AM 3813A: Nonlinear Differential Equations and Chaos
ASSIGNMENT #2 (Wednesday, September 25, 2024)
Due 5:00 p.m., Monday, October 7, 2024.
Note: For Questions 4, 5, 6 and 7, any computer programs
(such as Matlab, Maple, Python, R) can be used, and need to be included in the solution (e.g, using typing, screenshot, etc.)
1. Consider the map xn+1 = xn(3) . Find all the fixed points and classify their stability. Use the
following methods:
a) an analytic approach; and
b) a graphical technique.
2. Consider the map xn+1 = 3 xn − xn(3) .
a) Find all the fixed points and classify their stability.
b) Draw a cobweb starting at x0 = 1.9.
c) Draw a cobweb starting at x0 = 2.1.
d) Try to explain the dramatic difference between the orbits found in parts (b) and (c). For instance, can you prove that the orbit in (b) will remain bounded for all n? Or that |xn | → ∞ in (c)?
3. Consider the map xn+1 = μxn − xn(3) .
a) Find and classify all the fixed points as a function of μ .
b) Find the values of μ at which the fixed points bifurcate, and classify those bifurcations.
c) Derive the 2-cycle solutions and determine their stability. When it it superstable?
d) Plot a partial bifurcation diagram for the map. Indicate the fixed points, the 2-cycles, and their stability.
4. Write a small computer program and use computer to generate the orbit diagram of the logistic system. (See Fig. 4.19 in the notes on page 93, and seepage 92 for how to coding the program.)
5. Write a small computer program and use computer to generate the diagram of Lyapunov exponents of the logistic system. (See Fig. 4.23 in the notes on page 97, and see the same page for how to coding the program.)
6. Write a small computer program and use computer to generate the cobweb graph for the tent map for µ = 1.5 with the initial point x0 = 0.8 (see Eq. (4.59) on page 96).
7. Consider the tilt-tent map: xn+1 = f(xn ), where
Fig. 1 The tilt-tent map.
a) Write a small computer program and use computer to generate the cobweb graphs for this tilt-tent map with the following values of µ = 0.8, 1.2, 2.0, 3.0, 3.01, 3.5. Try to find the critical value of µ, leading to chaos.
b) Is it possible to find such critical value of µ analytically?