3. Nonlinear Dynamics and Chaos in Iterative Processes
Standard models for turbulent fluid flows in meteorological
theory cannot explain satisfactorily the observed multifractal
(space-time) structures in atmospheric flows(Tessier et
al.,1993,1996 References).
Traditionally, meteorological theory is based on the following
concepts. The turbulent atmospheric flows are governed by the
mutual interaction of a large number of factors, i.e. variables
such as pressure, temperature, moisture content, wind speed, etc.
The prediction of future flow pattern is based on mathematical
equations for the rate of change 
of component
variable X with time t. The rate of
change with time 
of any variable is generally a
nonlinear function of all the other interacting variables and
therefore analytical solution for X is not
available. The evaluation of any variable X with
time is then computed numerically from the iterative equation
(2)
where the subscript n denotes the time step and
the rate of change is assumed to be continuous for
small changes in time dt, an assumption based on
Newtonian continuum dynamics. The successive values of X are
then computed iteratively, a process known as numerical
integration. The prediction equation for the variable X has
intrinsic error feedback loop since the value of X at
each step is a function of its earlier value in such numerical
integration computational techniques. The fundamental (basic)
error in numerical computations is the round-off error of finite
precision computations.Blank(1994 References)
mentions that when solving differential or other dynamical
systems on a computer, the effects of finiteness(round-off) can
sometimes be very drastic.When we work with fixed precision
system, not all real numbers are even representable and
arithmetic does not have the properties that we are used
to(Corless et.al.,1990).Lorenz(1989 References)
has discussed chaotic behaviour when continuum equations are
solved numerically as difference equations.Climate modelling
concepts has come under criticism lately since uncertainty in
input parameter values can give drastically different
results(Kerr,1994).Mary Selvam (1993a References)
has shown that round-off error approximately doubles on an
average at each step of iteration. Such error doubling at each
step in numerical integration will result in the round-off error
propagating into the mainstream (digits place and above)
computation within 50 iterations using single precision (7th
decimal place accuracy) digital computers. In addition, any
uncertainties in specifying the initial value of the variable X
will also grow exponentially with time and give
unrealistic solutions. Numerical solutions are therefore
sensitively dependent on initial conditions. Deterministic
governing equations, namely evolution equations such as
Equation 2. which are precisely defined and
mathematically formulated give chaotic solutions because of
sensitive dependence on initial conditions. Finite precision
computer realizations of nonlinear mathematical models of
dynamical systems therefore exhibit deterministic chaos. Computed
model solutions are therefore mere mathematical artifacts of the
universal process of round-off error growth in iterative
computations(Mary Selvam,1997). Mary Selvam (1993a References)
has shown that the computed domain is the successive cumulative
integration of round-off error domains analogous to the formation
of large eddy domains as envelopes enclosing turbulent eddy
fluctuation domains such as in atmospheric flows. Computed
solutions, therefore qualitatively resemble real world dynamical
systems such as atmospheric flows with manifestation of
self-organized criticality. Self-organized criticality, i.e.
long-range spatiotemporal correlations, originates with the
primary perturbation domains corresponding respectively to
round-off error and dominant turbulent eddy fluctuations in model
and real world dynamical systems. Computed solutions, therefore,
are not true solutions. The vast body of literature investigating
chaotic trajectories in recent years (since 1980) document, only
the round-off error structure in finite precision computations.
Stewart(1992b References)
mentions that in the absence of analytical (true) solutions the
fidelity of computed solutions is questionable.
Historically,deterministic chaos in computed solutions was
identified nearly a century ago by Poincare in his study of the
three body problem (Poincare, 1892 References).
Lack of high speed computational machines precluded exhaustive
studies of nonlinear behavior and approximate linearized
solutions of nonlinear systems alone were studied. With the
advent of electronic digital computers in late 1950s, Lorenz
(1963 References)
identified deterministic chaos in a simple model of atmospheric
flows. Lorenz's result captured the attention of
scientists in all branches of science since a simple set of
equations exhibits chaotic behaviour similar to the complex,
irregular (unpredictable) fluctuations exhibited by real world
dynamical systems. Till then it was believed that complex
behavior results from complexity in the governing parameters and
the mathematical formulations. Lorenz's model
demonstrated that simple models can demonstrate complex behavior
of real world dynamical systems.
The computed trajectory is plotted graphically in phase space
of dimension m where m is the number
of variables representing the dynamical system. For example, a
particle in motion can be represented completely at any instant
by its position and momenta in the x, y and z directions,
i.e. 6-dimensional phase space. The line joining the successive
points in time gives the trajectory of the particle in phase
space. The trajectory traces the strange attractor
, so named because of its strange convoluted shape being the
final destination of all trajectories in the phase space. Two
trajectories, initially close together diverge exponentially with
time though still within the strange attractor
domain, thereby exhibiting sensitive dependence on initial
conditions or deterministic chaos. The strange attractor
exhibits selfsimilar fractal geometry similar to the space-time
fractal structure or self-organized criticality exhibited by real
world dynamical systems. Mary Selvam (1993a References)
has shown that the strange attractor has the
quasicrystalline structure of the quasiperiodic Penrose tiling
pattern. There is a very close similarly between the geometrical
patterns generated during iterative computations and those found
in nature (Jurgen et al., 1990; Stewart, 1992a References).
Iterative computations generate patterns strongly reminiscent of
plant forms and clearly these curious configurations show that
the rules responsible for the construction of elaborate living
tissue structures could be absurdly simple (Dewdney, 1986 References).
In summary, selfsimilar space-time structures or
self-organized criticality is ubiquitous to dynamical systems in
nature and also to mathematical models of dynamical systems which
incorporate finite precision iterative computations with
resultant feedback and magnification of round-off error
primarily, in addition to initial errors. Iterative computations
result in the cumulative addition (integration) of the
progressively increasing round-off error. Persistent
perturbations, though small in magnitude are therefore capable of
generating complex space-time structures with fractal selfsimilar
geometry because of feedback with amplification.