Chapter 9 Part II

Chapter 9 Part II - 736 CHAPTER 9 Discrete Dynamical...

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736 CHAPTER 9 Discrete Dynamical Systems 9.3 Nonlinear Iterative Equations: Chaos Again ! Attractors and Repellers 1. xx nn + = 1 2 Set = 2 , which yields 2 0 −= , which has two fixed points x e = 01 , . The cobweb diagram shows that equilibria 0 is attracting and that 1 is repelling. 1 1 x n –1 –1 x n +1 One repelling and one attracting fixed point 2. + =− 1 2 1 Set 2 1, which yields 2 10 −−= , which has two fixed points x e 1 2 5 2 . The cobweb diagram shows that both fixed points are repelling. Note that there is an attract- ing cycle that passes between them. 2 2 x n –2 –2 x n +1 Two repelling fixed points and one attracting cycle 3. + = 1 3 Set = 3 , which yields 3 0 , which has three fixed points x e 101 ,, . The cobweb diagram shows that 0 is attracting and that 1 and –1 are repelling. 2 2 x n –2 –2 x n +1 One attracting and two repelling fixed points
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SECTION 9.3 Nonlinear Iterative Equations: Chaos Again 737 4. xx nn + =− 1 cos Set = cos , which yields cos . −= 0 The roots of this equation must be found numerically using a computer. Graph yx = and cos to discover that they intersect only once. This value is approximately x e ≈− 074 . (radians). The cobweb diagram shows this to be an attracting fixed point. 2 2 x n –2 –2 x n +1 Attracting fixed point ! Hindsight 5. (a) Cobweb diagrams for ya y + = 1 are shown in the following figures for different values of a and for different initial conditions. In each case y = 0 is an equilibrium point. (i) a > 1. In this case, examine the iteration yy + = 1 2 . The line + = 1 2 (as a graph of y n + 1 versus y n ) has a slope larger than 1, which means the orbit goes mono- tonically toward ±∞ (diverges) for every starting point y n 0. The cobweb diagram indicates this fact. 4 4 y n –4 –4 y n +1 y 0 a > 1 (ii) a = 1. The line + = 1 (as a graph of y n + 1 versus y n ) is a 45-degree line, which means that solutions are constant as in- dicated by the cobweb diagram. In this case all points are equi- librium points. 4 4 y n –4 –4 y n +1 y 0 a = 1
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738 CHAPTER 9 Discrete Dynamical Systems (iii) 01 << a . The line yy nn + = 1 05 . (as a graph of y n + 1 versus y n ) has a smaller slope than the 45- degree line, which means that solutions converge to zero from all starting points. 4 4 y n –4 –4 y n +1 y 0 a (iv) a = 0. The line y n + = 1 0 (as a graph of y n + 1 versus y n ) is a horizontal line. All initial points converge to zero on the first iteration as, indicated by the cobweb diagram. 4 4 y n –4 –4 y n +1 y 0 a = 0 (v) −< < 10 a . The line + =− 1 . (as a graph of y n + 1 versus y n ) is a line with slope –0.5. All initial points converge to zero, as indi- cated by the cobweb diagram. 4 4 y n –4 –4 y n +1 y 0 a (vi) a 1. The line + 1 (as a graph of y n + 1 versus y n ) is a line with slope –1. All initial points cycle between yyyy 0101 ,,,, ! , as indicated by the cobweb dia- gram. 4 4 y n –4 –4 y n +1 y 0 = –1 a
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SECTION 9.3 Nonlinear Iterative Equations: Chaos Again 739 (vii) a <− 1. The line yy nn + =− 1 2 (as a graph of y n + 1 versus y n ) has a slope less than –1, which means the solution diverges for every starting point y 0.
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Chapter 9 Part II - 736 CHAPTER 9 Discrete Dynamical...

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