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Signatures of Quantum-like Chaos in Dow Jones Index and Turbulent Fluid Flows

A. M. Selvam

(Retired) Indian Institute of Tropical Meteorology, Pune 411 005, India

email: selvam@ip.eth.net

web site: http://amselvam.tripod.com/index.html

http://www.geocities.com/~amselvam

Abstract

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.

1.    Introduction

    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 19th 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=axb where a and b are constants (Thompson,1963; Strathmann, 1990; Jean, 1994; Stanley, Amaral, Buldyrev, Goldberger et al., 1996). This type of scaling implies a hierarchy of substructures and was used by D’Arcy Thompson for scaling anatomical structures, for example, how proportions tend to vary as an animal grows in size (West, 1990a). D’Arcy Thompson (1963, first published in 1917) in his book On Growth and Form has dealt extensively with similitude principle for biological modelling. Rapid advances have been made in recent years in the fields of biology and medicine in the application of scaling (fractal) concepts for description and quantification of physiological systems and their functions (Goldberger, Rigney and West, 1990; West, 1990a,b; Deering and West,1992; Skinner,1994; Stanley, Amaral, Buldyrev, Goldberger et. al., 1996). In meteorological theory, the concept of selfsimilar fluctuations was identified  and  introduced in the description of turbulent flows by Richardson (1965, originally published in 1922; see also Richardson, 1960), Kolmogorov (1941,1962), Mandelbrot (1975) (Kadanoff 1996) and others (see Monin and Yaglom ,1975 for a review).
    Self-organized criticality implies long-range space-time correlations or non-local connections in the spatially extended dynamical system. The physics underlying self-organized criticality is not yet identified. Prediction of the future evolution of the dynamical system requires precise quantification of the observed self-organized criticality. The author has developed a general systems theory (Capra, 1996 ) which predicts the observed self-organized criticality as a signature of quantum-like chaos in the macro-scale dynamical system (Mary Selvam, 1990; Mary Selvam, Pethkar and Kulkarni, 1992; Selvam and Fadnavis, 1998). The model also provides universal and unique quantification for the observed self-organized criticality in terms of the statistical normal distribution.
    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 self-organized criticality which is a signature of quantum-like chaos. Earlier studies by the author have identified quantum-like chaos exhibited by dynamical systems underlying the observed fractal fluctuations of the following data sets: (1) time series of meteorological parameters (Mary Selvam, Pethkar and Kulkarni,1992; Selvam and Joshi, 1995; Selvam et al.,1996; Selvam and  Fadnavis, 1998). (2) spacing intervals of adjacent prime numbers (Selvam and  Suvarna Fadnavis, 1998; Selvam, 2001a) (3) spacing intervals of adjacent non-trivial zeros of the Riemann zeta function (Selvam, 2001b)

2.    A General Systems Theory for Universal Quantification of Fractal Fluctuations of Dynamical Systems

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 R  form as envelopes enclosing small eddies of r.m.s circulation speed w*  and radius such that

(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 t .
    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 Central Limit Theorm (Ruhla, 1992). Therefore, the variance represents the probability densities. Such a result that the additive amplitudes of the eddies, when squared, represent the probabilities is an observed feature of the subatomic dynamics of quantum systems such as the electron or photon (Maddox 1988a, 1993; Rae, 1988 ). The fractal space-time fluctuations exhibited by dynamical systems are signatures of quantum-like mechanics. The cell dynamical system model provides a unique quantification for the apparently chaotic or unpredictable nature of such fractal fluctuations ( Selvam and Fadnavis, 1998). The model predictions for quantum-like chaos of dynamical systems are as follows.

(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) En in the dominant wavebands will be given by the relation

En=Ts(2+t )t n

(2)


where t is the golden mean equal to (1+Ö 5)/2 [@ 1.618] and Ts , the primary perturbation length scale. Considering the most representative example of turbulent fluid flows, namely, atmospheric flows, Ghil(1994) reports that the most striking feature in climate variability on all time scales is the presence of sharp peaks superimposed on a continuous background.

    The model predicted periodicities (or length scales) in terms of the primary perturbation length scale units are 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 -1 to 11. Peridicities close to model predicted have been reported in weather and climate variability (Burroughs 1992; Kane 1996).

