Homework 3
Math 323, Fall 2012
Due Date: Thursday, September 27
1. Let f : [0, 1] → [0, 1] be the function f (x) = 4x(1 − x).
(a) Find the endpoints of the LRR interval (shown in Figure 1.12 of the textbook).
Your answers must be correct to four decimal places.
(b) Find the point in [0, 1] with itinerary LRR (= LRR LRR LRR LRR · · · ). Your answer
must be correct to four decimal places.
(c) Draw a cobweb plot showing the orbit of the point you found in part (b).
(d) Use the Stability Test for Periodic Points to determine whether this 3-cycle is
stable or unstable.
2. Let T : [0, 1] → [0, 1] be the function T (x) =

2x
if 0 ≤ x ≤ 1/2
2(1 − x) if 1/2 < x ≤ 1.
(a) How many different 12-cycles does T have? Explain your reasoning.
(b) Find the period-12 point for T with itinerary LRLLRRLLLRRR. Express your
answer as a fraction in lowest terms.
(c) Draw a cobweb plot showing the orbit of the point you found in part (b).
3. Let f : R2 → R2 be the map defined by the following formula:
f(x, y) =
−x2 + y 2 + 0.9x − 0.6y, −2xy + 2x + 0.5y
(a) Find the two fixed points for f. Your answers must be accurate to four decimal
places.
(b) Use the ListLinePlot command to draw 200 points in the orbit of (1, 1). Use
the ImageSize -> 800 option to increase the quality of the plot.
(c) Use the ListPlot command to draw 500,000 points in the orbit of (1, 1). Use the
ImageSize -> 800 and PlotStyle -> PointSize[Tiny] options to increase the
quality of the plot.