Chaos Theory

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.