p img SRC= fig1.gif height=317 width=500 /center

!doctype html public -//w3c//dtd html 4.0 transitional//en html head meta http-equiv= Content- Type content= text/html; charset=iso-8859-1 meta name= deterministic chaos content= universal quantification in terms of Feigenbaum s constants meta name= keywords content= Feigenbaum s constants, deterministic chaos, universal spectrum for chaos meta name= GENERATOR content= Mozilla/4.7 [en] (Win95; I) [Netscape] title Universal quantification for deterministic chaos in dynamical systems /title /head body center font face= Arial,Helvetica Universal quantification for deterministic chaos in dynamical systems /font p font face= Arial,Helvetica A.Mary Selvam /font p i font face= Arial,Helvetica Indian Institute of Tropical Meteorology, Pune, India /font /i p font face= Arial,Helvetica (Retired) email: a href= selvam@ip.eth.net selvam@ip.eth.net /a /font br font face= Arial,Helvetica website: a href= http://www.geoc p i font face= Arial,Helvetica Applied Mathematical Modelling, 1993, Vol.17, 642-649 /font /i p font face= Arial,Helvetica Abstract /font p i font face= Arial,Helvetica A cell dynamical system model for deterministic chaos enables precise quantification of the round-off error growth,i.e., deterministic chaos in digital computer realizations of mathematical models of continuum dynamical systems. The model predicts the following: (a) The phase space trajectory (strange attractor) when resolved as a function of the computer accuracy has intrinsic logarithmic spiral curvature with the quasiperiodic Penrose tiling pattern for the internal structure. (b) The universal constant for deterministic chaos is identified as the steady-state fractional round-off error b k /b for each computational step and is equal to b 1 / /b /font b font face= Symbol t /font font face= Arial,Helvetica nbsp; sup 2 /sup /font /b font face= Arial,Helvetica nbsp; ( =0.382) where /font b font face= Symbol t /font font face= Arial,Helvetica nbsp;nbsp /font /b font face= Arial,Helvetica is the golden mean. b k /b being less than half accounts for the fractal(broken) Euclidean geometry of the strange attactor. (c) The Feigenbaum s universal constants b a /b and b d /b are functions of b k /b and, further, the expression b 2a sup 2 /sup = /b /font b font face= Symbol p /font font face= Arial,Helvetica d /font /b font face= Arial,Helvetica quantifies the steady-state ordered emergence of the fractal geometry of the strange attractor. (d) The power spectra of chaotic dynamical systems follow the universal and unique inverse power law form of the statistical normal distribution. The model prediction of (d) is verified for the Lorenz attractor and for the computable chaotic orbits of Bernoulli shifts, pseudorandom number generators, and cat maps. /font /i /center p font face= Arial,Helvetica b Keywords: /b deterministic chaos, strange attractor, Penrose tiling pattern, cell dynamical system, universal algorithm for chaos /font br nbsp; p b font face= Arial,Helvetica 1.nbsp;nbsp;n Introduction /font /b p font face= Arial,Helvetica Nonlinear mathematical models of dynamical systems are sensitively dependent on initial conditions, identified as deterministic chaos. Such deterministic chaos was first identified nearly a century ago by Poincare sup 1 /sup in his study of the three-body problem. The advent of digital computers in the 1950s facilitated numerical solutions of model dynamical systems, and in 1963 Lorenz sup 2 /sup identified sensitive dependence on initial conditions in a simple mathematical model of atmospheric flows. Deterministic chaos occurs in both continuum(differential equations) and discrete (maps) systems. Deterministic chaos is now an area of intensive research in all branches of science and other areas of human interest sup 3 /sup. Ruelle and Takens sup 4 /sup identified deterministic chaos as similar to turbulence in fluid flows; turbulence is as yet an unresolved problem. It is well-known that deterministic chaos is a direct result of the following inherent limitations of numerical solutions: (a) The differential equations of traditional Newtonian continuum dynamics are solved as difference equations introducing space-time discretizations with implicit assumption of subgrid scale homogeneity for the dynamical processes. (b) Approximations in the governing equations related to limitations of computer capacity. (c) Exact number representation is not possible at the data input stage itself because of the binary form for number representation in computer arithmetic sup 5 /sup. (d) Computer-accuracy-related round-off errors magnify exponentially with time the above-mentioned uncertainties and give solutions that are not totally realistic sup 6 /sup. The trajectory of the dynamical system in the phase space traces a strange attractor, so named because its strange convoluted shape is the final destination of all possible trajectories. Trajectories starting from two initially close points diverge exponentially with time though still within the strange attractor domain. The strange attractor has self-similar geometry. The word i fractal /i first coined by Mandelbrot sup 7 /sup means broken or fractured structure. The fractal dimension i D /i of such non-euclidean structure is given by the relation i D /i = i d /i ln i M /i / i d /i ln i R /i 1

where i M /i is the mass contained within a distance i R /i from a point within the fractal object. A constant value for i D /i indicates uniform distribution of mass with distance on a logarithmic scale for the length scale i R /i. Objects in nature in general possess a multifractal dimension sup 8 /sup. Selfsimilarity implies identical internal structure at all scales of magnification. The temporal fluctuations of dynamical systems have been investigated extensively and are found to exhibit a broad-band power spectrum sup 9 /sup. The physics of deterministic chaos is not as yet identified. In this paper, a cell dynamical system model for the growth of strange attractor pattern of deterministic chaos in dynamical systems is described by analogy with the formation of large eddy structures as envelopes enclosing turbulent eddies in fluid flows sup 10,11 /sup. /font p b font face= Arial,Helvetica 2.nbsp;nbsp;nbsp; Computer accuracy and round-off error /font /b p font face= Arial,Helvetica Round-off error is inherent to finite precision numerical computations and imposes a limit i dr /i to the resolution with which two quantities i R /i + i dr /i can be distinguished as separate, thereby introducing an uncertainty in computation equal to i dr /i in magnitude. Computer precision i dr /i is therefore analogous to yardstick length used in the measurement of distance of separation between two points. Two points cannot be distinguished as separate if they are closer together than the yardstick length i dr /i used for the measurement. The uncertainty in measurement of separation distance of two points is equal to the yardstick length i dr /i. Such round-off error is a direct consequence and the inevitable result of the necessity for discretization of space and time in traditional numerical computations and real-world measurements. In computer simulations