A typical way of getting started with studying Chaos theory is using a graphical way.

Suppose F(x) = x^{2} + a .

Draw an ordinaly graph on the field with x and y axis, the curve y = x^{2} + a .

And the line y = x , too.

Select a value of x_{0} to start.

Find a point (x_{0}, y_{0}) with y_{0} on the curve y = x^{2} + a . This is the intersection of the curve and a vertical line drawn from the point x_{0} 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(x_{0}). Let’s call it now x_{1}.

Then Find a point (x_{1}, y_{1}) with y_{1} on the curve y = x^{2} + a .

This y_{1} gives a value of F(F(x_{0}))

Let’s call this “as a result of function F^{2}(x_{0}).”

Repeat as above to watch how the result of F^{n}(x_{0}) 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 x_{0}.