Figure 1. The growth
of roundoff error structures in the phase space. The domain of the roundoff
error dR is represented by the circle OR_{2}R_{1}'R_{2}
on the left. The macroscale uncertainty domain of length scale R
is the sum of successive stages of such microscale roundoff 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 OR_{2}R_{1}'R_{2} corresponding to the measurement OR_{1} is interpreted as follows. One unit of measurement of yardstick length OR_{1} ( = dR ) implies two approximations: (a) a minimum measurable distance OR_{1} and (b) roundoff of all lengths less than OR_{1}' ( = 2dR ) as equal to OR_{1} ( = dR ). The domain of these two errors in the phase space is represented by a circle with center R_{1} and radius OR_{1} = dR because the projection of OR_{2} on OR_{1} for angles OR_{1}R_{2} less than or greater than 90 degrees respectively will be measured as equal to OR_{1} . The circle OR_{2}R_{1}'R_{2} therefore represents the total uncertainty domain for one unit of measurement of yardstick length OR_{1} = 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 roundoff error domain OR_{2}R_{1}'R_{2} 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 domain^{1012} . The mean square roundoff error circulation C^{2} at any instant around a circular path of length scale R is equal to the spatial integration of the microscopic domain error structures ( OR_{2}R_{1}'R_{2} ) 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
roundoff error structures ( Figure 1 ) of yardstick length dR.
The computational error structure (strange attractor) growth from microscopic
roundoff error domains may be visualized as follows. The strange attractor
domain is defined by the overall envelope of the microscopic scale roundoff
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 roundoff 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 intervals^{13}.
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 W_{n} and W_{n+1}
as the number of units of computation for the
n^{th}
and (n+1)^{th} intervals of computation equation
(2) can be written as
or
where r_{n} is
the uncertainty of yardstick length equal to (dR)_{n}
. The magnitude of the higher precision yardstick length r_{n}
increases with the computation. The incremental growth (dR)_{n}
in the yardstick length R_{n} is generated by W_{n}
units of computation at the n^{th} step and therefore
W_{n}
= (dR)_{n} , i.e., one unit of computation generates
one unit of uncertainty. The roundoff error growth for successive stages
of iteration is shown in Figure 2.
Figure 2. Visualization of roundoff error growth in successive iterations
The uncertainty r_{1} in the computation is equal to the number of units of computation W_{1}, i.e., r_{1} = 1 and is represented by A_{1}A_{2} in Figure 2. The computation length OA_{1} can be any radius of the sphere or circle in three or two dimensions respectively, with center O and radius OA_{1} . The computational domain W_{1}R_{1} is any rectangular crosssection OA_{1}B_{1}O^{'} of the cylinder with radius of base equal to OA_{1} and height A_{1}B_{1} (Figure 2). At the end of the first step of computation W_{1} = 1, R_{1} = 1 and r_{1} = 1. Therefore W_{2} = 1.254 from equation (3). The first step of computation generates the length domain R_{2} = R_{1} + r_{1} = 2 (OA_{2} = R_{2}) associated with W_{2} =1.254 units of computation (A_{2}B_{2} = W_{2}) and corresponding uncertainty, r_{2} = W_{2} = 1.254 (A_{2}A_{3} = r_{2}). Substitution in equation (3) gives W_{3}= 1.985. Similarly the values of W_{n} and R_{n} for the n successive iteration steps are computed from equation (3). The yardstick length R_{n} is equal to the cumulative sum of the yardstick lengths for the previous n intervals of computation, i.e., . The values of R_{n}, W_{n}, dR, W_{n+1}, and d q computed as equal to R_{n }/ R_{n+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.






2 3.254 5.239 8.425 13.546 21.780 35.019 56.305 90.530 
1.254 1.985 3.186 5.121 8.234 13.239 21.286 34.225 55.029 
1.254 1.985 3.186 5.121 8.234 13.239 21.286 34.225 55.029 
0.627 0.610 0.608 0.608 0.608 0.608 0.608 0.608 0.608 
1.985 3.186 5.121 8.234 13.239 21.286 34.225 55.029 88.479 
1.627 2.237 2.845 3.453 4.061 4.669 5.277 5.885 6.493 
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 OR_{0} is the initial yardstick length.
Figure 3. The quasiperiodic Penrose tiling pattern of the roundoff 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 roundoff 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 R_{0}R_{1}R_{2}R_{3}R_{4}R_{5} of the strange attractor follows the logarithmic spiral R = re ^{b }^{q} shown on the right where r = OR_{0} and b = tan a where a is the crossing angle.