(d) The ratio r/R also represents the increment dq  in phase angle q (Equation 1 ).  Therefore the phase angle q represents the variance. Hence, when the logarithmic spiral is resolved as an eddy continuum in conventional spectral analysis, the increment in wavelength is concomitant with increase in phase (Selvam and Fadnavis, 1998). Such a result that increments in wavelength and phase angle are related is observed in quantum systems and has been named 'Berry's phase' (Berry 1988; Maddox 1988b; Simon et al., 1988; Anandan, 1992). The relationship of angular turning of the spiral to intensity of fluctuations is seen in the tight coiling of the hurricane spiral cloud systems.

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 k is equal to 1/t2(@0.382) and is identified as the universal constant for deterministic chaos in fluid flows (Selvam and Fadnavis, 1998).The steady state emergence of fractal structures is therefore equal to

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 Von Karman ’s constant has the value equal to 0.38 as determined from observations (Wallace and Hobbs, 1977).

    In Equation 3, W represents the standard deviation of eddy fluctuations, since W  is computed as the instantaneous r.m.s. ( root mean square) eddy perturbation amplitude with reference to the earlier step of eddy growth. For two successive stages of eddy growth starting from primary perturbation w*  the ratio of the standard deviations Wn+1 and Wn is given from Equation 3 as (n+1)/n. Denoting by s the standard deviation of eddy fluctuations at the reference level (n=1) , the standard deviations of eddy fluctuations for successive stages of eddy growth are given as integer multiple of s , i.e., s , 2s , 3s , etc., and correspond respectively to

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 50 for the mean value of the distribution. Since the logarithm of the wavelength represents the r.m.s. value of eddy fluctuations the normalized standard deviation t is defined for the eddy energy as

(6)


where L is the period in years and T50 is the period up to which the cumulative percentage contribution to total variance is equal to 50 and t = 0. The variable logT50 also represents the mean value for the r.m.s. eddy fluctuations and is consistent with the concept of the mean level represented by r.m.s. eddy fluctuations. Spectra of time series of fluctuations of dynamical systems, for example, meteorological parameters, when plotted as cumulative percentage contribution to total variance versus t  follow the model predicted universal spectrum (Selvam and Fadnavis, 1998, and all  references therein). The literature shows many examples of pressure, wind and temperature whose shapes display a remarkable degree of universality (Canavero and Einaudi,1987).

    The periodicities (or length scales) T50 and T95 up to which the cumulative percentage contribution to total variances are respectively equal to 50 and 95 are computed from model concepts as follows.

    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 corresponding to cumulative percentage probability densities P equal to 50 and 95 respectively are equal to 0 and 2 from statistical normal distribution characteristics. Since t represents the eddy growth step n (Equation 5) the dominant periodicities (or length scales) T50 and T95 up to which the cumulative percentage contribution to total variance are respectively equal to 50 and 95 are obtained from Equation 2 for corresponding values of n equal to 0 and 2. In the present study of fractal fluctuations of Dow Jones Index, the primary perturbation length scale Ts is equal to unit time interval (days, months or years) and T50 and T95 are obtained as

T50 = (2+t )t0 @ 3.6 unit time interval

(7)
T95 = (2+t )t2 @ 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.

3.    Data and Analysis

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 11th 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.

3.1    Fractal Nature of Dow Jones Index Fluctuations on time scales of days to years

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



3.2    Continuous Periodogram Power Spectral Analyses

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 Lm which increase geometrically as Lm=2 exp(Cm) where C=.001 and m=0, 1, 2,....m . The data series Yt for the N data points was used. The periodogram estimates the set of Amcos(2pnmS-fm) where Am, nm and fm denote respectively the amplitude, frequency and phase angle for the mth periodicity and S is the time interval in days, months or years. The cumulative percentage contribution to total variance was computed starting from the high frequency side of the spectrum. The period T50 at which 50% contribution to total variance occurs is taken as reference and the normalized standard deviation tm values are computed as (equation 6) .

tm = (log Lm / log T50)-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 t  as given above. The period L is in time interval units (days, months or years). Periodicities up to T50 contribute up to 50% of total variance. The phase spectra were plotted as cumulative percentage normalized (normalized to total rotation) phase .

3.3    Power Spectral Analyses: Representative Variance and Phase Spectra

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


 
 



 3.4    Power Spectral Analyses: Summary of Results

The periodicities t50 up to which the cumulative percentage contribution to total variance is equal to 50 are shown for the three groups of Dow Jones Index data sets, namely, daily (115 data sets), monthly (11 data sets) and annual (5 data sets) in Figures 4 (a, b and c) respectively.

