Chaos in Dynamical Systems

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October 20, 2016
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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’.
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,