The logistic map is the most important toy example of nonlinear dynamics. Depending on the value of the parameter , various kinds of dynamic behavior emerge.

After the contraction theorem an iteration converges if the absolute value of the derivative of the iteration function has a value smaller than 1 at the attractor . This means geometrically that the slope of the graph of is smaller than the slope of the straight with the equation , which is used for the geometric construction of the iteration sequence and makes the theorem evidently clear.

To investigate the behavior of the logistic map for values of , it is helpful to not only have the graph of but also the iterates of the form ; up to the order of 6 is provided here.

The bifurcation for appears because switches between two values, the intersections of the graph of with the straight. It can be observed that at these values the contraction theorem is complied. This holds for all cases where the iteration has a finite number of solutions. Adjust the initial value for a distraction-free image.

— number of iterations — parameter of the logistic map