The successively larger
values of the yardstick lengths are then plotted as the radii OR_{1},
OR_{2},
OR_{3},
OR_{4},
and OR_{5} on either side of OR_{0}
such that the angle between successive radii are
p/5
so that the ratio of the successive yardstick lengths equals the golden
mean t
. 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^{14}. The larger yardstick length is therefore shown
to consist of microscale roundoff error domains OR_{0}R_{1}
where
OR_{0} = R_{0}R_{1} = dR.
The quantity dR is the imprecision inherent to the computational
system consisting of the model uncertainties and the roundoff 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 OR_{0} is the solid of revolution
generated by the rotation of the triangle OR_{1}R_{0}
about the axis
OR_{0} . It is seen from Table
1 and
Figure 3 that starting from either side of the initial
computational step
OR_{0} the computation W
proceeds in logarithmic spiral curves R_{0}R_{1}R_{2}R_{3}R_{4}R_{5}
such that one complete cycle is executed by the numerical computation after
five length steps of computation on either side of OR_{0}
, 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 t^{5n}
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 nonEuclidean
because of internal structure generated by succesive stages of roundoff
error growth as shown in the triangle OR_{0}R_{1}
(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 steadystate fractional roundoff
error k in the computed model at each stage of computation
is therefore given by
The parameter k also
represents the steadystate 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 steadystate 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/t^{2}@ 0.382
A steadystate fractional
roundoff 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 steadystate
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 Roepstroff^{15} 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 constants^{16} are functions of k . Dominant
coherent structures in numerical computation
W evolve
for yardstick length scale ratio z equal to t^{5n}
(n ranging from negative to positive integer values) as mentioned
earlier and are characterized by roundoff errorgenerated quasiperiodic
Penrose
tiling pattern for the internal structure. Numerical experiments have identified
the golden mean t
to be associated with deterministic chaos in dynamical systems^{17,18}.
Also, recent numerical investigations indicate that the strange attractor
can be defined completely as quasiperiodicities with fine structure^{19},
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.,
roundoff 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 roundoff errors, digital computer
simulations of orbits of chaotic atractors will always eventually become
periodic^{5}. The expected period in the case of fractal
chaotic attractors scales with roundoff^{20}. The universal quantification
of the roundoff error structure growth described in this paper is independent
of the magnitude of the roudoff 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 R_{0}R_{1}R_{2}R_{3}R_{4}R_{5}
(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
systems^{17}. 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 roundoff 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 roundoff 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 computerprecisionrelated
roundoff 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^{16} 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.
X_{n+1} = F(X_{n}) = LX_{n}(1  X_{n})
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
research^{16} 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
(n+2)
and (n+1), i.e.,
and similarly
. The value is
the L spacing between period doublings (n+2)
and (n+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
W
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 roundoff error domain W_{1}R_{1}
or W_{2}R_{2} where R_{1}
and R_{2} refer to the precision. From equation (2)
W_{1}^{2}R_{1}
= W_{2}^{2}R_{2} = constant. Therefore
From equations (4) and (5)
a = 1/k = t^{2}
The Feigenbaum's constant a therefore denotes the relative increase in the computed domain with respect to the yardstick length (roundoff error) domain and is equal to t^{2} (@2.618) and is inherently negative because the roundoff 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 W_{1}^{2}R_{1} = W_{2}^{2}R_{2} 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., counterrotating, 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
2a^{2} = 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 Delbourgo^{22}
give the relation
2a^{2} = 3d, which is almost identical
to the modelpredicted 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
2a^{2} = pd
refer to an infinitesimally small value for the computer roundofferror
(yardstick length), i.e., an infinitely large number of period doublings.
The modelpredicted and computed a and d are
therefore not identical.
1. Bernoulli shifts
2. Cat mapx > 3x mod 1,(x_{0}, x_{1},......, x_{n},...)with x_{0} = 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
X_{n+1} = 16807 X_{n} mod 2147483647; X_{0} = 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 P_{50})  1] where P is the period and P_{50} , 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 roundoff 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 W_{1} and W_{2} of respective yardstick lengths R_{1} and R_{2} 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 W_{1}, the successive steps of computation
have standard deviations W_{2} equal to s,
2s,
3s,
.....from equation (17) where z_{2} = z_{1}^{n}
and n = 1, 2, 3, .... for successive period doubling
growth sequences.
The important result
of the present study is the quantification of the roundoff 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 realworld dynamical systems and is recently identified as the temporal
signature of selforganized criticality^{26} and indicates
longrange temporal correlations or nonlocal connections. Sensitive dependence
on initial conditions, i.e., deterministic chaos, is therefore a
manifestation of selforganized criticality in model dynamical systems
and is a natural consequence of the spatial integration of microscopic
domain roundoff error structures as postulated by the cell dynamical system
model described in Section 3. The universal quantification for deterministic
chaos, or selforganized 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 selforganization of the nonlinear fluctuations
of all scales contributes to form the unique pattern of the normal distribution
(Figure 4).