B. Solomyak Math 480 Spring 2007
TEST PRACTICE PROBLEMS
1. Let x0= 0.8 and
xn+1 = 1.3xn(1 −xn), n ≥0.
What is the limit of the sequence xn? You have to justify your answer.
2. Consider the function F: [0,4] →[0,4] given by
F(x) = (2x, x ∈[0,2]
6−x, x ∈[2,4]
(a) Draw the graph of F2.
(b) Give a complete orbit analysis for F(x) (describe the eventual behavior of all orbits).
3. Consider the family Gλ(x) = x3+x2+λ, λ ∈(−∞,+∞).
(a) Find all λfor which Gλundergoes the saddle-node bifurcation, or prove that there
is no such λ.
(b) Find all λfor which Gλundergoes the period-doubling bifurcation, or prove that
there is no such λ.
(c) Make a rough sketch of the bifurcation diagram FOR ALL λ.
4. Let F(x) be a function which is continuous and one-to-one from the interval [a, b] into
itself. Prove that F(x) cannot have points of prime period larger than two.
5. Consider F(x) = x+x3−x4. Find the fixed points and classify them as attracting,
repelling, or neutral. If a fixed point is neutral, describe the dynamics in its neighborhood
as well as you can (e.g., weakly repelling from the left, weakly attracting from the right).
Finally, try to perform a complete orbit analysis (i.e., determine the fate of all orbits).
6. Consider the family of “tent-maps”
Tλ(x) = (λx, x ≤1/2;
λ−λx, x > 1/2.
(a) Describe the eventual behavior of all orbits under Tλ:R→Rfor 0 < λ ≤1.
