A.Mary Selvam

*Indian Institute of Tropical Meteorology,
Pune, India*

(Retired) email: selvam@ip.eth.net

website: http://www.geocities.com/amselvam

*Applied Mathematical Modelling , 1993,
Vol.17, 642-649*

Abstract

*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 k for each computational step and is
equal to 1 / t
^{2} ( =0.382) where
t
is the golden mean. k being
less than half accounts for the fractal(broken) Euclidean geometry of the
strange attactor. (c) The Feigenbaum's universal constants a and
d
are functions of k and, further, the expression 2a^{2}
= p
d 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.*

**Keywords:** deterministic chaos, strange
attractor, Penrose tiling pattern, cell dynamical system, universal algorithm
for chaos

**1. Introduction**

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^{1} 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^{2} 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^{3}
. Ruelle and Takens^{4} 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^{5}
. (d) Computer-accuracy-related round-off errors magnify exponentially
with time the above-mentioned uncertainties and give solutions that are
not totally realistic^{6} . 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 *fractal* first coined
by Mandelbrot^{7} means broken or fractured structure. The fractal
dimension *D* of such non-Euclidean structure is given by the relation
*D*
= *d* ln *M*/*d *ln *R* where *M *is the mass
contained within a distance *R* from a point within the fractal object.
A constant value for *D* indicates uniform distribution of mass with
distance on a logarithmic scale for the length scale *R*. Objects
in nature in general possess a multifractal dimension^{8} . Self-similarity
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^{9} . 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^{10,11} .

**2. Computer accuracy
and round-off error**

Round-off error is inherent to finite precision
numerical computations and imposes a limit *dR* to the resolution
with which two quantities *R* + *dR* can be distinguished as
separate, thereby introducing an uncertainty in computation equal to *dR*
in magnitude. Computer precision *dR* 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 *dR* used for the measurement. The uncertainty
in measurement of separation distance of two points is equal to the yardstick
length *dR*. 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., *X _{n+1}* =

**3. Cell dynamical
system model for deterministic chaos**

The round-off 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^{12}
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.

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 *dR* in all directions as illustrated in *Figure
1* by the circle *OR _{2}R_{1}'R_{2} *of
radius

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 *OR _{2}R_{1}'R_{2}*
on the left. The macroscale uncertainty domain of length scale

The uncertainty domain represented by the
circle *OR _{2}R_{1}'R_{2}* corresponding to
the measurement

The mean square value of *W* is then
obtained as

(1)

The above equation enables one to compute,
for any interval, the number of units Equation (1) is directly applicable to digital computations of nonlinear mathematical models where

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

(2)

Equation (2) is used to derive the progrssively
increasing magnitude

or

(3)

where

Figure 2. Visualization of round-off error growth in successive iterations

The uncertainty *r _{1}* in
the computation is equal to the number of units of computation

**Table 1.** The computed
spatial growth of the strange attractor design traced by dynamical systems
as shown in *Figure 1*.

R |
W_{n} |
dR |
dq |
W_{n+1} |
q |

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 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*. *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 *R _{0}R_{1}R_{2}R_{3}R_{4}R_{5}*
of the strange attractor follows the logarithmic spiral

The successively larger values of the yardstick
lengths are then plotted as the radii *OR _{1}*,

The computed result

(4)

*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 *d**q
*of the *W* units of computation
where *d**q
= 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. *k* represents the steady-state fractional
error due to uncertainty in initial conditions coupled with finite precision
in the computed model. *k* also gives quantitatively the fractional
departure from Euclidean geometrical shape of the computed strange attractor
. *k* is derived from equation (4) as

*k* = *1/*t^{2}
= 0.382

(5)

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 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. *k* , which is a
function of the golden mean t
, is hereby identified as the universal constant for deterministic chaos
in computer realizations of mathematical models of dynamical systems. *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}*
(

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

The incremental growth

(6)

Equation (6) can be integrated to obtain
the
*W* units of total computation starting with *w _{*}*
units of yardstick length

(7)

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}*
(

(8)

where *b* = *tan **a*
with *a
,
*the crossing angle equal to *dR/R*
. *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,

(9)

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
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 1.855*r* (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 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.

**4. Universal algorithm
for deterministic chaos incorporating Feigenbaum's universal constants**

The basic example with the potential to display the main features of the erratic behavior characterizing deterministic chaos is the Julia model given below

*X _{n+1}*
=

(10)

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

(11)

(12)

In the above equation
denotes the** X** spacing between adjoining period doublings
(

Feigenbaum's constant

Feigenbaum's constant

(13)

The total computational domain ** WR**
at any stage of computation may be considered to result from spatial integration
of round-off error domain

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 (round-off error) domain and is equal to

Further,

** 2a ^{2}**
= total variance of the fractional geometrical evolution of computed domain
for both clockwise and counterclockwise phase space trajectories.

The Feigenbaum's constant

Because ** W_{1}^{2}R_{1}
= W_{2}^{2}R_{2}** as explained above

The Feigenbaum's constant ** d**
is therefore obtained as

(14)

** W ^{4}**
represents the fourth moment about the reference level for the instantaneous
trajectory in the representative volume

(15)

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} = p
d*

(16)

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 model-predicted equation (16).

Feigenbaum's universal constants

The Feigenbaum's constants

**5. Universal quantification
for the power spectra of chaotic dynamical systems**

The temporal fluctuations of chatic dynamical
systems are found to exhibit broad-band power spectra^{9}. A complete
description of the temporal variability is therefore possible in terms
of the component periodicities and their phases^{19}. Such a description
in terms of cycles or periodicities of nonlinear fluctuations of real world
dynamical systems has been reported^{23}. 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 ** W** is obtained as the R.M.S. value of
inherent round-off error domains as given in equation (1), the mean square
variance

1. Bernoulli shifts

2. Cat mapwithx----> 3xmod 1,(x_{0},x_{1},......,x_{n},...)x_{0}= 0.1

3. Pseudorandom number generator: minimal standard Lehmer generatorwith initial points(0.1,0.0)F(x,y) = (x+ymod 1,x+ 2ymod 1)for all 0<=x,y< 1

X= 16807_{n+1}Xmod 2147483647;_{n}X= 0.1_{0}

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

(17)

Starting with reference level standard
deviation
** s
**equal to

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 (

**6. Conclusion**

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
** WR**
of the number of units of computation

**Acknowledgements**

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.

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