Math 5110/6830
Homework 4.1
1. Second-iterate equation for the logistic map
xn+1 = f 2 (xn )
has right hand side
f 2 (x) = r2 x(1 − (r + 1)x + 2rx2 − rx3 ).
√
r+1± (r−3)(r+1)
a) Show that x∗ = 0, r−1
,
are the fixed points of this map.
r
2r
b) Which of the fixed points form the 2-cycle of the original system? For
what values of r does it exist? What do the other points correspond to in the
original logistic equation?
c) Compute
d 2
dx f (x).
d) Verify that the
√ non-trivial 2-cycle is stable for 3 < r < 1 +
unstable for r > 1 + 6.
√
6, and
2. (MATLAB) a) For the logistic map xn+1 = rxn (1 − xn ) numerically
find x2 , x3 , . . . , x10 for r = 3.2 and x1 = 0.513. Plot xn vs. n. Did you find
the 2-cycle? Also do the cobwebbing (plot functions in Matlab, then cobweb by
hand or in Matlab) to see the same 2-cycle.
b) Repeat a) with r = 3.55 and x1 = 0.8874. What do you find?
c) Repeat a) with r = 3.8 and x1 = 0.5. What do you find?
3. (MATLAB. Extra credit for 5110 students. Required for 6830
students) Generate Feigenbaum diagram of the logistic map. To do this, fix
value of r (say, r = 0.1), then start with a hundred of different initial conditions
(100 of x1 values from 0 to 1), iterate each initial condition 50 times, and plot
x50 as a function of r. Repeat for different values of r, between 0 and 4 to
see the stable part of the bifurcation diagram, with stable fixed points, stable
2-cycles, 4-cycles and chaos. Label them in your graph (by hand is fine).
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