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Deterministic Chaos: A signature of Quantumlike Mechanics in Self-Organized Adaptive Network

A. Mary Selvam

Deputy Director (Retired)

Indian Institute of Tropical Meteorology, Pune 411 008, India

email: amselvam@eth.net

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

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

Proc. IEEE, NAECON 91, Dayton, U. S. A. May 20-24.

Abstract


    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.

Introduction


    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/t2  ( @ 0.382) where t  is the golden mean [(1+Ö5)/2]. (4) The universal Feigenbaum's constants (Feigenbaum, 1980) a and d are functions of k . (5) The steady state ordered emergence of fractal (broken) Euclidean geometry of the macro-scale dynamical system is quantified by the universal algorithm 2a2 = pd where the energy 2a2 released from the medium of propagation during stretching provides the spin angular momentum pd for a dominant cycle of evolution of the logarithmic spiral pattern for energy propagation, e.g. the latent heat of condensation in rising air parcels generates the strikingly spiral shaped hurricane cloud patterns. (6) The dynamical system is the macro-scale manifestation of the internal microscopic domain energy circulation networks and results in quantumlike mechanics for the macro-scale evolution. The apparent paradox of wave-particle duality of quantum mechanics for the sub-atomic domain is physically consistent in the context of atmospheric flows where formation of clouds occurs in updrafts with simultaneous dissipation of clouds in adjacent downdrafts giving rise to the observed discrete cellular geometry to individual clouds and to cloud ensembles. The bimodal (formation and dissipation) of the phenomenological form for energy manifestation is intrinsic to the bidirectional energy flow of eddy circulations. Quantumlike mechanics is therefore manifested as the commonplace occurrence of real world dynamical systems as discrete entities. In summary, quantumlike mechanics govern self-organized adaptive networks such as the ladder networks, i.e. networks formed by numerous repetitions of an elementary cell, (e.g., neural networks) with inherent quasiperiodic Penrose tiling pattern for the network with long-range spatiotemporal correlations. Recent studies on ladder networks on electrical circuits with equal magnitudes for longitudinal and transverse impedances indicate that electrical characteristics are strictly and surprisingly related to the Fibonacci numbers (D' Amico, M.Faccio and G. Ferri, 1990). The Fibonacci numbers are incorporated in the quasiperiodic Penrose tiling pattern formed by the ladder network.
    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 t equal to [(log l/ log l50) -1] where l is the period length in years and l50 represents the period up to which the cumulative percentage contribution to total variance is equal to 50. The above result, namely that power spectra of the temporal fluctuations of rainfall follow the universal and unique inverse power law form of the statistical normal distribution implies quantumlike mechanics for the dynamics of atmospheric flows and is also a signature of deterministic chaos.

Cell Dynamical System Model

    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 W and radius R of the growing internal circulation follow the Fibonacci mathematical number series. Therefore short-range circulation balance requirements generate successively larger circulation patterns with the precise geometry governed by the Fibonacci mathematical number series. The successively larger eddy radii may be subdivided again in the golden mean ratio. The internal structure of large eddy circulations is therefore made up of balanced small scale circulations tracing out the well known quasiperiodic Penrose tiling pattern identified as the quasicrystalline structure in condensed matter physics. The growing large eddy carries the turbulent eddies as internal circulations which contribute to their (large eddies) further growth. The turbulent eddy fluctuations mix environmental mass into the large eddy volume. The steady state fractional volume dilution k of the large eddy volume by environmental mixing is given as
(2)
Since the steady state fractional volume dilution of large eddy by inherent turbulent eddy fluctuations during successive length step increments is equal to 0.382 ,  i.e. less than half, the overall Euclidean geometrical shape of the large eddy is retained, e.g. cloud billows which resemble spheres. The variable k is hereby identified as the universal constant for deterministic chaos in dynamical systems and in the following it is shown that the universal Feigenbaum's constants are functions of k. The r.m.s. circulation speed W of the large eddy which grows from constant turbulence scale acceleration w* is given as
(3)
where z is the length scale ratio equal to R/r. The strange attractor design of energy circulation pattern in real world dynamical systems therefore consists of a nested continuum of logarithmic spiral circulations with quasiperiodic Penrose tiling pattern for the internal structure.
The variables W and W2 represent respectively the standard deviation and variance since W represents the instantaneous spatial average of the perturbation speed of the large eddy circulation. Therefore, for a constant turbulence scale acceleration w* , the ratio of the r.m.s. circulation speeds W1 and W2 of large eddies of radii R1 and R2 respectively represent the ratio of the standard deviations of large eddy fluctuations. From Equation 3
(4)
Starting with reference level standard deviation s equal to W1 , the successive dominant eddy growths have standard deviations W2 equal to s , 2s , 3s , etc. from Equation 1 where z1 = z2n and  n = 1, 2, 3, ...n for successive period doubling growth sequence since the successive R values follow the Fibonacci number series as shown earlier.

