direct applications in all other areas of science (Mary Selvam, 1998). The importance of scaling concepts were recognized nearly a century ago in biology and botany where the dependence of a property *y* on size *x* is usually expressed by the allometric equation *y=ax ^{b}* where

Continuous periodogram power spectral analyses of Dow Jones Index time series of widely different time scales (days, months, years) and data lengths (100 to 10000 in the case of daily data sets) agree with model prediction, namely, the power spectra follow the universal inverse power law form of the statistical normal distribution. Dow Jones Index time series therefore exhibit

As mentioned earlier (Section 1: Introduction) power spectral analyses of *fractal* space-time fluctuations exhibits inverse power law form, i.e., a selfsimilar eddy continuum. The *cell dynamical system model* (Mary Selvam, 1990; Selvam and Fadnavis, 1998, and all references contained therein; Selvam, 2001a, b) is a general systems theory (Capra, 1996) applicable to dynamical systems of all size scales. The model shows that such an eddy continuum can be visualised as a hierarchy of successively larger scale eddies enclosing smaller scale eddies. Eddy or wave is characterised by circulation speed and radius. Large eddies of root mean square (r.m.s) circulation speed ** W** and radius

(1)

Large eddies are visualised to grow at unit length step increments at unit intervals of time, the units for length and time scale increments being respectively equal to the enclosed small eddy perturbation length scale ** r** and the eddy circulation time scale

Since the large eddy is but the average of the enclosed smaller eddies, the eddy energy spectrum follows the statistical normal distribution according to the

(a) The observed *fractal* fluctuations of dynamical systems are generated by an overall logarithmic spiral trajectory with the quasiperiodic *Penrose tiling pattern* (Nelson, 1986; Selvam and Fadnavis, 1998) for the internal structure.

(b) Conventional continuous periodogram power spectral analyses of such spiral trajectories will reveal a continuum of periodicities with progressive increase in phase.

(c) The broadband power spectrum will have embedded dominant wave-bands, the bandwidth increasing with period length. The peak periods (or length* *scales) ** E_{n}** in the dominant wavebands will be given by the relation

*E _{n}=T_{s}(2+*t )t

(2)

where **t** is the ** golden mean** equal to

The model predicted periodicities (or length scales) in terms of the primary perturbation length scale units are *2.2*, *3.6*, *5.8*, *9.5*, *15.3*, *24.8*, *40.1*, *64.9*, *105.0*, *170.0*, *275.0*, *445.0 *and* 720.0* respectively for values of ** n** ranging from

(d) The ratio ** r/R** also represents the increment

The overall logarithmic spiral flow structure is given by the relation

(3)

where the constant ** k **is the steady state fractional volume dilution of large eddy by inherent turbulent eddy fluctuations . The constant

*1/k @2.62*

(4)

The model predicted logarithmic wind profile relationship such as Equation 4 is a long-established (observational) feature of atmospheric flows in the atmospheric boundary layer, the constant ** k**, called the

In Equation 3, ** W** represents the standard deviation of eddy fluctuations, since

**statistical normalized standard deviation t=0,1,2,3, etc.**

(5)

The conventional power spectrum plotted as the variance versus the frequency in log-log scale will now represent the eddy probability density on logarithmic scale versus the standard deviation of the eddy fluctuations on linear scale since the logarithm of the eddy wavelength represents the standard deviation, i.e., the r.m.s. value of eddy fluctuations (Equation 3). The r.m.s. value of eddy fluctuations can be represented in terms of statistical normal distribution as follows. A normalized standard deviation ** t=0** corresponds to cumulative percentage probability density equal to

(6)

where ** L** is the period in years and

The periodicities (or length scales) ** T_{50}** and

The power spectrum, when plotted as normalised standard deviation ** t** versus cumulative percentage contribution to total variance represents the statistical normal distribution (Equation 6), i.e., the variance represents the probability density. The normalised standard deviation values

*T _{50} = (2+t )t^{0 }@ 3.6 unit time interval*

(7)

*T _{95} = (2+t )t^{2 }@ 9.5 unit time interval*

(8)

The above model predictions are applicable to all real world and computed model dynamical systems. Continuous periodogram power spectral analyses of Dow Jones Index of widely different time scales and data lengths give results in agreement with the above model predictions.