of model dynamical systems, the negligibly small model input uncertainties are magnified by the round-off error and propagate into the mainstream computation with successive iterations because of the computational feedback logic inherent in such models, e.g., i X sub n+1 /sub /i = i F /i ( i X sub n /sub /i ) where the valuenbsp; i X sub n+1 /sub /i of the variable at the ( i n /i + i 1 /i )th interval is a function i F /i ofnbsp; i X sub n /sub /i. The uncertainty or its analogue, the yardstick length, i dr /i, increases exponentially with each stage of computation and results in deterministic chaos as explained earlier. In the following discussions, computer precision is treated in terms of the equivalent yardstick length for length measurement. The computational domain is defined as equal to the product i WR /i of the number of units of computation i W /i of yardstick length i R /i. /font p b font face= Arial,Helvetica 3.nbsp;nbsp;nbsp; Cell dynamical system model for deterministic chaos /font /b p font face= Arial,Helvetica The roundoff error structure growth, namely, the strange attractor pattern in computed model dynamical systems, is visualized as corresponding to the coherent structures such as large eddies (or helical vortex roll circulations) that form as the envelopes of enclosed turbulent eddies in planetary atmospheric boundary layer flows. Though turbulent eddy fluctuations are considered to be chaotic(random) and dissipative, they are an integral part of all organized coherent weather patterns such as cloud rows/streets and the hurricane spiral cloud pattern. Townsend sup 12 /sup has shown that large eddy circulations form as a chance configuration of turbulent eddy fluctuations in turbulent shear flows. Just as the small-scale fluctuations contribute to the organized growth of large-eddy fluctuation domains, so also the microscale round-off error structure domains contribute to form the total uncertainty domain in the phase space of the model dynamical system. /font br font face= Arial,Helvetica nbsp;nbsp;nbsp; Computational error is initiated with input data at the first step of numerical computation, i.e., one unit of computation generates one unit of uncertainty equal to the yardstick length i dr /i in all directions as illustrated in i Figure 1 /i by the circle i OR sub 2 /sub R sub 1 /sub R sub 2 /sub /i of radius i dr /i. /font p hr WIDTH= 100 center p img SRC= fig1.gif height=317 width=500 /center p br br br br br br br br br br br p font face= Arial,Helvetica Figure 1.nbsp;nbsp;nbsp; The growth of round-off error structures in the phase space. The domain of the round-off error i dr /i is represented by the circle i OR sub 2 /sub R sub 1 /sub R sub 2 /sub /i on the left. The macroscale uncertainty domain of length scale i R /i is the sum of successive stages of such microscale round-off error domains resulting from finite computer precision and shown by the close packing of circles of radii i dr /i on the right. /font br hr WIDTH= 100 p font face= Arial,Helvetica The uncertainty domain represented by the circle i OR sub 2 /sub R sub 1 /sub R sub 2 /sub /i corresponding to the measurement i OR sub 1 /sub /i is interpreted as follows. One unit of measurement of yardstick length i OR sub 1 /sub /i ( = i dr /i ) implies two approximations: (a) a minimum measurable distance i OR sub 1 /sub /i and (b) round-off of all lengths less than i OR sub 1 /sub ( = 2dR /i ) as equal to i OR sub 1 /sub /i ( = i dr /i ). The domain of these two errors in the phase space is represented 2

by a circle with center i R sub 1 /sub /i and radius i OR sub 1 /sub /i nbsp; = i dr /i because the projection of i OR sub 2 /sub /i on i OR sub 1 /sub /i for angles i OR sub 1 /sub R sub 2 /sub /i less than or greater than 90 degrees respectively will be measured as equal to i OR sub 1 /sub /i. The circle i OR sub 2 /sub R sub 1 /sub R sub 2 /sub /i therefore represents the total uncertainty domain for one unit of measurement of yardstick length i OR sub 1 /sub /i nbsp; = i dr /i. The precision decreases or the yardstick length i R /i increases for with successive stages of computation. The increased imprecision represented by increased yardstick length i R /i is composed of the microscale round-off error domain i OR sub 2 /sub R sub 1 /sub R sub 2 /sub /i as shown in i Figure 1 /i. Such microscopic error domain structures may be compared to turbulent eddy circulations, which contribute to form large eddy circulation patterns in fluid flows. i w sub * /sub /i units of computation of yardstick length i dr /i is equivalent to i W /i units of computation of a more imprecise larger yardstick length i R /i and is quantified by analogy with the formation of large eddy circulation structures as the spatial average of the turbulent eddy fluctuation domain sup 10-12 /sup. The mean square round-off error circulation i C sup 2 /sup /i at any instant around a circular path of length scale i R /i is equal to the spatial integration of the microscopic domain error structures ( i OR sub 2 /sub R sub 1 /sub R sub 2 /sub /i ) over the computational domain of length scale i R /i and is given as /font center p img SRC= Csqr.gif height=40 width=204 /center p font face= Arial,Helvetica The mean square value of i W /i is then obtained as /font center p img SRC= Eq1.gif height=61 width=370 /center div align=right font face= Arial,Helvetica (1) /font /div font face= Arial,Helvetica The above equation enables one to compute, for any interval, the number of units i dw /i of computation of decreased precision i R /i resulting from the spatial integration of i w sub * /sub /i units of inherent microscale round-off error structures ( i Figure 1 /i ) of yardstick length i dr /i. The computational error structure (strange attractor) growth from microscopic round-off error domains may be visualized as follows. The strange attractor domain is defined by the overall envelope of the microscopic scale round-off error domains, and incremental growth of strange attractor occurs in discrete length steps equal to the yardstick length i dr /i. Such a concept of strange attractor growth from microscopic round-off error domains envisions strange attractor growth in discrete length step increments i dr /i and is therefore analogous to i cellular automata /i computational technique where cell dynamical system growth occurs in discrete length step intervals sup 13 /sup. /font br font face= Arial,Helvetica nbsp;nbsp;nbsp; Equation (1) is directly applicable to digital computations of nonlinear mathematical models where i W /i units of imprecise computation of yardstick length i R /i are expressed in terms of i w sub * /sub /i units of a more precise (higher resolution) yardstick of length i dr /i. /font br font face= Arial,Helvetica nbsp;nbsp;nbsp; Each stage of numerical computation goes to form the higher precision earlier step for the next computational step. The magnitude of the number of units i w sub * /sub /i of higher precision earlier stage computation that forms the internal structure of the total computed domain is obtained from equation (1) as /font center p img SRC= Eq2.gif height=51 width=144 /center div align=right font face= Arial,Helvetica (2) /font /div font face= Arial,Helvetica Equation (2) is used to derive the progrssively increasing magnitude i w sub * /sub /i units of higher precision computation for successive steps of computation as follows. Denoting i W sub n /sub /i and i W sub n+1 /sub /i as the number of units of computation for the i n /i th and i (n+1 /i )th intervals of computation equation (2) can be written as /font center p img SRC= Eq3.gif height=56 width=198 p font face= Arial,Helvetica or /font p img SRC= Eq3b.gif height=56 width=168 /center div align=right font face= Arial,Helvetica (3) /font /div font face= Arial,Helvetica where i r sub n /sub /i is the uncertainty of yardstick length equal to ( i dr /i ) i sub n /sub /i. The magnitude of the higher precision yardstick length i r sub n /sub /i increases with the computation. The incremental growth ( i dr /i ) i sub n /sub /i in the yardstick length i R sub n /sub /i is generated by i W sub n /sub /i units of computation at the i n /i th step and therefore i W sub n /sub /i = ( i dr /i ) i sub n /sub /i, i.e., one unit of computation generates one unit of uncertainty. The round-off error growth for successive stages of iteration is shown in i Figure 2 /i. /font p hr WIDTH= 100 center p img SRC= fig2.gif height=318 width=500 p font face= Arial,Helvetica Figure 2.nbsp;nbsp;nbsp; Visualization of round-off error growth in successive iterations /font /center p hr WIDTH= 100 p font face= Arial,Helvetica The uncertainty i r sub 1 /sub /i in the com- 3

putation is equal to the number of units of computation i W sub 1 /sub /i, i.e., i r sub 1 /sub /i = 1 and is represented by i A sub 1 /sub A sub 2 /sub /i in i Figure 2 /i. The computation length i OA sub 1 /sub /i can be any radius of the sphere or circle in three or two dimensions respectively, with center i O /i and radius i OA sub 1 /sub /i. The computational domain i W sub 1 /sub R sub 1 /sub /i is any rectangular cross-section i OA sub 1 /sub B sub 1 /sub O sup /sup /i of the cylinder with radius of base equal to i OA sub 1 /sub /i and height i A sub 1 /sub B sub 1 /sub /i ( i Figure 2 /i ). At the end of the first step of computation i W sub 1 /sub /i = 1, i R sub 1 /sub /i = 1 and i r sub 1 /sub /i = 1. Therefore i W sub 2 /sub /i = 1.254 from equation (3). The first step of computation generates the length domain i R sub 2 /sub /i = i R sub 1 /sub /i + i r sub 1 /sub /i = 2 ( i OA sub 2 /sub /i = i R sub 2 /sub /i ) associated with i W sub 2 /sub /i = 1.254 units of computation ( i A sub 2 /sub B sub 2 /sub /i = i W sub 2 /sub /i ) and corresponding uncertainty, i r sub 2 /sub /i = i W sub 2 /sub /i = 1.254 ( i A sub 2 /sub A sub 3 /sub = r sub 2 /sub /i ). Substitution in equation (3) gives i W sub 3 /sub /i = 1.985. Similarly the values of i W sub n /sub /i nbsp; and i R sub n /sub /i for the i n /i successive iteration steps are computed from equation (3). The yardstick length i R sub n /sub /i is equal to the cumulative sum of the yardstick lengths for the previous i n /i intervals of computation, i.e.,nbsp; img SRC= Sumr.gif height=56 width=108 align=abscenter. The values of i R sub n /sub /i, i W sub n /sub /i, i dr /i, i W sub n+1 /sub /i, and i d /i /font i font face= Symbol q /font font face= Arial,Helvetica nbsp;nbsp /font /i font face= Arial,Helvetica computed as equal to i R sub n /sub / R sub n+1 /sub /i andnbsp; img SRC= Sumtheta.gif height=56 width=100 align=center are tabulated in i Table 1 /i. /font p font face= Arial,Helvetica b Table 1. /b nbsp;nbsp;nbsp; The computed spatial growth of the strange attractor design traced by dynamical systems as shown in i Figure 1 /i. /font br nbsp; center table BORDER COLS=6 WIDTH= 100 tr td center i font face= Arial,Helvetica R /font /i /center /td td center i font face= Arial,Helvetica W sub n /sub /font /i /center /td td center i font face= Arial,Helvetica dr /font /i /center /td td center i font face= Arial,Helvetica d /font font face= Symbol q /font /i /center /td td center i font face= Arial,Helvetica W sub n+1 /sub /font /i /center /td td center i font face= Symbol q /font /i /center /td /tr tr td center font face= Arial,Helvetica 1 /font br font face= Arial,Helvetica 2 /font br font face= Arial,Helvetica 3.254 /font br font face= Arial,Helvetica 5.239 /font br font face= Arial,Helvetica 8.425 /fon br font face= Arial,Helvetica 13.546 /font br font face= Arial,Helvetica 21.780 /font br font face= Arial,Helvetica br font face= Arial,Helvetica 56.305 /font br font face= Arial,Helvetica 90.530 /font /center /td td center font face= Arial,Helvetica 1 /font br font face= Arial,Helvetica 1.254 /font br font face= Arial,Helvetica 1.985 /font br font face= Arial,Helvetica 3.186 /font br font face= Arial,Helvetica 5.121 /fon br font face= Arial,Helvetica 8.234 /font br font face= Arial,Helvetica 13.239 /font br font face= Arial,Helvetica br font face= Arial,Helvetica 34.225 /font br font face= Arial,Helvetica 55.029 /font /center /td td center font face= Arial,Helvetica 1 /font br font face= Arial,Helvetica 1.254 /font br font face= Arial,Helvetica 1.985 /font br font face= Arial,Helvetica 3.186 /font br font face= Arial,Helvetica 5.121 /fon br font face= Arial,Helvetica 8.234 /font br font face= Arial,Helvetica 13.239 /font br font face= Arial,Helvetica br font face= Arial,Helvetica 34.225 /font br font face= Arial,Helvetica 55.029 /font /center /td td center font face= Arial,Helvetica 1 /font br font face= Arial,Helvetica 0.627 /font br font face= Arial,Helvetica 0.610 /font br font face= Arial,Helvetica 0.608 /font br font face= Arial,Helvetica 0.608 /fon br font face= Arial,Helvetica 0.608 /font br font face= Arial,Helvetica 0.608 /font br font face= Arial,Helvetica 0 br font face= Arial,Helvetica 0.608 /font br font face= Arial,Helvetica 0.608 /font /center /td td center font face= Arial,Helvetica 1.254 /font br font face= Arial,Helvetica 1.985 /font br font face= Arial,Helvetica 3.186 /font br font face= Arial,Helvetica 5.121 /font br font face= Arial,Helvetica 8.234 /fon br font face= Arial,Helvetica 13.239 /font br font face= Arial,Helvetica 21.286 /font br font face= Arial,Helvetica br font face= Arial,Helvetica 55.029 /font br font face= Arial,Helvetica 88.479 /font /center /td td center font face= Arial,Helvetica 1 /font br font face= Arial,Helvetica 1.627 /font br font face= Arial,Helvetica 2.237 /font br font face= Arial,Helvetica 2.845 /font br font face= Arial,Helvetica 3.453 /fon br font face= Arial,Helvetica 4.061 /font br font face= Arial,Helvetica 4.669 /font br font face= Arial,Helvetica 5 br font face= Arial,Helvetica 5.885 /font br font face= Arial,Helvetica 6.493 /font /center /td /tr /table /center p font face= Arial,Helvetica It is seen that the yardstick length i R /i and the corresponding 4

number of units of computation i W /i follow the Fibonacci mathematical number series. The progressive increase in imprecision represented by the increasing magnitude for the yardstick length can be plotted in polar coordinates as shown in i Figure 3 /i where i OR sub 0 /sub /i is the initial yardstick length. /font p hr WIDTH= 100 br nbsp; br nbsp; br br center p img SRC= fig3.gif height=308 width=500 /center p br br br br br br br br br br br p font face= Arial,Helvetica b Figure 3. /b nbsp;nbsp;nbsp; The quasiperiodic Penrose tiling pattern of the round-off error structure growth in the strange attractor. The phase space trajectory is represented by the product i WR /i of the number of units of computation i W /i of yardstick length i R /i. i R /i represents the round-off error in the computation. The successive values of i W /i and i R /i follow the Fibonacci mathematical number series, and the strange attractor pattern represented in this manner consists of the quasiperiodic Penrose tiling pattern. The overall envelope i R sub 0 /sub R sub 1 /sub R sub 2 /sub R sub 3 /sub R sub 4 /sub R sub 5 /sub /i of the strange attractor follows the logarithmic spiral i R /i = i re sup b /sup /i /font i sup font face= Symbol q /font /sup /i font face= Arial,Helvetica shown on the right where i r /i = i OR sub 0 /sub /i and i b /i = tan /font font face= Symbol a /font font face= Arial,Helvetica nbsp; where /font font face= Symbol a /font font face= Arial,Helvetica nbsp; is the crossing angle. /font p hr WIDTH= 100 p font face= Arial,Helvetica The successively larger values of the yardstick lengths are then plotted as the radii i OR sub 1 /sub /i, i OR sub 2 /sub,or sub 3 /sub /i, i OR sub 4 /sub, /i and i OR sub 5 /sub /i on either side of i OR sub 0 /sub /i such that the angle between successive radii are /font font face= Symbol p /font fon face= Arial,Helvetica / 5 so that the ratio of the successive yardstick lengths equals the golden mean /font font face= Symbol font size=+1 t /font /font font face= Arial,Helvetica. The radii can be further subdivided into the golden mean ratio so that the internal structure of the polar diagram displays the quasiperiodic Penrose tiling pattern sup 14 /sup. The larger yardstick length is therefore shown to consist of microscale round-off error domains i OR sub 0 /sub R sub 1 /sub /i where i OR sub 0 /sub /i = i R sub 0 /sub R sub 1 /sub /i = i dr /i. i dr /i is the imprecision inherent to the computational system consisting of the model uncertainties and the round-off error of the digital computer. /font br font face= Arial,Helvetica nbsp;nbsp;nbsp; The computed result i WR /i is represented by a rectangle of sides i W /i and i R /i, and therefore the phase space trajectory can also be resolved into the quasiperiodic Penrose tiling pattern. The spatial domain of the yardstick length i OR sub 0 /sub /i is the solid of revolution generated by the rotation of the triangle i OR sub 1 /sub R sub 0 /sub /i about the axis i OR sub 0 /sub /i. It is seen from i Table 1 /i and i Figure 3 /i that starting from either side of the initial computational step i OR sub 0 /sub /i the computation i W /i proceeds in logarithmic spiral curves i R sub 0 /sub R sub 1 /sub R sub 2 /sub R sub 3 /sub R sub 4 /sub R sub 5 /sub /i such that one complete cycle is executed by the numerical computation after five length steps of computation on either side of i OR sub 0 /sub /i, i.e., clockwise and counterclockwise rotation. Denoting the yardstick length scale ratio i R/dR /i by i Z /i, dominant periodicities or cycles occur in thenbsp; i W /i units of computation for i Z /i values in multiples of /font font face= Symbol font size=+1 t /font /font font face= Arial,Helvetica i sup 5n /sup /i where i n /i ranges from positive to negative integer values. The internal structure of the phase space trajectory, i.e., the strange attractor, therefore consists of the quasiperiodic Penrose tiling pattern. The overall envelope of the computation i W /i follows the logarithmic spiral pattern. The incremental units of computation i dw /i of yardstick length i R /i at any stage of computation is non-euclidean because of internal structure generated by succesive stages of round-off error growth as shown in the triangle i OR sub 0 /sub R sub 1 /sub /i ( i Figure 3 /i ). The incremental units of computation i dw /i of yardstick length i R /i at any stage of computation have intrinsic internal structure consisting of discrete spatial domains of total size i w sub * /sub dr /i generated by i w sub * /sub /i units of discrete yardstick length i dr /i, which represents the uncertainty in initial conditions, i.e., the error generated by assuming that the minimum separation distance between two arbitrarily close points is equal to i dr /i. At each stage of computation, the computed spatial domain i RdW /i contains smaller domains of total size i w sub * /sub dr /i representing the uncertainty in input conditions, i.e., the error domains relating to the finite size for yardstick length. The steady-state fractional round-off error i k /i in the computed model at each stage of computation is therefore given by /font center p img SRC= Eq4.gif height=51 width=113 /center div align=right font face= Arial,Helvetica (4) /font /div 5

p br font face= Arial,Helvetica i k /i also represents the steady-state measure of the departure from Euclidean shape of the computed model, namely, the strange attractor. The successive computational steps generate angular turning i d /i /font i font face= Symbol q /font font face= Arial,Helvetica nbsp; /font /i font face= Arial,Helvetica of the i W /i units of computation where i d /i /font i font face= Symbol q /font font face= Arial,Helvetica nbsp; = dr/r /font /i font face= Arial,Helvetica, which is a constant equal to /font font face= Symbol font size=+1 t /font /font font face= Arial,Helvetica nbsp;, the golden mean ( i Figure 3 /i ). Further, the successive values of the i W /i units of computation of yardstick length i R /i follow Fibonacci mathematical number series. i k /i represents the steady-state fractional error due to uncertainty in initial conditions coupled with finite precision in the computed model. i k /i also gives quantitatively the fractional departure from Euclidean geometrical shape of the computed strange attractor. i k /i is derived from equation (4) as /font center p font face= Arial,Helvetica i k /i = i 1/ /i /font font face= Symbol font size=+1 t /font /font font face= Arial,Helvetic = 0.382 /font /center div align=right font face= Arial,Helvetica (5) /font /div p br font face= Arial,Helvetica A steady-state fractional round-off error of 0.382 and the associated quasiperiodic Penrose tiling pattern for the strange attractor are intrinsic to digital computations of nonlinear mathematical models of dynamical systems even in the absence of uncertainty in input conditions for the model. Because the steady-state fractional departure from Euclidean shape of the strange attractor design traced in the phase space by i W /i units of computation is equal to 0.382, i.e., less than half, the overall Euclidean geometrical shape of the strange attractor is retained. Beck and Roepstroff sup 15 /sup also find the universal constant 0.382 for the scaling relation between length of periodic orbits and computer precision in numerical computations. i k /i, which is a function of the golden mean /font font face= Symbol font size=+1 t /font /font font face= Arial,Helvetica, is hereby identified as the universal constant for deterministic chaos in computer realizations of mathematical models of dynamical systems. i k /i is independent of the magnitude of the precision of the digital computer and, also, the spatial and temporal length steps used in model computations. In Section 4 it is shown that the Feigenbaum s universal constants sup 16 /sup are functions of i k /i. Dominant coherent structures in numerical computation i W /i evolve for yardstick length scale ratio i Z /i equal to /font font face= Symbol font size=+1 t /font /font font face= Arial,Helvetica i sup 5n /sup /i ( i n /i ranging from negative to positive integer values) as mentioned earlier and are characterized by round-off error-generated quasiperiodic Penrose tiling pattern for the internal structure. Numerical experiments have identified the golden mean /font font face= Symbol font size=+1 t /font /font font face= Arial,Helvetica to be associated with deterministic chaos in dynamical systems sup 17,18 /sup. Also, recent numerical investigations indicate that the strange attractor can be defined completely as quasiperiodicities with fine structure sup 19 /sup, i.e., a continuum. /font br font face= Arial,Helvetica nbsp;nbsp;nbsp; Traditional computational techniques are digital in concept, i.e., they require a unit or yardstick for the computation and thereby lead inevitably to approximations, i.e., round-off errors. Because the computed quantity structure can be infinitesimally small in the limit, there exists no practical lower limit for the yardstick length. Therefore, numerical computations in the long run give results that scale with computer precision and also give quasiperiodic structures. Numerical experiments show,that, due to round-off errors, digital computer simulations of orbits of chaotic atractors will always eventually become periodic sup 5 /sup. The expected period in the case of fractal chaotic attractors scales with round-off sup 20 /sup. The universal quantification of the round-off error structure growth described in this paper is independent of the magnitude of the roud-off error, the time and space increments, and the details of the nonlinear differential equations and, therefore, is universally applicable for all computed model dynamical systems. /font br font face= Arial,Helvetica nbsp;nbsp;nbsp; The incremental growth i dw /i units of numerical computation of yardstick length i R /i can be expressed in terms of i w sub * /sub /i units of more precise yardstick length i dr /i as follows from equation(4): /font center p img SRC= Eq6.gif height=51 width=128 /center div align=right font face= Arial,Helvetica (6) /font /div p br font face= Arial,Helvetica Equation (6) can be integrated to obtain the i W /i units of total computation starting with i w sub * /sub /i units ofnbsp; yardstick length i r /i,where, as mentioned earlier, i dr /i represents the uncertainty in initial conditions of the computational system at the beginning of the computation. /font center p img SRC= Eq7.gif height=51 width=111 /center 6

div align=right font face= Arial,Helvetica (7) /font /div p br font face= Arial,Helvetica The i W /i units ofnbsp; computation and therefore i R /i follow a logarithmic spiral with i Z /i being the yardstick length scale ratio, i.e., i Z /i = i R/dR /i. The logarithmic spiral i R sub 0 /sub R sub 1 /sub R sub 2 /sub R sub 3 /sub R sub 4 /sub R sub 5 /sub /i ( i Figure 3 /i ) is given quantitatively in terms of the yardstick length i R /i as /font center p img SRC= Eq8.gif height=31 width=91 /center div align=right font face= Arial,Helvetica (8) /font /div p br font face= Arial,Helvetica where i b /i = i tan /i /font i font face= Symbol a /font /i font face= Arial,Helvetica nbsp;nbsp; with /font i font face= Symbol a /font font face= Arial,Helvetica nbsp;, /font /i font face= Arial,Helvetica the crossing angle equal to i dr/r /i. /font i font face= Symbol a /font font face= Arial,Helvetica nbsp; /font /i font face= Arial,Helvetica is therefore equal to 1/ /font font face= Symbol font size=+1 t /font /font font face= Arial,Helvetica nbsp; as shown earlier and, because i b /i is equal to i /i /font i font face= Symbol a /font font face= Arial,Helvetica nbsp; /font /i font face= Arial,Helvetica in the limit for small increments i dw /i in computation, /font center p img SRC= Eq9.gif height=30 width=91 /center div align=right font face= Arial,Helvetica (9) /font /div p br font face= Arial,Helvetica The yardstick length i R /i, which represents uncertainty in initial conditions, therefore grows exponentially with progress in computation. The separation distance i r /i of two arbitrarily close points at the beginning of the computation grows to i R /i at the end of the computation with the angular turning of the trajectories being equal to /font i font face= Symbol p /font font face= Arial,Helvetica nbsp; /nbsp; 5 /font /i font face= Arial,Helvetica. The exponential divergence of two arbitrarily close points is given quantitatively by the exponent 1/ /font font face= Symbol font size=+1 t /font /font font face= Arial,Helvetica nbsp; equal to 0.618 and is identified as the Lyapunov exponent conventionally used to measure such divergence in computed dynamical systems sup 17 /sup. For each lengthnbsp; of computation with unit angular turning (equal to /font i font face= Symbol p /font font face= Arial,Helvetica nbsp; /nbsp; 5 /font /i font face= Arial,Helvetica ) the initial yardstick length i r /i increases to 1.855 i r /i (from equation (9)) at the end of the computation, i.e., the yardstick length (or round-off error) approximately doubles for each iteration when the phase space trajectory is expressed as the product i WR /i where i W /i units of computation of yardstick length i R /i follow the Fibonacci mathematical number series as a natural consequence of the cumulative addition of round-off error domains. Hammel i et al. /i sup 21 /sup mention that it is not unusual that the distance between two close points roughly doubles on every iterate of numerical computation. The Lyapunov exponent equal tonbsp; 1/ /font font face= Symbol font size=+1 t /font /font font face= Arial,Helvetica (=nbsp; 0.618) is intrinsic to numerically computed systems even in the absence of uncertainty in initial conditions for the numerical model. When uncertainty in input conditions exists for the model dynamical system, the initial yardstick length i r /i effectively becomes larger and, therefore, larger divergence of initially close trjectories occurs for a shorter length step of computation as seen from equation (9). The generation of strange attractor in computer realizations of nonlinear mathematical models is a direct consequence of computer-precision-related round-off errors. The geometrical structure of the strange attractor is quantified by the recursion relation of equation (2). Equation (2) is hereby identified as the universal algorithm for the generation of the strange attractor pattern with underlying universality quantified by the Feigenbaum s universal constants sup 16 /sup i a /i and i d /i in computer realizations of nonlinear mathematical models of dynamical systems. In the following section it is shown quantitatively that equation (2) gives directly the universal characteristics such as the Feigenbaum s constants identifying deterministic chaos of diverse nonlinear mathematical models. /font p b font face= Arial,Helvetica 4.nbsp;nbsp;nbsp; Universal algorithm for deterministic chaos incorporating Feigenbaum s universal constants /font /b p font face= Arial,Helvetica The basic example with the potential to display the main features of the erratic behavior characterizing deterministic chaos is the Julia model given below /font center p b i font face= Arial,Helvetica X sub n+1 /sub /font /i font face= Arial,Helvetica = i F /i ( i X sub n /sub /i ) = i LX sub n /sub /i (1 - i X sub n /sub ) /i /font /b /center div align=right font face= Arial,Helvetica (10) /font /div p br font face= Arial,Helvetica The above nonlinear model represents the population values of the parameter i X /i at different time periods i n /i, and i L /i parameterizes the rate of growth of i X /i 7

for small i X /i. Feigenbaum s research sup 16 /sup nbsp; showed that the two universal constants i a /i and i d /i are independent of the details of the nonlinear equation for the period doubling sequences /font center p img SRC= Eq11.gif height=40 width=234 /center div align=right font face= Arial,Helvetica (11) /font /div center b i font face= Arial,Helvetica a = -2.5029 /font /i /b p img SRC= Eq12.gif height=40 width=345 /center div align=right font face= Arial,Helvetica (12) /font /div center b i font face= Arial,Helvetica d = 4.6692 /font /i /b /center p font face= Arial,Helvetica In the above equationnbsp; img SRC= DXn1.gif height=35 width=76 align=center denotes the b i X /i /b spacing between adjoining period doublings ( i n /i +2) and ( i n /i +1), i.e.,nbsp; img SRC= DXn1b.gif height=35 width=210 align=center and similarlynbsp; img SRC= Dxn.gif height=35 width=181 align=center.nbsp; img SRC= Ln2.gif height=35 width=125 align=center is the b i L /i /b spacing between period doublings ( i n /i +2) and ( i n /i +1). The universal recursion relation quantifying deterministic chaos in nonlinear mathematical models, namely, equation (2) is analogous to the Julia model, because the macroscale computation structure b i W /i /b is determined by the microscopic yardstick length b i dr /i /b. The Feigenbaum s constants b i a /i /b and b i d /i /b for the universal period doubling route to chaos may be derived directly from the universal recursion relation (equation (2)) as shown in the following. The universalnbsp; relation (equation (2)) is used for computing quantitatively the successive length step increments in the magnitude of the number of units of computation b i w sub * /sub /i /b of yardstick length b i dr /i /b incorporated in the computation. /font br font face= Arial,Helvetica nbsp;nbsp;nbsp; Feigenbaum s constant b i a /i /b is given by the successive spacing ratios b i W /i /b for adjoining period doublings. b i W /i /b and b i R /i /b are respective successive spacing ratios, because bynbsp; definition b i W /i /b and b i R /i /b are computed as incremental growth steps b i dw /i /b and b i dr /i /b for each stage of computation. /font br font face= Arial,Helvetica nbsp;nbsp;nbsp; Feigenbaum s constant b i a /i /b nbsp; is obtained as the successive spacing ratios of b i W /i /b, i.e., /font center p img SRC= Eq13.gif height=56 width=203 /center div align=right font face= Arial,Helvetica (13) /font /div p br font face= Arial,Helvetica The total computational domain b i WR /i /b at any stage of computation may be considered to result from spatial integration of round-off error domain b i W sub 1 /sub R sub 1 /sub or b i W sub 2 /sub R sub 2 /sub /i /b where b i R sub 1 /sub /i /b and b i R sub 2 /sub /i /b refer to the precision. From equation (2) b i W sub 1 /sub sup 2 /sup R sub 1 /sub = W sub 2 /sub sup 2 /sup R sub = constant. Therefore /font center p img SRC= Aval.gif height=58 width=160 /center p font face= Arial,Helvetica From equations (4) and (5) /font center p b i font face= Arial,Helvetica a = 1/k = /font font face= Symbol font size=+1 t /font /font font face= Arial,Helvetica nbsp; sup 2 /sup /font /i p font face= Arial,Helvetica The Feigenbaum s constant b i a /i /b therefore denotes the relative increase in the computed domain with respect to the yardstick length (round-off error) domain and is equal tonbsp;nbsp;nbsp;nbsp;nbsp; /font b i font face= Symbol font size=+1 t /font /font font face= Arial,Helvetica nbsp; sup 2 /sup /font /i /b font face= Arial,Helvetica nbsp; (=2.618) and is inherently negative because the round-off error has a negative sign by convention. /font br font face= Arial,Helvetica nbsp Further, /font center b i font face= Arial,Helvetica a sup 2 /sup = variance of a = /font font face= Symbol font size=+1 t /font /font font face= Arial,Helvetica nbsp; sup 4 /sup /font /i /b /center p b i font face= Arial,Helvetica 2a sup 2 /sup /font /i /b font face= Arial,Helvetica = total variance of the fractional geometrical evolution of computed domain for both clockwise and counterclockwise phase space trajectories. /font br font face= Arial,Helvetica nbsp;nbsp;nbsp; The Feigenbaum s constant b i d /i /b is the successive spacing ratios of b i R /i /b for the universal recursion relation(equation (2)) for the numerical computation. /font p font face= Arial,Helvetica Because b i W sub 1 /sub sup 2 /sup R sub 1 /sub = W sub 2 /sub sup 2 /sup R sub 2 /sub /i /b as explained above /font center p img SRC= R2byr1.gif height=63 width=123 /center p font face= Arial,Helvetica The Feigenbaum s constant b i d /i /b is therefore obtained as /font center p img SRC= Eq14.gif height=63 width=116 /center div align=right font face= Arial,Helvetica (14) /font /div p br b i font face= Arial,Helvetica W sup 4 /sup /font /i /b font face= Arial,Helvetica rep- 8

resents the fourth moment about the reference level for the instantaneous trajectory in the representative volume b i R sup 3 /sup /i /b of the phase space. b i d /i /b is, therefore, equal to the relative volume intermittency of occurrence of Euclidean structure in the phase space during each computational step, i.e., /font b i font face= Symbol p /font font face= Arial,Helvetica / 5 /font /i /b font face= Arial,Helvetica radian angular rotation as shown earlier in i Table 1 /i for the quasiperiodic Penrose tiling pattern traced by the strange attractor. For one complete cycle(period) of computation, five length steps of simultaneous clockwise and counterclockwise, i.e., counter-rotating, computations are performed. Therefore, for one complete cycle of computation the relative volume intermittency of occurrence of Euclidean structure in the computed phase space domain is /font b i font face= Symbol p /font font face= Arial,Helvetica d /font /i /b font face= Arial,Helvetica. The universal recursion relation for deterministic chaos(2) can be written as /font center p img SRC= Eq15.gif height=60 width=229 /center div align=right font face= Arial,Helvetica (15) /font /div p br font face= Arial,Helvetica The reformulated universal algorithm for numerical computation at equation (15) can now be written in terms of the universal Feigenbaum s constants (equations (13) and (14)) as /font center p b i font face= Arial,Helvetica 2a sup 2 /sup = /font font face= Symbol p /font font face= Arial,Helvetica d /font /i /b /center div align=right font face= Arial,Helvetica (16) /font /div p br font face= Arial,Helvetica The above equation states that the relative volume intermittency of occurrence of Euclidean structure for one dominant cycle of computation contributes to the total variance of the fractional Euclidean structure of the strange atractor in the phase space of the computed domain. Numerical computations by Delbourgo sup 22 /sup give the relation b i 2a sup 2 /sup = 3d /i /b, which is almost identical to the model-predicted equation (16). /font br font face= Arial,Helvetica nbsp;nbsp;nbsp; Feigenbaum s universal constants b i a = 2.503 /i /b and b i d = 4.6692 /i /b (equations (11) and (12)) have been determined by numerical computations at period doublings i n /i, i n+1 /i and i n+2 /i where i n /i is large. At large i n /i, computational difficulties in resolution of adjacent period doublings impose a limit to the accuracy with which b i a /i /b and b i d /i /b can be estimated. /font br font face= Arial,Helvetica nbsp;nbsp;nbsp; The Feigenbaum s constants b i a /i /b and b i d /i /b computed from the universal algorithm b i 2a sup 2 /sup = /i /b /font b i font face= Symbol p /font font face= Arial,Helvetica d /font /i /b font face= Arial,Helvetica refer to an infinitesimally small value for the computer round-off-error(yardstick length), i.e., an infinitely large number of period doublings. The model-predicted and computed b i a /i /b and b i d /i /b are therefore not identical. /font p b font face= Arial,Helvetica 5.nbsp;nbsp;nbsp; Universal quantification for the power spectra of chaotic dynamical systems /font /b p font face= Arial,Helvetica The temporal fluctuations of chatic dynamical systems are found to exhibit broad-band power spectra sup 9 /sup. A complete description of the temporal variability is therefore possible in terms of the component periodicities and their phases sup 19 /sup. Such a description in terms of cycles or periodicities of nonlinear fluctuations of real world dynamical systems has been reported sup 23 /sup. In the following, it is shown that the power spectra of the nonlinear fluctuations of chaotic dynamical systems can be quantified in terms of the universal characteristics of the statistical normal distribution. Because the successive number of units of computation b i W /i /b is obtained as the R.M.S. value of inherent round-off error domains as given in equation (1), the mean square variance b i W sup 2 /sup /i /b for the continuum of computed b i W /i /b will follow normal distribution characteristics. The model predictions are in agreement with continuous periodogram analysis sup 24 /sup of the Lorenz attractor sup 25 /sup and the computable chaotic orbits of (a) Bernoulli shifts, (b)cat maps, and (c) pseudorandom number generators sup 5 /sup. The details of the mathematical equations of the computable chaotic dynamical systems sup 5 /sup used is given in the following: /font p font face= Arial,Helvetica 1.nbsp;nbsp;nbsp; Bernoulli shifts /font blockquote blockquote font face= Arial,Helvetica nbsp; i x /i - 3 i x /i mod 1,( i x /i sub 0 /sub, i x /i sub 1 /sub,..., i x /i sub n /sub,...) /font /blockquote font face= Arial,Helvetica with i x /i sub 0 /sub = 0.1 /font /blockquote font face= Arial,Helvetica 2.nbsp;nbsp;nbsp; Cat map /font blockquote blockquote font face= Arial,Helvetica i F /i i y /i ) = ( i x /i + i y /i mod 1, i x /i + 2 i y /i mod 1) /font blockquote blockquote font face= Arial,Helvetica for all 0lt;= i x /i, i y /i lt; 1 /font /blockquote /blockquote /blockquote font face= Arial,Helvetica with initial points(0.1,0.0) /font /blockquote font face= Arial,Helvetica 3.nbsp;nbsp;nbsp; Pseudorandom number generator: minimal standard Lehmer generator /font blockquote blockquote 9

center i font face= Arial,Helvetica X sub n+1 /sub /font /i font face= Arial,Helvetica = 16807 i X sub n /sub /i mod 2147483647; i X sub 0 /sub /i = 0.1 /font /center /blockquote /blockquote p br hr WIDTH= 100 center p img SRC= fig4.gif height=363 width=500 /center font face= Arial,Helvetica b Figure 4. /b nbsp;nbsp;nbsp; The results of continuous periodogram analysis of the Lorenz attractor and the computable chaotic orbits of the Bernoulli shifts, pseudorandom number generators and cat maps. The data series are samples of different sections of each chaotic orbit, and averages of up to 50 successive values were used for the study. Details of the data series corresponding to the spectra numbered 1-10 in the figure are listed in the following: (1) Lorenz attractor, 100 successive 25-point averages of i Y /i from fifth point. (2) Lorenz attractor, 100 successive 25-point averages of i X /i nbsp; from fifth point.nbsp; (3) Lorenz attractor, 100 successive 25-point averages of i Y /i from 4001th point. (4) Bernouille shifts, 100 successive three-point averages starting from 301th value. (5)nbsp; Bernouille shifts, 100 successive two-point averages starting from 501th value. (6) Bernouille shifts, 100 successive three-point averages starting from 1001th value. (7) Pseudorandom numbers, 100 values from 301th value. (8) Pseudorandom numbers, 100 values from 101th value. (9) Cat map, 100 values of i X /i from 101th value. (10) Cat map, 50 values of i Ynbsp; /i from 101th value. /font br nbsp; p hr WIDTH= 100 br nbsp; p font face= Arial,Helvetica nbsp;nbsp;nbsp; The power spectra of the above chaotic dynamical systems( i Figure 4 /i ) are found to be the same as the normal probability density distribution with the normalized variance representing the eddy probability density corresponding to the normalized standard deviation i t /i equal to [(log i P /i /log i P /i sub 50 /sub ) - 1] where i P /i is the period and i P /i sub 50 /sub, the period up to which the cumulative percentage contribution to total variance is equal to 50. The above relation for the normalized standard deviation i t /i in terms of the periodicities follows directly from equation (7) because by definition b i W /i /b and b i W sup 2 /sup /i /b represent respectively the standard deviation and variance as a direct consequence of b i W /i /b being computed as the instantaneous average round-off error domains for each stage of computation. Therefore, for a constant value of b i w sub * /sub /i /b, the number of units of computation of precision b i dr /i /b, the ratio of the R.M.S. units of computation b i W sub 1 /sub /i /b and b i W sub 2 /sub /i /b of respective yardstick lengths b i R sub 1 /sub /i /b andnbsp; b i R sub 2 /sub /i /b will give the ratios of the standard deviations of the unit b i W /i /b of computation. From equation (7) /font center p img SRC= Eq17.gif height=58 width=113 /center div align=right font face= Arial,Helvetica (17) /font /div p br font face= Arial,Helvetica Starting with reference level standard deviation /font b i font face= Symbol s /font font face= Arial,Helvetica nbsp; /font /i /b font face= Arial,Helvetica equal to b i W sub 1 /sub /i /b, the successive steps of computation have standard deviations b i W sub 2 /sub /i /b equal to /font b i font face= Symbol s /font /i /b font face= Arial,Helvetica nbsp;, 2 /font b i font face= Symbol s /font font face= Arial,Helvetica /font /i /b font face= Arial,Helvetica, 3 /font b i font face= Symbol s /font font face= Arial,Helvetica /font /i /b font face= Arial,Helvetica,...from equation (17) where b i Z sub 2 /sub = Z sub 1 /sub sup n /sup /i /b and i n /i = 1, 2, 3,...for successive period doubling growth sequences. /font br font face= Arial,Helvetica nbsp;nbsp;nbsp; The important result of the present study is the quantification of the round-off error structure, namely, the strange attractor in model dynamical systems in terms of the universal and unique characteristics of the statistical normal distribution. The power spectra of the Lorenz attractor and the computable chaotic orbits of the Bernouille shifts, pseudorandom number generators, and cat map exhibit ( i Figure 4 /i ) the universal inverse power law form of the statistical normal distribution. The inverse power law form for the power spectra of the temporal fluctuations is ubiquitous to real-world dynamical systems and is recently identified as the temporal signature of self-organized criticality sup 26 /sup and indicates long-range temporal correlations or non-local connections. Sensitive dependence on initial conditions, i.e., deterministic chaos, is therefore a manifestation of self-organized criticality in model dynamical systems and is a natural consequence of the spatial integration of microscopic domain round-off error structures as postulated by the cell dynamical system model described in Section 3. The universal quantification for deterministic chaos, or self-organized criticality in terms of the unique inverse power law form of the statistical normal distribution identifies the universality underlying numerical computations of chaotic dynamical systems. The total pattern of fluctuations of chaotic dynamical systems is predictable, because self-organization of the nonlinear fluctuations of all scales contributes to form the unique pattern of the normal distribution 10

( i Figure 4 /i ). /font p b font face= Arial,Helvetica 6.nbsp;nbsp;nbsp; Conclusion /font /b p font face= Arial,Helvetica In summary, the cell dynamical system model for round-off error growth in computer realizations of nonlinear dynamical systems visualizes the computer precision (round-off error) as analogous to yardstick length in length measurement. The computed domain consists of the cumulative integrated mean of enclosed round-off error domains. The computed domain, namely, the phase space trajecory, is the product b i WR /i /b of the number of units of computation b i W /i /b of precision (yardstick length) b i R /i /b. The phase space trajectory thus defined traces an overall logarithmic spiral pattern with the golden mean winding number and quasiperiodic Penrose tiling pattern for the internal structure, implying long-range temporal correlations, namely, sensitive dependence on initial conditions. The universality underlying deterministic chaos is quantified in terms of the following universal constants, which are functions of the golden mean /font b i font face= Symbol font size=+1 t /font /font /i /b font face= Arial,Helvetica nbsp; : (1) The constant b i k /i /b nbsp; for deterministic chaos equal to b 1 / /b /font b i font face= Symbol font size=+1 t /font /font /i /b font face= Arial,Helvetica nbsp; b i sup 2 /sup /i /b represents the steady-state fractional round-off error for each computational step. b i k /i /b also represents the fractional departure from Euclidean geometry of the strange attractor. (2) The Lyapunov exponent is equal to b 1 / /b /font b i font face= Symbol font size=+1 t /font /font /i /b font face= Arial,Helvetica. (3) The Feigenbaum s constants b i a /i /b and b i d /i /b define the algorithm for deterministic chaos as b 2 i a sup 2 /sup = /i /b /font b i font face= Symbol p /font font face= Arial,Helvetica d /font /i /b font face= Arial,Helvetica, which states that the relative volume intermittency of occurrence equal to /font b i font face= Symbol p /font font face= Arial,Helvetica d /fo face= Arial,Helvetica nbsp; of fractional Euclidean structure contributes to the total variance equal to b 2 i a sup 2 /sup /i /b of the strange attractor. b i a /i /b is equal to /font b i font face= Symbol font size=+1 t /font /font /i /b font face= Arial,Helvetica nbsp; b i sup 2 /sup /i /b nbsp;nbsp; and represents the fractional Euclidean structure of the strange attractor. (4) The power spectra of computed chaotic dynamical systems follow the universal inverse power law form of the statistical normal distribution. Continuous periodogram power spectral analysis of Lorenz attractor, Bernouille shifts, pseudorandom number generators, and cat maps are in agreement with model predictions. /font p b font face= Arial,Helvetica Acknowledgements /font /b p font face= Arial,Helvetica The author is grateful to Dr.A.S.R.Murty for his keen interest and encouragement during the course of this study. Thanks are due to Shri R.Vijayakumar for assistance with computer graphics and to Shri M.I.R.Tinmaker for typing the manuscript. /font p b font face= Arial,Helvetica References /font /b p font face= Arial,Helvetica 1nbsp;nbsp;nbsp Poincare, H. i Les meyhodes nouvelle de la mecanique celeste /i. Gautheir-Villars,nbsp;nbsp;nbsp;nbsp; Paris, 1892 /font br font face= Arial,Helvetica 2nbsp;nbsp;nbsp; Lorenz, E. N. i J /i. i Atmos.Sci /i. (1963), b 20 /b, 130 /font br font face= Arial,Helvetica 3nbsp;nbsp;nbsp; Gleick, J. i Chaos: Making a New Science. /i nbsp; Viking, New York, 1987 /font br font face= Arial,Helvetica 4nbsp;nbsp;nbsp; Ruelle, D. and Takens, F. i Commun. Math. Phys /i. 1971, b 20 /b, 167 /font br font face= Arial,Helvetica 5nbsp;nb Palmore, J. andnbsp; Herring, C. i Physica D /i 1990, b 42 /b, 99 /font br font face= Arial,Helvetica 6nbsp;nbsp;nbsp Stewart, I. i Nature /i 1992, b 355 /b, 16 /font br font face= Arial,Helvetica 7nbsp;nbsp;nbsp; Mandelbrot, B. B. PAGEOPH 1989, b 131 /b (1/2), 5 /font br font face= Arial,Helvetica 8nbsp;nbsp;nbsp; Stanley, H. E. and Meakin, P. i Nature /i 1988, b 335 /b, 405 /font br font face= Arial,Helvetica 9nbsp;nbsp;nbsp; Procaccia, I. i Nature /i 1988, b 333 /b, 618 /font br font face= Arial,Helvetica 10nbsp; Selvam, A. Mary, i Can. J. Phys /i. 1990, b 68 /b, 831 /font br font face= Arial,Helvetica 11nbsp; Selvam, A. Mary, Pethkar, J. S. and Kulkarni, M. K. i Int. J. Climatol /i. 1992, b 12 /b, 137 /font br font face= Arial,Helvetica 12nbsp; Townsend, A. A. i The Structure of Turbulent Shear Flow /i. Cambridge University Press, Cambridge, 1956 /font br font face= Arial,Helvetica 13nbsp;nbsp; Oona, Y. and Puri, S. i Phys. Rev. A /i. 1988, b 38 /b, 434 /font br font face= Arial,Helvetica 14nbsp;nbsp;nbsp; Janssen, T. i Phys. Rep. /i 1988, b 168 /b, 1 /font br font face= Arial,Helvetica 15nbsp;nbsp;nbsp; Beck, C. and Roepstorff, G. i Physica D /i 1987, b 25 /b, 173 /font br font face= Arial,Helvetica 16nbsp;nbsp;nbsp; Feigenbaum, M. J. i Los Alamos Sci. /i 1980, b 1 /b, 4 /font br font face= Arial,Helvetica 17nbsp;nbsp;nbsp; McCauley, J. L. i Physica Scripta /i 1988, b T20 /b, 1 /font br font face= Arial,Helvetica 18nbsp;nbsp;nbsp; Stewart, I. i New Scientist /i 1992, b 135 /b, 14 /font br font face= Arial,Helvetica 19nbsp;nbsp;nbsp; Cvitanovic, P. i Phys. Rev. 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