Chaos in Dynamical Systems
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Date
October 20, 2016
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Abstract
The behavior of dynamical system has become an interesting field of endeavor.
Periodicity, fixed points and importantly chaos of systems have evolved as
an integral part of mathematics and especially in dynamical system. We
tend to consider asymptotic behavior of systems especially in the area of
chaos. No universally accepted definition exist for chaos but we consider the
various routes to chaos including transitivity, expansivity, topological entropy,
Lyapunov exponent, dense orbits, period doubling , period three point and
sensitive dependence to initial conditions. A combination of each of these
guarantees a type of chaos. We study the various distinct routes to chaos
and how various kinds of chaos are interrelated. Properties of an unknown
map can be associated with that of the known via topological conjugacy, hence
properties of unknown maps can always be studied in terms of the unknown.
The tent map and logistic maps are two known chaotic maps. We explore how
numerical values are used to determine chaos especially in terms of Lyapunov
exponents with respect to known maps like the tent map and logistic maps.
’Chaos is when the present determines the future but the approximate
present does not approximately determine the future.‘Edward Lorenz’.
Description
A thesis submitted to the Department of Mathematics,
Kwame Nkrumah University of Science and Technology In
partial fulfillment of the requirement for the Degree of Master of Philosophy in Pure Mathematics,