Dow Jones Index values were obtained from Dow Jones Industrial Average History File, Dow Jones closing prices starting in 1900: 3 Jan 1900 to 5 June 2000 (27523 trading days). Data from: Department of Statistics at Carnegie Mellon Univ., (http://www.stat.cmu.edu/cmu-stats) Quote.com (http://quote.com) Yahoo! (http://quote.yahoo.com). The normalised day- to- day changes in the Dow Jones Index values were computed as percentages of the earlier day value. Monthly and annual mean values were then computed from the normalised day- to- day changes in the Dow Jones Index. A total of 27,500 daily values of normalised Dow Jones Index were used for the study. Starting from the Dow Jones Index values on day numbers 1, 10001, and 20001 respectively, the number of days used for the spectral analyses were in increments of 100 days up to 2500 days (25 data sets) and thereafter, in increments of 500 days till 10000 days (15 data sets) giving a total of 115 data sets. A total of 1200 monthly mean values of Dow Jones Index were available for the study. A total of 11 data sets were subjected to spectral analyses. Starting from the first month, the number of months used for the spectral analyses for the first 10 data sets were in increments of 100 months till 1000 months and the 11^{th} data set contains 1200 months. A total of 100 annual mean Dow Jones Index values were available for the study. Power spectral analyses were done for 5 data sets. Starting from the first year, the number of years used for the 5 data sets were in increments of 20 years.

The normalised daily, monthly and annual fluctuations of Dow Jones Index exhibit irregular *fractal* fluctuations as shown for representative data sets in Figure 1.

Figure 1: *Fractal* fluctuations of normalised daily, monthly and annual fluctuations in Dow Jones Index for the 100-year (1900 to 1999) data set

The broadband power spectrum of space-time fluctuations of dynamical systems can be computed accurately by an elementary, but very powerful method of analysis developed by Jenkinson (1977) which provides a quasi-continuous form of the classical periodogram allowing systematic allocation of the total variance and degrees of freedom of the data series to logarithmically spaced elements of the frequency range (*0.5*, *0*). The periodogram is constructed for a fixed set of *10000*(** m**) periodicities

*t _{m} = (log L_{m} / log T_{50})-1*

The cumulative percentage contribution to total variance, the cumulative percentage normalized phase (normalized with respect to the total phase rotation) and the corresponding ** t** values were computed. The power spectra were plotted as cumulative percentage contribution to total variance versus the normalized standard deviation

The variance and phase spectra along with statistical normal distributions are shown in Figure 2 for representative data sets of normalised daily, monthly and annual Dow Jones Index. The 'goodness of fit' (statistical chi-square test) between the variance spectra and statistical normal distribution is significant at less than or equal to 5% level for all the daily and monthly spectra. In the case of annual data sets, the variance spectra follows normal distribution for all data sets except for the first set consisting of the first 20-years (1900 to 1919). Phase spectra are close to the statistical normal distribution, with the 'goodness of fit' being statistically significant for all monthly and annual data sets and 66% of daily data sets. Further, in all the cases, the 'goodness of fit' between variance and phase spectra are statistically significant (chi-square test) for individual dominant wavebands, in particular, for longer periodicities. A representative example of daily data set is shown, where, though the phase spectrum does not follow normal distribution (Figure 2), the phase and variance spectra are the same in dominant wavebands ( Figure 3).

Figure 2: Representative spectra of variance and phase along with statistical normal distribution for normalised daily, monthly and annual fluctuations of Dow Jones Index. The variance spectra for all data sets and phase spectra for monthly and annual data sets follow the model predicted statistical normal distribution . The phase spectrum does not follow the normal distribution for the sample *daily* data set, but the variance and phase spectra are the same in individual dominant eddies as shown in Figures 3 for the same data set

Figure 3: A representative example of *daily* data set is shown, where, though the phase spectrum does not follow normal distribution (Figure 2), the phase and variance spectra are the same in dominant wavebands as shown below.The variance and phase spectra being the same is a signature of *Berry*'s phase in quantum systems

The periodicities ** t_{50}** up to which the cumulative percentage contribution to total variance is equal to

Figure 4a: The period ** t_{50}** up to which the cumulative percentage contribution to total variance is equal to

Figure 4b: The period ** t_{50}** up to which the cumulative percentage contribution to total variance is equal to

Figure 4c: The period ** t_{50}** up to which the cumulative percentage contribution to total variance is equal to

The power spectra exhibit dominant wavebands where the normalised variance is equal to or greater than *1*. The dominant peak periodicities were grouped into class intervals *2 - 3*, *3 - 4*, *4 - 6*, *6 - 12*, *12 - 20*, *20 - 30*, *30 - 50*, *50 - 80*, *80 - 120*, *120 - 200*, *200 - 300*, *300 - 6000*, *600 - 1000*, and *greater than 1000* . These class intervals include the model predicted (Equation 2) dominant peak periodicities (or length scales) *2.2*, *3.6*, *5.8*, *9.5*, *15.3*, *24.8*, *40.1*, *64.9*, *105.0*, *170.0*, *275.0*, *445.0*, *720.0*, (in days, months or years) for values of ** n** ranging from

Figure 5: Class interval-wise average percentage frequency of occurrence of dominant periodicities for daily (115 data sets), monthly (11 data sets) and annual (5 data sets) of normalised Dow Jones Index

Figure 6: Class interval-wise average percentage frequency of occurrence of significant dominant periodicities for daily (115 data sets), monthly (11 data sets) and annual (5 data sets) of normalised Dow Jones Index. The number of dominant statistically significant (less than or equal to 5%) periodicities are computed as percentages of the total number of dominant wavebands in each class interval.

Figure 7: Class interval-wise average percentage frequency of occurrence of dominant wavebands which exhibit *Berry*'s phase (variance and phase spectra are the same) for daily (115 data sets), monthly (11 data sets) and annual (5 data sets) of normalised Dow Jones Index. The number of dominant wavebands which exhibit *Berry*'s phase are computed as percentages of the total number of dominant wavebands in each class interval

The Dow Jones Index time series (daily, monthly and annual) exhibit *fractal* fluctuations (Figure 1) generic to dynamical systems in nature. The *fractal* fluctuations are basically a zig-zag pattern of successive upward and downward swings on all time scales in the Dow Jones Price Index. Such irregular fluctuations may be visualised to result from the superimposition of a continuum of eddies. Power spectral analysis is commonly applied to resolve the component periodicities and their phases. Continuous periodogram power spectral analyses of the *fractal* fluctuations in Dow Jones Index time series (daily, monthly and annual) follow closely the following model predictions given in Section 2.

The observed inverse power law form for power spectra of *fractal* fluctuations is a signature of* self-organized criticality* in dynamical systems. The author had shown earlier (Selvam and Suvarna Fadnavis, 1998; selvam 2001a,b) that (a) *self-organized criticality* can be quantified in terms of the universal inverse power law form of the statistical normal distribution and (b) *self-organized criticality* of selfsimilar *fractal* fluctuations implies long-range space-time correlations and is a signature of quantum-like chaos in macro-scale dynamical systems of all space-time scales.

Power spectra of normalised daily, monthly and annual fluctuations of Dow Jones Index time series follow the model predicted universal and unique inverse power law form of the statistical normal distribution. Inverse power law form for power spectra of temporal fluctuations imply long-range temporal correlations, or in other words, persistence or long-term memory of short-term fluctuations. The long-time period fluctuations carry the signatures of short-time period fluctuations. The cumulative integration of short-term fluctuations generates long-term fluctuations (eddy continuum) with two-way ordered energy feedback between the fluctuations of all time scales (Equation 1 ). The eddy continuum acts as a robust unified whole fuzzy logic network with global response to local perturbations. Increase in random noise or energy input into the short-time period fluctuations creates intensification of fluctuations of all other time scales in the eddy continuum and may be noticed immediately in shorter period fluctuations. Noise is therefore a precursor to signal.

Real world examples of noise enhancing signal has been reported in electronic circuits (Brown, 1996). Man-made, urbanisation related, greenhouse gas induced global warming (enhancement of small-scale fluctuations) is now held responsible for devastating anomalous changes in regional and global weather and climate in recent years (Selvam and Fadnavis, 1998).

Average class interval-wise distribution of wavebands give the following results. (a) The periodicities *t _{50}* up to which the cumulative percentage contribution to total variance is equal to

The apparently irregular fractal fluctuations of the Dow Jones Index as a representative example in this study and dynamical systems in general, self-organize spontaneously to generate the robust geometry of logarithmic spiral with the quasiperiodic

The author is grateful to Dr. A. S. R. Murty for his keen interest and encouragement during the course of this study.