Mathematics contains an important study
field under the name Chaos theory.
Chaos theory studies the
concept and behavior
of highly insensitive
dynamical systems.
It also studies behavior of dynamic
systems in initial conditions,
which often turns out to be super
sensitive at a very high level.
In Chaos theory, this concept
is referred as Butterfly
effect, which is the main field
of study in this theory.
From which, various branches are spread and
constantly progressing and being developed.
Various initial conditions are made because
of some numerical errors in computations.
These errors provide wildly diverging
results for some dynamic systems.
This makes it almost impossible to predict
the behavior of long-term rendering.
This happens even when behavior of
this system is determined by initial
conditions of very same system and no
random elements are involved in process.
Dynamic systems with such conditions
are known as deterministic.
In simple words, it can be said that such
deterministic behavior or say nature
of any kind of dynamic system is not
able enough to make them predictable.
Such deterministic behavior is known
as deterministic chaos or just chaos.
The whole theory of chaos is
based on this simple fact.
Each concept of chaos theory is
based on these handful statements.
Thus, an attempt was made by
Edward Lorenz in order to describe
the main concept of chaos
theory in a single definition.
According to him:
“Present can determine the
future, but approximate
present cannot determine
approximate future.”
Many natural systems such
as weather, climate,
etc follow the rules
of chaos theory.
They possess the same chaotic behavior
as described in chaos theory.
Not only natural system, but some
artificial systems, or system that
contains artificial components also
follows the same chaotic behavior.
Road traffic is a great example of
such artificial system since it
contains multiple artificial components
that are not a part of nature.
Chaotic mathematical model is
analyzed in order to understand
such behaviors of natural and
artificial dynamic systems.
For such analyzing process,
analyzing techniques such
as recurrence plots and
Poincare maps are implemented.
Following is a list of
fields and disciplines in
which chaos theory is
applied or is applicable:
· Meteorology
· Sociology
· Physics
· Environmental science
· Computer science
· Engineering
· Economics
· Biology
· Ecology
· Philosophy
These are the field, in
which chaos theory has been
successfully applied, that
too with expected results.
There are many other fields,
in which research is
still going on about
application of chaos theory.
Deterministic systems are
main subject of chaos theory.
Chaos theory holds a great concern
about these deterministic systems.
Especially those systems, this
behavior can be predicted
that is within the scope of
principle, and related concepts.
In the beginning, such chaotic
systems tend to be predictable.
Later on, they turn out to be random.
It means that chaotic systems
are predictable until
a specific amount of time
from their generation.
This amount of time can be
derived from three aspects:
· Level of expected uncertainty in
the forecast or simply prediction
· Accuracy in the measurement of the
current or last available stage
· Lyapunov time, which is the time scale
fully dependant on system dynamics
Following are some
examples of Lyapunov time:
· Chaotic electric circuits
[1 millisecond, almost]
· Weather system [several
days, yet unproven]
· The great solar system [50 million years,
most accurate since we cannot test it]
In chaotic systems, uncertainty in the
forecast keeps increasing exponentially.
The increment level keeps
rising as the time flows on.
The more time is passed the more
uncertainty should be expected.
Thus, according to mathematical prospective,
if the forecast time is doubled
the uncertainty will increase more than
square value of its current level.
According to this concept, chaos
theory says that accurate
prediction is hard to achieve if
it is made after a time interval,
which is double or triple
than Lyapunov time value.
A system is said to be random if any
meaningful prediction cannot be made.
According to this description,
anyone can understand that
chaos theory is all about
predictions and its accuracy.
Many concepts in chaos theory describe
how technical predictions can be made,
some other describes how
inaccuracy can be reduced
and accuracy can be increased
in those predictions.
This is the main base for
each concept of the theory.
The learner of this theory
will study how this
concept is applied in various
subjects and fields.
Sources, for chaos theory can be
found in many other theories.
Such as pure mathematics,
pure physics, theory of
computer science, theory
of environmental science,
economic theory, biology and
many other concepts and fields.
Chaotic Dynamics
Generally, chaos means
the state of disorder.
Ignoring its general meaning,
Chaos theory defines the
term “Chaos” more descriptively
using better precision.
No mathematical definition
of chaos is yet developed.
Those available are not
accepted by mathematicians.
Definition of chaos given by Robert L.
Devaney
is the most commonly used
definition presently.
