Takehome Midterm
Math 323, Fall 2012
Due Date: Saturday, October 20
Rules: This is a midterm exam, not a homework assignment. You must solve the problems
entirely on your own, and you should not discuss the problems with any other students
in the class, or with anyone on the internet. Most of these problems require the use of
Mathematica, and you should feel free to use Mathematica’s online help system, as well as
other Mathematica resources available on the web.
1. Let f : R → R be the following function:



1−x
if x ≤ −4,


f (x) = (11 − x)/3 if −4 < x ≤ 2,



7 − 2x
if x > 2.
(a) This function has a single attracting cycle. Find it, and prove that it is attracting.
(b) Determine the basin of attraction for the cycle you found in part (a). Justify your
answer.
2. Let T : [0, 1] → [0, 1] be the tent map

2x
if 0 ≤ x ≤ 1/2,
T (x) =
2(1 − x) if 1/2 < x ≤ 1.
(a) Find a point p such that 0.4 < p < 0.4001 and T 14 (p) = 0. Express your answer
as a fraction in lowest terms.
832
is periodic under T . What is its period?
3391
832
(c) Find a point q in the orbit of
such that 0.65 < q < 0.66. Express your
3391
answer as a fraction in lowest terms.
(b) The point
3. Let C be the circle x2 + y 2 = 1 in the xy-plane, and let f : C → C be the map
f (θ) = 2θ (mod 1), where 0 ≤ θ < 1. The following figure shows a certain 8-cycle for
the function f :
Find the values of θ for the eight points of this cycle, expressing your answers as
fractions in lowest terms. Justify your answer.
4. Let f : R2 → R2 be the following function:
f(x, y) =
0.9x, −0.29x2 + 1.1y .
Note that f has a fixed point at (0, 0).
(a) Use the Jacobian of f to show that the fixed point at (0, 0) is hyperbolic.
(b) Use the ListLinePlot command to graph the orbits of several points on the line
segment [−2, 2] × {2}. Use at least 50 different starting points, and make sure to
graph at least 50 points in each orbit.
(c) Find an equation (in terms of x and y) for the unstable manifold of the point (0, 0).
Justify your answer.
(d) Find an equation (in terms of x and y) for the stable manifold of the point (0, 0).
Justify your answer.
5. Let f : [0, 2] → [0, 2] be the map

x + 1 if 0 ≤ x ≤ 1
f (x) =
4 − 2x if 1 < x ≤ 2.
For the following questions, express your answers as fractions in lowest terms.
(a) Find a point p ∈ [0, 2] whose orbit under f has Lyapunov number 2.
√
(b) Find a point q ∈ [0, 2] whose orbit under f has Lyapunov number 2.
√
(c) Find a point r ∈ [0, 2] whose orbit under f has Lyapunov number 5 8.