Homework 4
Math 323, Fall 2012
Due Date: Thursday, October 4
1. Use Mathematica to draw a bifurcation picture for the functions f (x) = sin(kx), where
2 ≤ k ≤ π. I recommend the following approach:
• Start by creating a list of values for k in the interval [2, π]. Use 100 values of k
while experimenting, and then increase to 800 values for the final notebook.
• For each value of k, compute the values x1001 , . . . , x2000 in some typical orbit
(ignoring x1 , . . . , x1000 ), and make a list of the points (k, x1001 ), . . . , (k, x2000 ).
• Use the ListPlot command to plot all of the points together.
Use PlotStyle -> {PointSize[Tiny], Blue, Opacity[0.2]} and ImageSize->800
to improve the quality of the ListPlot. Also, remember to increase to using 800 different values for k.
2. Let f : R2 → R2 be the function f(x, y) =
0.4x2 + y, 0.5(x + 1)y .
(a) Consider the following grid of rectangles:
H-1,1L
H-1,-1L
H3,1L
H3,-1L
Use the ParametricPlot command to draw the image of this grid under the
function f.
(b) Find the three fixed points of f.
(c) Use the eigenvalues of the Jacobian to classify each fixed point as a source, a sink,
or a saddle.
(d) Make a plot of the first 50 points in the orbit of (2.45, 0.015) under f. Briefly
describe how this orbit is affected by each of the three fixed points.