According to this definition, the
dynamic system, which possess
following properties can be
defined as chaotic system:
· It should respond sensitively
in various initial conditions
· It should be act as mixed system
according to the concepts of topology
· It must possess periodic
orbits with noticeable density
Following is a list of chaotic dynamics:
Sensitivity to initial conditions:
Sensitivity to initial condition means that
each point of a typical chaotic system
is closely related with
other point in different
future paths that are
known or trajectories.
Thus, it is derived that if
trajectory faces even a small change,
future path is likely to be
affected at a much higher
level, and the amount of the
modification is high too.
Because of the modification of future
path, future behavior will change too.
As described above, sensitivity
possessed by initial
conditions are referred
as butterfly effect.
This name has a historical reason.
In 1972, Edward Lorenz
submitted a paper to
American Association for
Advancement of Science.
The title of that paper
was “Does the flap of a
butterfly’s wings in Brazil
set off a tornado in Texas?”
In the title described, flapping
wing represents that initial
condition is facing
a small change
that is causing multiple
subsequent events that are
able enough to initiate a
phenomena of large scale.
If the butterfly had not
flapped its wings, the
Status of initial system
might have been different.
If a finite amount of
information is taken about
the system in the
begging of the process,
then after a certain time
has passes, the system
will not remain as predictable
as it used to be.
This is a consequence of
sensitivity to initial conditions.
This effect can be observed in cases where
prediction can be made about a week ahead.
Weather can be a great example
to understand this effect
since it can be predicted one
week ahead or even more.
This does not mean that it
is completely impossible to
predict about event that are
likely to happen in far future.
However, there are some heavy
restrictions to do so.
For instance, using weather
prediction, we can determine the
lowest and the highest possible
temperature throughout the year,
but we cannot define exactly
which day is going to be
the hottest or the coolest
day for that very same year.
Sensitivity of initial
conditions can be measured
according to a mathematical
term named Lyapunov exponent.
To do so, two starting trajectories
in the phase space are assumed.
These trajectories are
infinitesimally close.
In this equation, t represents time
and λ represents exponent of Lyapunov.
Orientation of the initial separation
vector determines the separation rate.
This is the actual reason for
which Lyapunov spectrum exists.
The number of dimensions is derived from
the number of Lyapunov exponents since
these both are equivalent numbers, which
means both have same values or count.
If number of dimensions
and Lyapunov exponents
are different, then large
one is preferred more.
For example, Maximal Lyapunov
exponent, which is known
as MLE helps to derive the
predictability of any system.
If the value of MLE is positive, the
corresponding system is considered chaotic.
Measure theoretical mixing and
K-system properties are great examples
of other properties that correspond
to initial condition sensitivity.
Topological Mixing:
Concept of topological mixing is also
known as topological transitivity.
Topological mixing means that with the flow
of the time the system will evolve with it.
It indirectly means that
existing open set or regions
will be overwritten with
any other new region.
These regions or open sets correspond
to phase shape of the very same system.
Mixing is a mathematical concept.
According to the concept of
mixing standard intuition and
colored dyes or fluid mixing are
examples of a chaotic system.
In chaos theory, topological
mixing is referred multiple times.
It describes that chaos
corresponds only with
sensitivity found in
initial conditions.
However, it is impossible
to define chaos by only
using sensitive dependence
of initial conditions.
To understand this concept,
refer to this example:
Consider a dynamic system is developed by
doubling the initial value subsequently.
In such system, any pair of point
will be separated as the time passes.
Thus, the system will have
sensitive dependence on
initial value, almost
everywhere in the system.
However, this example is not based on
topological mixing, hence has no chaos.
Although, the behavior in example is
extremely simple because all points
rather than zero will be infinite
value both positively and negatively.
Density of periodic orbits:
If a chaotic system has periodic
orbits with noticeable density,
it means that periodic orbit
has covered every point in
that system space with a close
arbitrarily approaches.
Using Swarovski’s theory as a base
Li and Yorke managed to prove that
every one-dimensional system that
holds a regular periodic cycle
also corresponds to the regular
cycles of other length.
This kind of system also
corresponds to chaotic orbits.
This research was done in
1975 by these two theorists.
Strange attractors:
One-dimensional logistic map
that is a dynamic system defined
with x - > 4 x (1-x) are said to
possess chaotic behavior everywhere.
