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Chapter Two One Dimensional Chaos In this chapter , we study many methods of describing the way in which iterates of neighboring points separate from another : sensitive dependence on initial conditions , Lyapunov exponent and the transitivity. These notions are fundamental to the concept of chaos, which also will appear in the present section. Section one: Sensitive Dependence on Initial Conditions Before defining the sensitive dependence on initial conditions, we adopt a notation that henceforth will facilitate our discussion. We will write π: π½ β π½ signifies that the domain of π is π½ and the range is contained in π½. Definition : Let π½ be an interval , and suppose that π: π½ β π½. Then π has sensitive dependence on initial conditions at x , or just sensitive dependence at π₯ if there is π > 0 such that for each πΏ > 0, there is π¦ in π½ and a positive integer π such that |π₯ β π¦| < πΏ and |π π (π₯) β π π (π¦)| > π , that is: β π > 0 β πΏ > 0 β π¦ β π½ β π β π β |π₯ β π¦| < πΏ and |π π (π₯) β π π (π¦)| > π If π has sensitive dependence on initial conditions at each ππ π½ , we say that π has sensitive dependence on initial conditions on π½ , or that f has sensitive dependence . The β initial conditionsβ in the definition refer to the given , or initial points π₯ and π¦. the definition says that f has sensitive dependence on initial conditions if arbitrarily close to any given point π₯ in the domain of π there is a point and an π β π‘β iterate that is farther from the π β π‘β iterate of π₯ than a distance π. This has practical significance , because in such instance higher iterate of an approximate value of π₯ may not resemble the true iterate of π₯. To illustrate sensitive dependence on initial conditions, we turn to bakerβs function: Example 1 : Consider the bakerβs function B, given by: π΅(π₯) = { 2π₯ πππ 2π₯ β 1 0β€π₯β€ 1 πππ 2 2 <π₯β€1 1 Show that after 10 iterate of 1 3 and 0.333 are farther than 1 2 Solution Notice 1 3 iterate of 1 3 2 1 2 3 3 3 3 1 1 and π΅2 ( ) = so that the 3 3 alternate between and . To compare the iterate of iterate 1 3 0.333 1 is periodic point of period 2 , that is, B( ) = 1 3 and 0.333 we make the following table: 1 2 3 4 5 6 7 8 9 10 2 3 0.666 1 3 0.332 2 3 0.664 1 3 0.328 2 3 0.656 1 3 0.312 2 3 0.624 1 3 0.248 2 3 0.496 1 3 0.992 Therefore the tenth iterate of farther a part than a distance 1 3 1 1 and 0.333 are, respectively , and 0.992 which are 3 2 Example 2: Show that the tent function T has sensitive dependence on initial conditions on [0,1]. Solution Let π₯ be any number in [0,1] Claim: if π£ is any dyadic rational number (of the form π 2π in lowest terms) in [0,1] and w is any irrational number in [0,1], then there is a positive integer π such that 1 |π π (π£) β π π (π€)| > β¦(1) 2 Toward that goal, if π£ = π 2π then π π (π£) = 1 and π π+π (π£) = 0 for all π > 0 By contrast if π€ is any irrational number in [0,1] then since π doubles each number 1 in (0, ), there exists an π > π such that π π (π€) > 2 1 2 . 1 1 2 2 Since π > π , it follows that π π (π£) = 0 so that |π π (π£) β π π (π€)| > |0 β | > So the claim is valid . Next , let πΏ > 0 then there exists a dyadic rational π£ and an irrational number in [0,1]. such that |π₯ β π£| < πΏ πππ |π₯ β π€| < πΏ therefore (1) implies that 1 < |π π (π£) β π π (π€)| = |π π (π£) β π π (π₯) + π π (π₯) β π π (π€)| 2 β€ |π π (π₯) β π π (π£)| + |π π (π₯) β π π (π€)| 1 1 1 4 4 4 So either |π π (π₯) β π π (π£)| > or |π π (π₯) β π π (π€)| > . thus if we let π = then π has sensitive dependence on initial conditions at the arbitrary number π₯ , and hence on [0,1]. Basically , the reason that π has sensitive dependence on initial conditions if π₯ β 1 1 1 then |παΏ½(π₯)| = 2,so that distances between pairs of numbers in (0, ) or ( , 1) are 2 2 2 doubled in T