A typical way of getting started with studying Chaos theory is using a graphical way.
Suppose F(x) = x2 + a .
Draw an ordinaly graph on the field with x and y axis, the curve y = x2 + a .
And the line y = x , too.
Select a value of x0 to start.
Find a point (x0, y0) with y0 on the curve y = x2 + a . This is the intersection of the curve and a vertical line drawn from the point x0 on the x – axis.
From that point, draw a horizontal line until it meets with the line y = x.
The x value of the intersection is the very F(x0). Let’s call it now x1.
Then Find a point (x1, y1) with y1 on the curve y = x2 + a .
This y1 gives a value of F(F(x0))
Let’s call this “as a result of function F2(x0).”
Repeat as above to watch how the result of Fn(x0) to migrate, as n increases. Will it be increased or decreased forever, or come to a definite point, or go round among certain points, or?
The results are varied with the parameter a and an initial value x0.