However, sometimes this
kind of chaotic behavior
is only found in certain
subset of phase space.
The most interesting
cases are those when
chaotic behavior takes
place upon an attractor.
Consequently, orbits were lead to the
very same coverage of this chaotic
region by a massive group of initial
conditions or circumstances.
Described here are the most
applied dynamics of chaos theory.
Other less applied yet important
chaotic dynamics are listed below:
· Chaotic system at its minimum complexity
· Jerk systems
Concept of Spontaneous Order
Spontaneous order is also
referred as self-organization.
In chaos theory, it is the
spontaneous emergence of order.
This concept finds its origins
in many theories such as
physics, biology, social
network and economics also.
In biological and physical
processes, this term is often used.
Spontaneous order is used to define
emergence of various social orders
that are derived from a combination
of individuals with self-interest.
Those individuals should not try
to create an order intentionally.
Some famous examples for systems that
evolve spontaneous order are listed below:
· Evolution of earth
· Evolution of language
· Evolution of crystal structure
· Evolution of internet
· Free market economy
Organizations have made spontaneous
orders to distinguish.
Turning into scale-free network was
the main reason for them to be
distinguished, since organizations
follow hierarchical networks.
Moreover, organizations are most likely to
be a part of spontaneous social orders.
Here, it is noticeable that
spontaneous orders cannot be a
part of hierarchical networks
like these organizations.
There is a one thing that
separates both the concepts.
Humans are responsible for creating
and controlling organizations.
Spontaneous orders are created
and controlled by themselves.
They cannot be controlled
by anyone and anyhow.
Following the fact, economics
and social studies describe
that spontaneous orders are
result of human actions;
they are not resulted from human design.
According to the concept
of emergent behavior,
spontaneous orders are
considered as a kind of synonym.
Self-interested spontaneous
order is nothing but just an
instance of this very same concept
called emergent behavior.
History:
Murray Rothbard describes
that main idea of
spontaneous order was
presented between 369-286 BC.
Yes! It is that old.
Zhuangzi is considered the
constructer of this concept.
He ignored or literally rejected to
accept Confucianism’s authoritarianism.
He believed that there has
never been such thing
that tried or attempted
to govern humanity.
Instead he said that there
was always been a concept,
which lets humankind roam alone
and free of restriction.
He managed to describe how good
order is developed automatically
and spontaneously, when we
let the facts let alone.
Later, this concept was
developed further by Proudhon.
In the field of market, first every
serious studies of spontaneous
order were done by the theorists
of Scottish enlightenment.
The modern concept was described
by Adam Ferguson in around 1767.
He also proved (with different
approach) that phenomenon
of Spontaneous order is the
result of human actions.
They cannot be developed by
executing any human design.
This concept was later refined
by Austrian School of Economics.
This school was led by Carl Menger,
Ludwig von Mises and Friedrich Hayek.
They managed to get this concept at center
of their social and economical fields.
Following are some examples of
Spontaneous order in various fields:
Game Studies:
Modern game studies share
a strong connection
with the concept of
spontaneous order.
In 1940s, important results were
disclosed by Johan Huizinga.
He wrote that forces of
civilized life share
their origins in both
myth and rituals.
He termed civilized life
as massive instinctive.
He also wrote that law, order,
commerce, profit, craft, art,
poetry, wisdom, science, etc all
are the roots of soil of play.
Following the results of book
named The Fatal Conceit,
Hayek also presented his
thoughts about this concept.
He termed games as instance in fact,
clear instance of a progress.
He said that in such progress common
rules are followed by all participating
elements, their conflicting different
purposed result as overall order.
Markets:
Hayek is one of those
many liberals who argued
that market economics
indeed a spontaneous order.
He also claimed that economics are superior
order than any other order and humans
cannot design great model than this due
to lack of specifics and information.
This information or specifications
cannot be derived from statistical data,
because statistical data
is gathered by abstracting
the particulars away from
the proper situation.
According to the concept
of market economy, price
can be aggregated from
required information.
There is only one condition
that people must be set free to
use their individual knowledge
rather than derived theory.
Everyone is then allowed
to deal with commodity
or substitute in order
to make decision.
This decision will be
based on more information
than an individual person
can personally gather.
It means that information
cannot be conveyed
using a statistical
centralized authority.
