The tent map and logistic maps are two known chaotic maps. 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 deterministic nature of these systems does not make them predictable and this is due to chaos theory. Later it was proven that if a system is transitive with dense periodic orbits then obviously sensitivity dependence to initial condition is guaranteed. Devaney has one of the most popular and accepted definitions of chaos in which such systems must exhibit sensitive dependence to initial conditions, topological transitivity, and dense periodic orbits. Several approaches and conditions are factored before the construction of any definition of chaos. They discussed how a dynamical system with period three orbits gives an assurance that the system is chaotic. Yorke et al., 1976, concluded that period three implies chaos.
Most dynamical systems are considered chaotic depending on the either the topological or metric properties of the system. Chaos is one of the few concepts in mathematics which cannot usually be defined in a word or statement. Quite often it has been studied as an abstract concept in mathematics. Introductionĭynamical systems are part of life. Systems with at least two of the following properties are considered to be chaotic in a certain sense: bifurcation and period doubling, period three, transitivity and dense orbit, sensitive dependence to initial conditions, and expansivity. This research presents a study on chaos as a property of nonlinear science. The behavior of systems such as periodicity, fixed points, and most importantly chaos has evolved as an integral part of mathematics, especially in dynamical system.
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