1. This question deals once again with the quadratic mapping fc(x) = x2 + c, c ∈ R which you
examined in Problem Set No. 3 (PS3). Recall that fc has real fixed points for c ≤ 1/4 and real
two-cycles for c ≤ −3/4.

(a) For c < 1/4, let us restrict our attention to the action of the map fc(x) to the interval
Ic= [−x¯2(c), x¯2(c)], where ¯ x2(c) once again (as in PS3) denotes the rightmost (repulsive)
fixed point of fc. For c = −1, sketch the graph of fc(x) on a set of x-y axes along with the line
y = x. Also draw a box which encloses the square region Rc = {(x, y) | |x| ≤ x¯1(c), |y| ≤ x¯1(c)}.

The graph of the function fc(x) inside the region Rc should look like an inverted picture of the
logistic map fa(x) = ax(1 − x) inside the region [0, 1]2. As c decreases from −1, the area of
the rectangular region Rc will increase and the bottom of the parabolic graph (i.e., the global
minimum of fc(x) at x = 0) will approach the bottom horizontal boundary of the region.

(b) At a certain value of c < −1, the minimum point of fc(x) (at x = 0) will touch the bottom
horizontal boundary of the region Rc. At this special value of c, to be denoted as c = c∗, the
function fc maps the interval Ic onto itself in a two-to-one way. (Recall that the logistic
map f4(x) = 4x(1 − x) is a two-to-one map of [0, 1] onto itself. A knowledge of the dynamics
of f4(x) on [0, 1] will help in this question.) As a result, the region Rc∗ will be a square region
with center (0, 0). The “sides” of Rc∗ will be the intervals Ic along the x- and y-axes.

i. Find this special value of c, which we shall denote as c∗.

ii. Sketch the graph of fc∗ inside the region Rc∗. Show graphically that for a typical y-value
inside Rc∗, there exist two x-values, say x1 and x2, such that fc∗(x1) = fc∗(x2) = y.

iii. Sketch the graph of fc2∗(x). How many period-2 points are there?

iv. On the basis of the graphs of fc∗(x) and fc2∗(x), what do you conjecture to be the number
of periodic points of period n for fc∗(x)?

v. What do you conjecture to be the nature of these periodic points, i.e. repulsive or attrac
tive? No explanation is necessary.

vi. What do you conjecture to be the distribution of all periodic points of fc∗? A brief answer
should be sufficient.

vii. Recalling the behaviour of the logistic map fa on [0,1] for a = 4, what do you conjecture
to be the behaviour of the mapping fc∗, i.e. the behaviour of sequences generated by the
iteration of fc∗? A very short answer – one word starting with “c” – is sufficient.

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