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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