By Hari Srivastava, H. L. Manocha

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X(n) n 3 2 4 5 2. Newton's Method of Computing the Square Root of a Positive Number: The equation x 2 =a can be written in the form x = ~ (x + ajx). This form leads to Newton's method x(n + 1) = ~2 [x(n) + _a_] . x(n) (a) Show that this difference equation has two equilibrium points -,j{i and ,j{i. (b) Sketch a stair step diagram for a = 3, x(O) = I, and x(O) = -1 (c) What can you conclude from (b)? 3. C. Pielou [ 1] referred to the following equation as the discrete logistic equation: x(n + 1) = ax(n) 1 + f3x(n) , a> 1, f3 < 0.

41 42 1. 4. 449499 ... 544090 ... 564407 ... 568759 ... 569692 ... 569891 ... 449499 ... 094591 ... 020313 ... 004352 ... 00093219 ... 00019964 ... 752027 ... 656673 ... 667509 ... 668576 ... 669354 ... 22 Cycles To find the 4 cycles we solve F:(x) = x. The computation now becomes unbearable and one should resort to a computer to do the work. 544090 .... 544090 .... When ll = tL 2 , the 22 cycle bifurcates into a 2 3 cycle. The new 23 cycle is attracting for tL 3 < ll :::: tL 4 for some number tL 4 • This process of double bifurcation continues indefinitely.

X(n + 1) = 5- (6/x(n)). 7. Let f(x) = ax 3 - bx + 1, where a, b which {0, I} is an attracting 2 cycle. E R. 5 Periodic Points and Cycles 37 Consider the so-called Baker's function defined as follows. 1 for 0 < 2 - x -<- B(x)= 12x 2x- 1 1 for- < x < 1 - 2 Problems 8, 9, and 10 are concerned with the Baker's function B(x) on [0, 1]. 8. * (Hard) Draw the Baker's function B(x ). Then find the number of n-periodic points of B. 9. Sketch the graph of B 2 and then find the 2 cycles of the Baker's function B.