Figure 4a: The period t50 up to which the cumulative percentage contribution to total variance is equal to 50 for 115 data sets of normalised daily Dow Jones Index. Details are also given of data set length, data mean and data standard deviation
 
 




Figure 4b: The period t50 up to which the cumulative percentage contribution to total variance is equal to 50 for 11 data sets of normalised monthly Dow Jones Index. Details are also given of data set length, data mean and data standard deviation
 




Figure 4c: The period t50 up to which the cumulative percentage contribution to total variance is equal to 50 for 5 data sets of normalised annual Dow Jones Index. Details are also given of data set length, data mean and data standard deviation
 



    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 - 600, 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 -1 to 11. Class interval-wise percentage frequency of occurrence of dominant periodicities were computed. In each class interval, the number of dominant statistically significant (less than or equal to 5%) periodicities and also the number of dominant wavebands which exhibit Berry's phase (variance and phase spectra are the same) are computed as  percentages of the total number of dominant wavebands in each class interval. The class interval-wise mean and standard deviation of the above computed frequency distribution of dominant periodicities, significant dominant periodicities and dominant periodicities exhibiting Berry's phase (see Section 2) were then computed for the three groups of data sets of daily (115 data sets), monthly (11 data sets) and annual (5 data sets) Dow Jones Index time series. The average class interval-wise distribution of dominant periodicities, significant dominant periodicities and dominant periodicities exhibiting Berry's phase respectively  are shown in Figures 5, 6 and 7.



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 5c: 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


4.    Results and Discussion

The Dow Jones Index time series (daily, monthly and annual) exhibit fractal fluctuations (Figures 1a, b, c) 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.
  • (1) The variance spectra follow statistical normal distribution for all the three (daily, monthly and annual) data groups except for the first data set of length 20 years (1900 to 1919) annual mean Dow Jones Index time series.
  • (2) Phase spectra follow normal distribution for 66% of daily data sets and for 100% data sets of the monthly and annual data groups.
  • (3) The period t50 up to which the cumulative percentage contribution to total variance is equal to 50% is very close to model predicted value of about 3.6 time units (daily, monthly or annual) for annual and monthly data sets. The daily data sets showed higher values for data set numbers 81 to 108. Incidentally these data sets 81 to 108 correspond to the period just prior to and following the oil shock of the year 1973 (Chen 1996a,b).Though the data length varied from 100 to 10000 for daily data sets, the value of t50 was relatively constant and close to the model predicted values (Figures 4a, b, c).
  • (4) The power spectra exhibit dominant wavebands with peak periodicities close to model predicted values (Equation 2). The average class interval-wise distribution of dominant and also dominant significant peak periodicities for all the three data groups (Figures 5a,b; 6a,b; 7a,b) show a maximum for the shorter periodicities up to 4 - 6 time units (days, months or years). This result is consistent with observed and model predicted value of t50 equal to 3.6 time units (equation ) as shown in item (3) above.
  • (5) Berry's phase (Section 2), namely, the variance spectra following closely the phase spectra is exhibited for almost all dominant wavebands for longer periodicities (Figures 5c; 6c; 7c) the 'goodness of fit' (chi-square test) being significant at less than or equal to 5%.
  • Conclusions

    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 wavebandsgive the following results. (a) The periodicities t50 up to which the cumulative percentage contribution to total variance is equal to 50 are close to model predicted value of 3.6  for the data groups of widely different time scale units (days, months, years) and different data lengths ( 100 to 10000 for daily data sets). (b) A majority of dominant wavebands occur within the shorter period class interval up to 5 time scale interval units consistent with observed and model predicted value that periodicities up to 3.6 time interval units alone contribute up to 50% of the total variance. (c) Variance and phase spectra are the same (Berry's phase in quantum mechanics) in a majority of dominant wave bands for larger periodicities. The dominant wave bandwidth increases progressively with period length (Section 2).
        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 Penrose tiling pattern for the internal structure. Conventional power spectral analyses resolves such a logarithmic spiral geometry as an eddy continuum with embedded dominant wavebands, the peak periodicities being functions of the golden mean and the primary perturbation time scale. Power spectral analyses of Dow Jones Index also exhibits the model predicted dominant wavebands. These dominant periodicities are intrinsic to the selfsimilar fractal fluctuations (space-time) of dynamical systems in general. Such cycles are referred to as business cycles or endogeneous cycles in market economics.

    Acknowledgements

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

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