Any interference made by central
authority will affect the price since
no one will have information about
all the particulars involved.
Anarchism:
Many anarchists argue that state is
artificially formed by ruling elite.
They also think that if we
eliminate these ruling elite,
true spontaneous order would
arise as consequences.
According to anarchist, prospective,
voluntary co-operations of
individual particulars would be also
involved in such constructed order.
Oxford dictionary of sociology describes
that according to anarchist vision,
it is compatible if multiple symbolic
ineractionists combine their work.
It also refers spontaneous order to
be a particular view of society.
Sobornost:
Russian Slavophile movements
and Fyodor Destoyevsky’s
work describes the concept of
spontaneous order briefly.
The main concept of organic
social manifestation is described
under the concept of sobornost
as a concept in Russia.
Leo Tolstoy used sobornost as a base
in Christian anarchism ideology.
Uniting force behind the peasant,
serf obshchina in pre-soviet
Russia is described using
the concept of Sobornost.
Recent developments:
Friedrich Hayek is said to be
the most famous researcher
and theorist in the field of
social spontaneous orders.
He made an argument, in which
he said that common law
and brain are two separate
types of spontaneous orders.
He termed this argument as catallaxy.
In The republic of science, science
was described as a spontaneous order.
Michael Polanyi was the one
who made such argument.
This concept was further pursued and
developed by Bill Butos in multiple papers.
Thomas McQuade also contributed a lot
with his work on these researches.
History of Chaos Theory
Henri Poincare is said to be the earliest
proponent in the field of chaos theory.
Henri Poincare was pursuing studies in
three body problems in early 1880s.
During that time span, he discovered
that some orbits could be non-periodic.
He also found that such
orbits do not increase
infinitely and nor they
approach a fixed point.
In 1898, an influential study of chaotic
motion was published by Jacques Hadamard.
The motion he studies
was of a free particle
that is frictionless
gliding on a surface.
This surface tends to
have negative curvature.
Such a surface is known
as Hadamard’s Billiards.
He managed to prove that all
trajectories are unstable.
Ergodic theory was the
first field to provide
a start to the concept
of chaos theory.
Subsequently, George Birkhoff
pursued studies of chaos
theory in the field of nonlinear
differential equations.
Following is the list of every
researcher included in that research:
· George David Birkhoff
· Andrey Nicolaevich Kolmogorov
· Mary Lucy Catwright
· John Edensor Littlewood
· Stephen Smale
However, they could not successfully
observe chaotic planetary motion.
Non-periodic oscillation in radio
circuits and turbulence in
fluid motion were encountered
by many of these researchers.
However, they could not describe
it properly since they had no
proper theory to tell them what
they were actually looking at.
Chaos theory failed to be
formalized in the early
20th century, despite of
multiple initial insights.
The complete formalization of chaos
theory was done in pre 20th century.
It became possible when some scientists
observed that linear theory failed
to describe obtained behavior of certain
experiments such as logistic map.
Linear theory was considered a
prevailing system theory of that time.
“Noise” was attributed as measure
imprecision and considered as a complete
component of corresponding system according
to theorists studying in chaos field.
Simply refer to them as chaos theorists.
At that time, they were not
known as chaos theorists
since there was no fully
developed chaos theory!
Electronic computer acted as main catalyst
in order to develop chaos theory.
Repeated iteration of simple
mathematical formulas is
included in mathematical
concepts of chaos theory.
Such iteration is impractical or
even impossible to do by hand.
With the help of electrical computers,
the task of iteration was made practical
and easy, figures and images helped
to visualize the sunning system.
Random transitional phenomena
were discovered in 1961.
However, they were published in 1970.
This phenomenon was
discovered in Chihiro Hayashi
laboratory that operates
in Kyoto University.
These phenomena were noticed
by Yoshisuke Ueda, a
graduate student on 27
November in the year of 1961.
However, his adviser did
not agree to disclose the
researches without solid
explanations and proofs.
The concept was finally
disclosed in 1970, with
proper description to explain
the very same concept.
Yes! It took almost 9 years
to find explanation for
a fact that was already
discovered 9 years ago.
In 1979, Pierre Hohenberg
organized a symposium.
During that very same
symposium, Albert J.
Libchaber disclosed his
experimental observations.
In his researches, he
observed bifurcation cascade.
It was proven the very same
concepts that lead to successful
implementation of chaos in Rayleigh
Benard convection systems.
