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

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

(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)

(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.

Figure 1:

*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

Figure 3: A representative example of

Figure 4b: The period

Figure 4c: The period

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

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

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

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