## ASSIGNMENT 2 SOLUTIONS

Assignment: Page 34: Nos. 1 A,C,F; 4 A,B,D; 5. Page 50: Nos. 1 A,b,f,j; 2 B,C; 4 A,C,E.

Page 34

Page 50

• 1. For each of the following functions, find all fixed points and classify them as attracting, repelling, or neutral.

a) F(x) = x^2-x/2

x^2-x/2 = x => x^2 - 3x/2 = 0 => x (x-3/2) = 0 => x=0 or x=3/2.

Therefore, the fixed points of F are {0, 3/2}.
F'(x) = 2x-1/2.
F'(0) = -1/2 => 0 is attracting and oscillating.
F'(3/2) = 5/2>1 => 3/2 is repelling.

b) F(x) = x(1-x)

3x(1-x) = x => 2x - 3x^2 = 0 => x(2-3x) = 0 => x=0 or x=2/3.

Therefore, the fixed points of F are {0,2/3}.
F'(x) = 3 - 6x.
F'(0) = 3 >1 => 0 is repelling.
F'(2/3) = 3 - 6(2/3) = -1 => 2/3 is neutral.

f) S(x) = (pi/2) sin x.
The fixed points of S are {0, +/-pi/2}.
S'(x) = (pi/2) cos x.
S'(0) = pi/2>1 => 0 is repelling.
S'(+/-pi/2) = (pi/2) cos(+/-pi/2) = 0 => +/-pi/2 are superattracting.

j) T(x) = 2x if x <=1/2
2-2x if x>1/2.
Setting each piece of this two part function equal to x yields that the fixed points of T are {0, 2/3}. Also,

T'(x) = 2 if x<1/2
-2 if x>1/2

since T is pecewise linear, but the derivative of T is not defined at x=2. Therefore, T'(0) = 2 and so 0 is repelling. And since T'(2/3)=-2, 2/3 is also repelling. In fact, no periodic points for T can be attracting.

• 2. For each of the following functions, zero lies on a periodic orbit. Classify this orbit as attracting, repelling, or neutral.

b) C(x) = (pi/2) cos x.
Since C(0) = pi/2 and C(pi/2) = 0, {0,pi/2} is a 2-cycle.
C'(x) = - (pi/2) sin x.
(C^2)'(0) = C'(0) C'(pi/2) = 0 (- pi/2) = 0, and so this orbit is superattracting.

c) F(x) = - (1/2)x^3 - (3/2) x^2 + 1.
Since F(0) = 1, F(1) = -1, and F(-1) = 0, we have that {0, +/-1} is a 3-cycle.
F'(x) =-(3/2) x^2 - 3x
(F^3)'(0) = F'(0) F'(-1) = 0 (-9/2) (3/2) = 0, and so this period 3 orbit is superattracting.

• 4. Each of the following functions has a neutral fixed point. Find this fixed point and, using graphical analysis with an accurate graph, determine if it is weakly attracting, weakly repelling, or neither.

• 5. Suppose that F has a neutral fixed point at p with F'(p)=1. Suppose also that F''(p)>0. What can you say about p: is p weakly attracting, weakly repelling , or neither? Use graphical analysis and the concavity of the graph of F near p to support your answer.

Since F''(p)>0, the graph of F is concave up at the neutral point p. Graphical analysis clearly shows that p is weakly attracting on the left and weakly repelling on the right.

• 6. Repeat exercise 5, but this time assume that F''(p)<0.

When F''(p)<0, the graph of F is concave down at the neutral point p. In this case, p is weakly repelling on the left and weakly attracting on the right.