Homework #15

Reading: pg. 655-708

1. Let f:[1,5]->[1,5] be the continuous piecewise linear map defined by f(1)=3, f(2)=5, f(3)=4, f(4)=2, f(5)=1 and f(x) is linear between successive integers. Show that f has a period 5 cycle but no period 3 cycle. What is the implication of Sharkovskii's Theorem for this map? Find the fixed point(s) of f(x).

2. Let f:[0,2*pi]->[0,2*pi] be defined by f(x)=2x mod 2*pi (that is, f(x)=2x if 2x is in [0,2*pi), and f(x)=2x-2*pi otherwise). Prove that f is chaotic on [0, 2*pi]. (Note: View f as a circle map, that is, a map defined on the unit circle, which takes an angle theta and doubles it.) Show that f is in fact expansive (not just SDIC) on [0,2*pi].

3. a. Define code space (on two symbols) to be the space of all infinite sequences s=(b1,b2,b3,b4,...) where b(i) is either zero or one. Prove that periodic points of the shift map are dense in code space and that the shift map admits a dense orbit, that is, there exists a sequence s=(b1,b2,b3,b4,...) such that the successive iterates of f come arbitrarily close to any point in code space.

b. Show that the shift map on code space has SDIC. (Note that if two elements of code space differ in their first position then the distance between them is 1/2.) Conclude that the shift map is chaotic on code space.

4. a. Generate the bifurcation diagram for the logistic map over the parameter interval [0,4]. Include your program.

b. Use your diagram to estimate the Feigenbaum constant.

5. The Henon map on R2 is defined by x(n+1)=1+y(n)-ax(n)^2, y(n+1)=bx(n), where a and b are parameters. Take a=1.4 and b=.3 and iterate some (x0,y0) to generate the Henon strange attractor (see pg.663 of the text). Submit a plot of it. What are the fixed points (in R2) of the Henon map?

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