Universal Feigenbaum's constants

    The universal period doubling route to chaos has been extensively studied by Feigenbaum (1980) who found that two universal constants a and d describe the approach to turbulence independent of the details of the nonlinear equations describing the physical system. Experimental studies have identified the universal constants a and d in real world dynamical systems also (Gleick 1987). Delbourgo's (1986) computations show that the universal constants a and d follow the relation 2a2 = 3d over a wide domain. The physical concepts of the large eddy growth by period doubling process enables to derive Feigenbaum's universal constants a and d and their mutual relationship as functions inherent to the scale invariant eddy energy structure of the dynamical system as follows. The function a may be defined as
(5)
The constant a is equal to 1/k where k represents the steady state fractional volume dilution of large eddy by turbulent eddy fluctuations across unit cross-section on the large eddy envelope. Therefore a represents the steady state fractional mass dispersion by dilution and a2 represents the corresponding fractional energy flux into the environment. Let d represent the ratio of the spin angular momenta of the total mass of the large and turbulent eddies
(6)
Therefore  2a2 = 3d
(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 a and d to give 2a2 = 3d . Therefore the universal algorithm at Equation 1 is a statement of the law of conservation of energy for the period doubling growth sequence. The variable 2a2 represents the total bidirectional eddy energy flow into the environment and is equal to the spin angular momentum of the large eddy. The property of inertia enables propagation of turbulence scale perturbation in the medium where translational kinetic energy is generated during dilation by release of intrinsic latent energy potential of the medium, e.g. the buoyant energy generation by water vapour condensation in the updraft regions of the atmospheric boundary layer.

Deterministic chaos and quantumlike mechanics in atmospheric flows

    Since the large eddy is the sum total of the smaller scales, the large eddy energy content is equal to the sum of all its individual component eddy energies and therefore the kinetic energy distribution is normal and the kinetic energy of any component eddy expressed as a fraction of the energy content of the largest eddy in the hierarchy will represent the cumulative normal probability density distribution, i.e. the eddy energy probability density distribution is equal to the square of the eddy amplitude. Therefore, the atmospheric eddy continuum energy structure follows laws similar to quantum mechanical laws for subatomic dynamics. Therefore, the eddy continuum energy spectrum, i.e. the variance versus frequency values which are conventionally plotted on a log-log scale will represent the normal probability density distribution on a logarithmic scale versus the normalized standard deviation t as shown in the following. In the case of the eddy continuum energy spectrum, t will be obtained directly from Equation 4 as
    The above described analogy of quantumlike mechanics for atmospheric flows is similar to the concept of a sub-quantum level of fluctuations whose space-time organization gives rise to the observed manifestation of subatomic phenomena, i.e. quantum systems as order out of chaos phenomena (Grossing 1989). Further, numerical simulations and analytic solutions show that the power spectral density of the stochastic system and the deterministic chaotic system are indistinguishable (Stone 1990) in agreement with the above result, namely normal distribution characteristics for the power spectra of strange attractors of dynamical systems.

Data and analysis

    The summer monsoon (June-September) rainfall for 29 meteorological sub-divisions for 115 years (1871-1985) was taken from Parthasarathy et al. (1987). The data was subjected to a quasi-continuous periodogram spectral analyses (Jenkinson 1977). The cumulative percentage contribution to total variance (P), the cumulative percentage normal probability density and the corresponding normalized standard deviation t values are plotted in Figure 1. It is seen that the cumulative percentage contribution to total variance closely follows the cumulative normal probability density distribution. The "goodness of fit" was tested using the chi-square test. The horizontal lines in the figure indicate the values of P above which the fit is good at 95% level of significance.
Figure 1

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