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

The mean flow in the planetary atmospheric boundary layer (ABL) possesses an upward momentum flux of surface frictional origin. This turbulence scale upward momentum flux is progressively amplified by the exponential decrease of atmospheric density with height coupled with buoyant energy generation in microscale fractional condensation by deliquescence on hygroscopic nuclei even in an unsaturated environment. The incessant upward momentum flux generates helical vortex roll (large eddy) circulations manifested as cloud rows/streets and meso-scale (~100 kms) cloud clusters (MCC) in global cloud cover pattern. Townsend (1956) has shown that the spatial integration of inherent turbulent eddies gives rise to large eddy circulations. The root mean square (r.m.s) circulation speed *W* of the large eddy of radius *R* is therefore expressed in terms of the r.m.s. circulation speed w_{*} of dominant turbulent eddy of radius *r* as follows.

(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)

Therefore (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: