The observed long-range
spatiotemporal correlations of real world dynamical systems is governed
by quantumlike mechanics with inherent non-local connections. In summary,
microscopic scale local fluctuations form a unified self-organized adaptive
network manifested as the macro-scale dynamical system with implicit ordered
energy flow between the larger and smaller scales. Such a concept of ladder
networks may find applications in the design of artificial intelligence
systems.

Long-range spatiotemporal
correlations manifested as the self-similar fractal geometry to the spatial
pattern concomitant with inverse power law form for the power spectrum
of temporal fluctuations are ubiquitous to real world dynamical systems
and such non-local connections are now identified as signatures of self-organized
criticality (Bak, Tang and Wiesenfeld, 1988) or deterministic chaos. The
physics of deterministic chaos is not yet identified. A striking example
of macro-scale dynamical system exhibiting the signatures of deterministic
chaos is the planetary atmospheric boundary layer atmospheric flow
structure where, the co-operative existence of fluctuations ranging in
size from the planetary scale of thousands of kilometers to the turbulence
scale of a few millimeters gives rise to coherent weather systems with
long-range spatiotemporal correlations such as the El-Nino/Southern Oscillation
cycle of period 2-7 years marked by episodes of abnormal warming off the
coast of Peru associated with devastating changes in the global climate
pattern (Mary Selvam, 1990). The recently identified self-similar fractal
geometry to the global cloud cover pattern and the inverse power law form
for the atmospheric eddy energy spectrum (Lovejoy and Schertzer, 1986)
are signatures of deterministic chaos in real world atmospheric flows.
A cell dynamical system model for atmospheric flows (Mary Selvam, 1990)
applicable to real world dynamical systems shows that quantumlike mechanics
govern atmospheric flow structure and is manifested as the observed long-range
spatiotemporal correlations. In summary, the following model predictions
are applicable to all real world dynamical systems: (1) the energy flow
structure in macro-scale dynamical systems consists of a nested continuum
of vortex roll (large eddy) circulations with overall logarithmic spiral
envelope enclosing internal circulations tracing the quasiperiodic Penrose
tiling pattern such that short-range energy circulation balance requirements
impose long-range orientational order in the spatial pattern, (2) The model
envisages the co-operative existence of a continuum of fluctuations with
ordered energy flow between the larger and smaller scales resulting in
the mixing of the environment into the macro-scale dynamical system. (3)
The universal constant *k* for deterministic chaos is identified as
the steady state fractional volume dilution of the macro-scale dynamical
system by inherent small-scale spatiotemporal fluctuations. The value of
*k*
is equal to 1/*t ^{2}*
( @ 0.382) where

Continuous periodogram analysis of the time series of 115 years (1871-1985) summer monsoon (June-September) rainfall over the Indian region show that the power spectra of the temporal fluctuations are the same as the normal distribution with the square of the eddy amplitude representing the eddy probability density corresponding to the normalized standard deviation

(1)

The growth of large eddy circulations from
turbulence scale buoyant energy generation therefore occurs in unit length
step increments in unit intervals of time, the turbulence scale yardsticks
for length and time being used. Such a concept of large eddies as the macroscale
envelope of a self-sustaining network of small scale circulations is analogous
to the concept of 'cellular automata' computational technique where the
macroscale dynamical system is assumed to consist of identical unit cells
with arbitrary rules for evolution of the ensemble (Oona and Puri, 1988).
The cellular automata computational technique described in this paper for
growth of large eddies from microscopic domain turbulent fluctuations is
based on the governing Equation 1 which is physically consistent and mathematically
rigorous. Further, the growth of large eddies by successive length step
increments equal to the turbulent eddy length scale doubling is identified
as the universal period doubling route to chaos. Such a concept envisages
the growth of an eddy continuum starting from the turbulence scale with
the power spectrum of the temporal fluctuations following the inverse power
law form which is a signature of deterministic chaos. Equation 1 therefore
implies a two-way ordered energy flow between the larger and smaller scales
and is a statement of the law of conservation of energy for the dynamical
system, namely atmospheric flows. In summary, the spatio-temporal growth
of dynamical systems in general occurs by the propagation of inherent small
scale fluctuations which are sustained by energy released from the medium
of propagation during stretching. The energy circulation pattern in a dynamical
system consists of a continuum of vortices within vortices. Equation 1
is hereby identified as the universal algorithm for deterministic chaos
in real world dynamical systems. Computations show that the successive
values of the circulation speed (2)

Since the steady state fractional volume dilution
of large eddy by inherent turbulent eddy fluctuations during successive
length step increments is equal to (3)

where The variables

(4)

Starting with reference level standard deviation
s
equal to (5)

The constant (6)

(7)

Equation 7 is in agreement with Delbourgo's
(1986) results. Further, the universal algorithm for deterministic chaos
at Equation 1 can now be reformulated in terms of D' Amico, A., M. Faccio and G. Ferri, 1990:

Delbourgo, R., 1986:

Feigenbaum, M. J., 1980:

Gleick, J., 1987:

Grossing, G., 1989:

Jenkinson, A. F., 1977: Met. O 13 Branch Memorandum No. 57, 1-23.

Lovejoy, S., and D. Schertzer, 1986:

Mary Selvam, A., 1990: Can. J. Phys.

Oona, Y., and S. Puri, 1988:

Parthasarathy, B., N. A. Sontakke, A. A. Munot and D. R. Kothawale, 1987:

Philander, G., 1989:

Stone, E. F., 1990:

Townsend, A. A., 1956: