Figure 1. The growth
of round-off error structures in the phase space. The domain of the round-off
error dR is represented by the circle OR2R1'R2
on the left. The macroscale uncertainty domain of length scale R
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 dR on the right.
The uncertainty domain represented by the circle OR2R1'R2 corresponding to the measurement OR1 is interpreted as follows. One unit of measurement of yardstick length OR1 ( = dR ) implies two approximations: (a) a minimum measurable distance OR1 and (b) round-off of all lengths less than OR1' ( = 2dR ) as equal to OR1 ( = dR ). The domain of these two errors in the phase space is represented by a circle with center R1 and radius OR1 = dR because the projection of OR2 on OR1 for angles OR1R2 less than or greater than 90 degrees respectively will be measured as equal to OR1 . The circle OR2R1'R2 therefore represents the total uncertainty domain for one unit of measurement of yardstick length OR1 = dR . The precision decreases or the yardstick length R increases with successive stages of computation. The increased imprecision represented by increased yardstick length R is composed of the microscale round-off error domain OR2R1'R2 as shown in Figure 1. Such microscopic error domain structures may be compared to turbulent eddy circulations, which contribute to form large eddy circulation patterns in fluid flows. The parameter w* units of computation of yardstick length dR is equivalent to W units of computation of a more imprecise larger yardstick length R and is quantified by analogy with the formation of large eddy circulation structures as the spatial average of the turbulent eddy fluctuation domain10-12 . The mean square round-off error circulation C2 at any instant around a circular path of length scale R is equal to the spatial integration of the microscopic domain error structures ( OR2R1'R2 ) over the computational domain of length scale R and is given as
The mean square value of W is then obtained as
The above equation enables one to compute, for any interval, the number of units dW of computation of decreased precision R resulting from the spatial integration of w* units of inherent microscale round-off error structures ( Figure 1 ) of yardstick length dR. 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 dR. Such a concept of strange attractor growth from microscopic round-off error domains envisions strange attractor growth in discrete length step increments dR and is therefore analogous to cellular automata computational technique where cell dynamical system growth occurs in discrete length step intervals13.
Equation (1) is directly applicable to digital computations of nonlinear mathematical models where W units of imprecise computation of yardstick length R are expressed in terms of w* units of a more precise (higher resolution) yardstick of length dR.
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 w* of higher precision earlier stage computation that forms the internal structure of the total computed domain is obtained from equation (1) as
Equation (2) is used to derive the progrssively increasing magnitude w* units of higher precision computation for successive steps of computation as follows. Denoting Wn and Wn+1 as the number of units of computation for the nth and (n+1)th intervals of computation equation (2) can be written as
where rn is the uncertainty of yardstick length equal to (dR)n . The magnitude of the higher precision yardstick length rn increases with the computation. The incremental growth (dR)n in the yardstick length Rn is generated by Wn units of computation at the nth step and therefore Wn = (dR)n , i.e., one unit of computation generates one unit of uncertainty. The round-off error growth for successive stages of iteration is shown in Figure 2.
Figure 2. Visualization of round-off error growth in successive iterations
The uncertainty r1 in the computation is equal to the number of units of computation W1, i.e., r1 = 1 and is represented by A1A2 in Figure 2. The computation length OA1 can be any radius of the sphere or circle in three or two dimensions respectively, with center O and radius OA1 . The computational domain W1R1 is any rectangular cross-section OA1B1O' of the cylinder with radius of base equal to OA1 and height A1B1 (Figure 2). At the end of the first step of computation W1 = 1, R1 = 1 and r1 = 1. Therefore W2 = 1.254 from equation (3). The first step of computation generates the length domain R2 = R1 + r1 = 2 (OA2 = R2) associated with W2 =1.254 units of computation (A2B2 = W2) and corresponding uncertainty, r2 = W2 = 1.254 (A2A3 = r2). Substitution in equation (3) gives W3= 1.985. Similarly the values of Wn and Rn for the n successive iteration steps are computed from equation (3). The yardstick length Rn is equal to the cumulative sum of the yardstick lengths for the previous n intervals of computation, i.e., . The values of Rn, Wn, dR, Wn+1, and d q computed as equal to Rn / Rn+1 and are tabulated in Table 1.
Table 1. The computed
spatial growth of the strange attractor design traced by dynamical systems
as shown in Figure 1.
It is seen that the yardstick length R and the corresponding number of units of computation W 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 Figure 3 where OR0 is the initial yardstick length.
Figure 3. 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 WR of the number of units of computation W of yardstick length R. Yardstick length R represents the round-off error in the computation. The successive values of W and R 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 R0R1R2R3R4R5 of the strange attractor follows the logarithmic spiral R = re b q shown on the right where r = OR0 and b = tan a where a is the crossing angle.
The successively larger
values of the yardstick lengths are then plotted as the radii OR1,
and OR5 on either side of OR0
such that the angle between successive radii are
so that the ratio of the successive yardstick lengths equals the golden
. The radii can be further subdivided into the golden mean ratio
so that the internal structure of the polar diagram displays the quasiperiodic
tiling pattern14. The larger yardstick length is therefore shown
to consist of microscale round-off error domains OR0R1
OR0 = R0R1 = dR.
The quantity dR is the imprecision inherent to the computational
system consisting of the model uncertainties and the round-off error of
the digital computer.
The computed result WR is represented by a rectangle of sides W and R, and therefore the phase space trajectory can also be resolved into the quasiperiodic Penrose tiling pattern. The spatial domain of the yardstick length OR0 is the solid of revolution generated by the rotation of the triangle OR1R0 about the axis OR0 . It is seen from Table 1 and Figure 3 that starting from either side of the initial computational step OR0 the computation W proceeds in logarithmic spiral curves R0R1R2R3R4R5 such that one complete cycle is executed by the numerical computation after five length steps of computation on either side of OR0 , i.e., clockwise and counterclockwise rotation. Denoting the yardstick length scale ratio R/dR by z, dominant periodicities or cycles occur in the W units of computation for z values in multiples of t5n where n 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 W follows the logarithmic spiral pattern. The incremental units of computation dW of yardstick length R 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 OR0R1 (Figure 3). The incremental units of computation dW of yardstick length R at any stage of computation have intrinsic internal structure consisting of discrete spatial domains of total size w*dR generated by w* units of discrete yardstick length dR , 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 dR. At each stage of computation, the computed spatial domain RdW contains smaller domains of total size w*dR 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 k in the computed model at each stage of computation is therefore given by
The parameter k 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 dq of the W units of computation where dq = dR/R, which is a constant equal to t , the golden mean (Figure 3 ). Further, the successive values of the W units of computation of yardstick length R follow Fibonacci mathematical number series. The parameter k represents the steady-state fractional error due to uncertainty in initial conditions coupled with finite precision in the computed model. The parameter k also gives quantitatively the fractional departure from Euclidean geometrical shape of the computed strange attractor . The parameter k is derived from equation (4) as
k = 1/t2@ 0.382
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 W units of computation is approximately equal to 0.382, i.e., less than half, the overall Euclidean geometrical shape of the strange attractor is retained. Beck and Roepstroff15 also find the universal constant 0.382 for the scaling relation between length of periodic orbits and computer precision in numerical computations. The parameter k , which is a function of the golden meant, is hereby identified as the universal constant for deterministic chaos in computer realizations of mathematical models of dynamical systems. The parameter k 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 constants16 are functions of k . Dominant coherent structures in numerical computation W evolve for yardstick length scale ratio z equal to t5n (n 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 t to be associated with deterministic chaos in dynamical systems17,18. Also, recent numerical investigations indicate that the strange attractor can be defined completely as quasiperiodicities with fine structure19, i.e., a continuum.
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 periodic5. The expected period in the case of fractal chaotic attractors scales with round-off20. 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.
The incremental growth dW units of numerical computation of yardstick length R can be expressed in terms of w* units of more precise yardstick length dR as follows from equation(4):
Equation (6) can be integrated to obtain the W units of total computation starting with w* units of yardstick length r , where, as mentioned earlier, dR represents the uncertainty in initial conditions of the computational system at the beginning of the computation.
The W units of computation and therefore R follow a logarithmic spiral with z being the yardstick length scale ratio, i.e., z = R/dR . The logarithmic spiral R0R1R2R3R4R5 (Figure 3) is given quantitatively in terms of the yardstick length R as
where b = tan a with a , the crossing angle equal to dR/R . The angle a is therefore equal to 1/t as shown earlier and, because b is equal to a in the limit for small increments dW in computation,
The yardstick length R, which represents uncertainty in initial conditions, therefore grows exponentially with progress in computation. The separation distance r of two arbitrarily close points at the beginning of the computation grows to R at the end of the computation with the angular turning of the trajectories being equal to p/5 . The exponential divergence of two arbitrarily close points is given quantitatively by the exponent 1/t approximately equal to 0.618 and is identified as the Lyapunov exponent conventionally used to measure such divergence in computed dynamical systems17. For each length of computation with unit angular turning (equal to p/5 ) the initial yardstick length r increases to 1.855r (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 WR where W units of computation of yardstick length R follow the Fibonacci mathematical number series as a natural consequence of the cumulative addition of round-off error domains. Hammel et al.21 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 to 1/t (@ 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 r 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 constants16 a and d 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.
