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
or
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.
|
|
|
|
|
|
|
|
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 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,
OR2,
OR3,
OR4,
and OR5 on either side of OR0
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 pattern14. The larger yardstick length is therefore shown
to consist of microscale round-off error domains OR0R1
where
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
(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 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).