For this useful research in the
field of chaos theory, following
researchers were awarded Wolf Prize
in Physics in the year of 1986:
· Albert J. Libchaber
· Mitchell J. Feigenbaum
New York academy of science jointed
its operation with national
institute of mental health and
the office of naval research.
It was the first ever
conference based on chaos
in the joint field of
biology and medicine.
It was formed in 1986.
In which, eye tracking model was presented
by Bernardo Huberman among schizophrenics.
As consequence, physiologies
were updated or say renewed
in 1980s with implemented
applications of chaos theory.
For instance, studies
of pathological cardiac
cycles were included in
the field of physiology.
Distinguishing random from chaotic data
In practical implementation or research, no
time series is contained with pure signal,
which makes it difficult to
tell whether physical or other
kind of process is random or
chaotic by only using data.
It is hard to get pure signal to do so.
Despite the sound is preset as
truncation error, some form
of corrupting noise is most
likely to be encountered.
Thus, it can be said that any
kind of real time series
contains some randomness,
even if it is deterministic.
Almost every method used to
distinguish deterministic
and stochastic processes are
based on a simple fact.
That, any deterministic
system makes progress
in the same way it did
at given starting point.
Further, one can follow given process
if a time series is provided for test:
1. Consider a state for testing purpose
2. Compare and find a time series
with the nearest possible state
3. Compare time evolutions of both states
The difference found between
time evolution of determined
state and nearest possible
state is defined as error.
Error found in any
deterministic system tends to
remain small or keeps
incrementing as the time passes.
The increasing error level is
said to have chaos behavior.
Generally, almost every measure
taken to determine time series is
dependent on the difference between
closest state and given test state.
Correlation dimension and Lyapunov
exponents are great example for this fact.
One have to depend on phase
space and methods such as
Poincare plots to derive and
understand state of a system.
To do so, one needs to consider
an embedding dimension.
Using this dimension as a
base, propagation between
two nearby states is
investigated and explained.
If the error tends to be random,
the dimension level is increased.
At such point, if dimension can
be increased the analysis is
done because one can easily
determine error by increasing it.
It many sound very simple but it is not.
Because as the dimension is increased,
the data and calculations required are
increased constantly in order to derive
a suitable and close candidate as well.
If the dimension level is kept low,
deterministic data will tend to be random.
However in chaos, theory there is no
restriction to keep the dimension at a
higher level, the method will work the
same with any level of dimension.
If an external fluctuation attend a
nonlinear deterministic system, serious
and permanent distortions are much
likely to encounter by its trajectories.
Moreover, the inherent nonlinearity
causes noise to be amplified.
It turns out to reveal very
new dynamic properties.
There is a huge risk of failure included
in attempt and experiments to separate
noise from the deterministic skeleton
or to isolate the deterministic part.
The situation becomes harder
if the deterministic component
turns out to possess a
nonlinear feedback system.
When interaction between nonlinear
deterministic component and
noise takes place, it results
as displaying nonlinear series.
This nonlinear series is
able enough to derive
dynamics that are used
to test nonlinearity.
Such result are sometimes
not achieved or derived.
Philosophy also describes
how deterministic chaotic
system can be distinguished
from stochastic system.
It has managed to prove that
these two aspects are most
likely to be equivalent in
observational prospective.
Applications of Chaos theory
In the beginning era of
Chaos theory, it was
mainly used in order to
observe weather patterns.
Now days, it is also applicable
to other various situations too.
Following is a list of
fields, in which chaos
theory is proven applicable
in a proper manner:
· Geology
· Mathematics
· Microbiology
· Biology
· Computer science
· Economics
· Engineering
· Finance
· Algorithmic trading
· Meteorology
· Philosophy
· Physics
· Politics
· Population dynamics
· Psychology
· Robotics
Despite of containing a
noticeable number of fields,
this list cannot be said a
complete or comprehensive
list since research
is still going on
and many new fields are proven
eligible to apply chaos theory.
Some of the important fields from
list above are covered below:
Computer Science:
Use of chaos theory in
computer science is not a new
concepts, since it is been
practically used since years now.
It is also described in cryptography.
Chaos theory models an
encryption, secrete or symmetric
key that operates based on
diffusion and confusion.
Information in image or any other
form can be encrypted more
efficiently by implementing
chaos theory in DNA computing.
