The Problem.
Part II. A study of the quadratic map Q(x) = x^2 + b.

Do all computations with the parameter b as an arbitrary number between -2 and 1/4 inclusive unless indicated otherwise and concentrate on values of x in the interval [-(1 + Sqrt[1 - 4 b])/2, (1 + Sqrt[1 - 4 b])/2].

1) Compute the fixed points of Q(x).

2) Compute the slope of Q(x) at each fixed point.

3) Verify that for all -2 <= b <= 1/4, if |x0| > (1 + Sqrt[1 - 4 b])/2 then the orbit of x0 will diverge to positive infinity.

4) A bifurcation occurs at b = 1/4. Describe the dynamics of ALL orbits (consider all real numbers x0) for b close to and on both sides of 1/4.

5) Let p+ > p- be the two fixed points which exist for all b < 1/4. Find the b-values (for b < 1/4) where each fixed point switches stability (if any.)

6) Graph Q^2(x) = Q(Q(x)) along with x for various values of b between -2 and 1/4. Can you tell for what b-value two period-2 points of Q(x) emerge?

7) Determine formulas for the period-2 points of Q(x) (i.e., the new fixed points of Q^2(x) which are not fixed points of Q(x).)

8) Find the b-value(s) for which the period-2 points switch(s) from attracting to repelling.

9) Let b=-1.3 and plot Q(x), Q^2(x), Q^3(x), and Q^4(x) along with the replacement line y = x. List how many periodic points Q(x) has of every period less than or equal to 4. Do you think that Q(x) has any higher periodic orbits (when b = -1.3?) Why or why not?