4.2 Tent Map II

Question 1

What is the notation for the regular tent map, and one that has had p iterations

Answer 1

M(x) is the regular tent map

M^p (x) is the p iterated map

Question 2

Describe the coordinates for M(x) when x = :
0, 1/2, 1
1/4 3/4

Answer 2

M(0) = M(1) = 0
M(1/4) = M(3/4) = 1/2
M(1/2) = 1

Question 3

What is M^2(x) equal to for the intervals, and state the gradients
0 < x < 1/4
1/4 < x < 1/2
1/2 < x < 3/4 
3/4 < x < 1

Answer 3

0 < x < 1/4 - L^2. Gradient = 4
1/4 < x < 1/2 - R L. -4
1/2 < x < 3/4 - R^2. 4
3/4 < x < 1 - L R. -4

Question 4

What equation relates x twiddle N, x_N, epsilon_0 and e^Nλ?

Answer 4

x_N twiddle - x_N = epsilon_0 e^Nλ

Question 5

For p iterates of the tent map, what is the width and centre point of each mini triangle/tent?

Answer 5

Width: 2^(-p + 1)

Centre point = 2^-p