Chaos theory also provided
benefits to the field of robotics.
In traditional robotics,
robots interact with their
surrounding in a trial
and error type model.
Such model had its own limitations.
Chaos theory was used to build
a predictive model, which is
considered as an important
revolution in the field of robotics.
Biped robotics also managed to
take advantage of chaos theory.
Chaotic dynamics were used, in order to
develop passive walking biped robots.
In order to develop passive
walking biped robots, whole
concept of passive walking
was based on this particular
theory, in the field of robotics
and especially biped robots.
Biology:
Since hundreds of years, population
model is being used by many biologists.
The purpose of population
model is to keeping
track of population
of various species.
Almost every population model was considered
continuous, until scientists managed
to implement population models with
chaotic behavior for various species.
Population model of Canadian lynx is a
great example for such behavior, since it
was observed that population of this
particular specie possessed chaotic behavior.
Chaotic behavior was also observed
in various systems of ecology.
Hydrology is a great example of
ecological chaotic behavior.
Many scientists argue that there
is much left to learn about
hydrological behavior from
prospective of chaos theory.
Cardiotocography is another example for
application of chaos concepts in ecology.
To obtain accurate information, fetal
surveillance is used as ideal balance.
The process is enforced to be
noninvasive as far as possible.
Chaotic modeling allows
getting better models of
warning signs that are a
part of fetal hypoxia.
Application in other fields:
In the process of
manufacturing polymers, it
is essential to predict
solubility of gas.
In such process, wrong
points are much likely to be
covered if particle swarm
optimization (PSO) is used.
Implementation of chaos allowed
developing improved version of PSO.
This update version
allows the simulation to
keep flowing without being
stuck at any point.
In celestial mechanics, asteroids
are observed in order to predict
their behavior such like when they
reach earth or other location, etc.
Applying chaos theory in such
observation allows getting
better predictions about
behavior of such asteroids.
As known, Pluto has five moons in total.
Four of these five moons are
observed rotating chaotically.
Chaos theory is also used in quantum
physics and electronic engineering.
It is used greatly in studies of large
arrays present in Josephson junctions.
Coalmines are always dangerous, since
natural gas leak often cause many deaths.
It was tough to predict that
exactly when they occur.
Recently, it had been proven that these gas
leaks tend to possess chaotic tendencies.
If scientist manage to model
these tendencies properly, gas
leaks are much likely to be
predicted with noticeable accuracy.
This theory is applicable even outside
the concepts of natural science.
A model for career counseling can be
determined by including a chaotic
interpretation of the relationship exists
between job market and its employees.
This model will help a lot to
provide better suggestions to those
people who are struggling to make
appropriate career decisions.
According to theory of chaos, organizations
are complex adaptive systems.
These systems are said to contain natural
fundamental structures that are nonlinear.
These structures correspond
to various forces.
These internal and external forces
are said to be the source of Chaos.
In the field of verbal theories,
the chaos metaphor is used.
To describe small work group
complexity, mathematical and
psychological aspects are provided
related to human behavior.
These aspects are used to
derive useful insights
to study complexity
of small work groups.
This kind of research takes the concept
beyond the concept of metaphor itself.
Traffic forecasting also receives great
help from applications of chaos theory.
It allows to predict various
kind of behavior of traffic like
when traffic will start, end,
what level it will possess,
how thick it will be can
be predicted before
its occurrence by
applying chaos theory.
A basic overview of broad
Chaos theory was provided.
You should have already
got an idea that chaos
theory was all about
predictions and its accuracy.
The applications of chaos theory are not
limited as shoved in the final chapter.
Since, many researches are
going on in order to make it
possible to apply chaos theory
in more and more fields.
Until the date, most of
the scientific fields
are proven applicable
of chaos theory.
Many are still to be proven.
However, chaotic behavior
is very common to
find in any field of
studies and researches.
All theorists need to do
that to find practical
proofs in order to prove
the observation legit.
Late 20th century was proven a
huge bang for theory of chaos.
As it can be seen in the chapter
about theory, many major
researches are done in the time
span of late 20th century.
Most of the modern
researchers are following
the researches done
in previous century.
Thus, it can be said
that base of the chaos
theory was constructed
in that span of time.
Researchers are still trying and
putting their efforts to take
this theory at a next level with
more accurate and legit results.
