Chaos Theory - E-book - ePub

Mathematics contains an important study field under the nameChaos theory. Chaos theory studies the concept and behaviorof highly insensitive dynamical systems. It also studies behaviorof dynamic systems in initial conditions, which often turns outto be super sensitive at a very high level. In Chaos theory, thisconcept is referred as Butterfly effect, which is the main field ofstudy in this theory.
From which, various branches are spreadand constantly progressing and being developed. Various initialconditions are made because of some numerical errors incomputations. These errors provide wildly diverging results forsome dynamic systems. This makes it almost impossible topredict the behavior of long-term rendering. This happens evenwhen behavior of this system is determined by initial conditionsof very same system and no random elements are involved inprocess.
Dynamic systems with such conditions are known asdeterministic. In simple words, it can be said that suchdeterministic behavior or say nature of any kind of dynamicsystem is not able enough to make them predictable. Suchdeterministic behavior is known as deterministic chaos or justchaos. The whole theory of chaos is based on this simple fact. Each concept of chaos theory is based on these handfulstatements.
Thus, an attempt was made by Edward Lorenz inorder to describe the main concept of chaos theory in a singledefinition. According to him:"Present can determine the future, but approximate presentcannot determine approximate future."Many natural systems such as weather, climate, etc follow therules of chaos theory. They possess the same chaotic behavioras described in chaos theory. Not only natural system, butsome artificial systems, or system that contains artificialcomponents also follows the same chaotic behavior.
Road trafficis a great example of such artificial system since it containsmultiple artificial components that are not a part of nature. Chaotic mathematical model is analyzed in order to understandsuch behaviors of natural and artificial dynamic systems. Forsuch analyzing process, analyzing techniques such as recurrenceplots and Poincare maps are implemented. Following is a list of fields and disciplines in which chaos theoryis applied or is applicable:MeteorologySociologyPhysicsEnvironmental scienceComputer scienceEngineeringEconomicsBiologyEcologyPhilosophyThese are the field, in which chaos theory has beensuccessfully applied, that too with expected results.
There aremany other fields, in which research is still going on aboutapplication of chaos theory.