The following recurrence relation is called the Logistic Equation:Write a program that per...

The following recurrence relation is called the Logistic Equation:

Write a program that performs forward evaluation of this recurrence relation, with arbitrary values for

x(0)

and

gain.

Use a

for

loop to iterate through

n

evaluations. After each evaluation, print the value of

x

to six decimal places.

Using

n = 100

and the starting value,

x(0) = 0.2,

describe the qualitative behavior of

x(k)

as

k

increases from zero, for

gain

in each of these three ranges:

0.0<gain<1.01.0<gain<2.03.0<gain

Change the total number of iterations to

n = 1000,

and describe the behavior in this range:

2.0<gain<2.569945...

(the Feigenbaum Point)

In this range, there are subordinate ranges between thresholds 2.0, 2.4478 …, 2.5438 …, 2.5643 …, 2.5687 …, etc., where each of these ranges is smaller than the previous one by a common scaling factor of 4.6692 .… This scaling factor, called the Feigenbaum Constant, is a universal constant, like pi, and it appears in many natural phenomena. For the clearest picture of what's happening, focus on gains just before subordinate range boundaries, like gains of 2.44, 2.54, 2.56, and 2.568.

For gains in the range between the Feigenbaum Point and 3.0, the dynamics are chaotic. When the

gain

equals 3.0,

x

-values become uniformly distributed in the range,

0 < x < 4/3,

and the behavior seems to be random. But of course, it is not random. It is deterministic, because each value depends on the previous value in accordance with the Logistic Equation’s formula.