Xn+1 = F(Xn) = LXn(1 - Xn)
The above nonlinear model represents the population values of the parameter X at different time periods n , and L parameterizes the rate of growth of X for small X . Feigenbaum's research16 showed that the two universal constants a and d are independent of the details of the nonlinear equation for the period doubling sequences
In the above equation
denotes the X spacing between adjoining period doublings
and (n+1), i.e.,
. The value is
the L spacing between period doublings (n+2)
and (n+1) . The universal recursion relation quantifying
chaos in nonlinear mathematical models, namely, equation (2) is analogous
to the Julia model, because the macroscale computation structure
is determined by the microscopic yardstick length
dR . The
Feigenbaum's constants a and
d 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 universal relation (equation (2)) is used for computing quantitatively
the successive length step increments in the magnitude of the number of
units of computation
w* of yardstick length dR
incorporated in the computation.
Feigenbaum's constant a is given by the successive spacing ratios W for adjoining period doublings. W and R are respective successive spacing ratios, because by definition W and R are computed as incremental growth steps dW and dR for each stage of computation.
Feigenbaum's constant a is obtained as the successive spacing ratios of W , i.e.,
The total computational domain WR at any stage of computation may be considered to result from spatial integration of round-off error domain W1R1 or W2R2 where R1 and R2 refer to the precision. From equation (2) W12R1 = W22R2 = constant. Therefore
From equations (4) and (5)
a = 1/k = t2
The Feigenbaum's constant a therefore denotes the relative increase in the computed domain with respect to the yardstick length (round-off error) domain and is equal to t2 (@2.618) and is inherently negative because the round-off error has a negative sign by convention.
2a 2 = total variance of the fractional geometrical evolution of computed domain for both clockwise and counterclockwise phase space trajectories.
Because W12R1 = W22R2 as explained above
The Feigenbaum's constant d is therefore obtained as
W 4 represents the fourth moment about the reference level for the instantaneous trajectory in the representative volume R 3 of the phase space. The parameter d is, therefore, equal to the relative volume intermittency of occurrence of Euclidean structure in the phase space during each computational step, i.e., p/5 radian angular rotation as shown earlier in Table 1 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 pd . The universal recursion relation for deterministic chaos (2) can be written as
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
2a2 = pd
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 Delbourgo22 give the relation 2a2 = 3d, which is almost identical to the model-predicted equation (16).
Feigenbaum's universal constants a = 2.503 and d = 4.6692 (equations (11) and (12)) have been determined by numerical computations at period doublings n, n+1 and n+2 where n is large. At large n, computational difficulties in resolution of adjacent period doublings impose a limit to the accuracy with which a and d can be estimated.
The Feigenbaum's constants a and d computed from the universal algorithm 2a2 = pd 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 a and d are therefore not identical.
1. Bernoulli shifts
2. Cat mapx ----> 3x mod 1,(x0, x1,......, xn,...)with x0 = 0.1
3. Pseudorandom number generator: minimal standard Lehmer generatorF(x, y) = (x + y mod 1, x + 2y mod 1)with initial points(0.1,0.0)for all 0<= x, y < 1
Xn+1 = 16807 Xn mod 2147483647; X0 = 0.1
The power spectra of the above chaotic dynamical systems (Figure 4) 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 t equal to [(log P/log P50) - 1] where P is the period and P50 , the period up to which the cumulative percentage contribution to total variance is equal to 50. The above relation for the normalized standard deviation t in terms of the periodicities follows directly from equation (7) because by definition W and W 2 represent respectively the standard deviation and variance as a direct consequence of W being computed as the instantaneous average round-off error domains for each stage of computation. Therefore, for a constant value of w* , the number of units of computation of precision dR , the ratio of the r.m.s. units of computation W1 and W2 of respective yardstick lengths R1 and R2 will give the ratios of the standard deviations of the unit W of computation. From equation (7)
Starting with reference level standard deviation s equal to W1, the successive steps of computation have standard deviations W2 equal to s, 2s, 3s, .....from equation (17) where z2 = z1n and n = 1, 2, 3, .... for successive period doubling growth sequences.
The important result of the present study is the quantification of the round-off error structure, namely, the strange attractor in mathematical 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 Bernouilli shifts, pseudorandom number generators, and cat map exhibit (Figure 4) 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 criticality26 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 (Figure 4).