The graphical iteration method is a way to compute successive iterations of a map starting from an initial value . Given , its iterate is the ordinate of the point of the graph of having abscissa , therefore, to compute the sequence of iterations it is necessary to bring the various ordinates along the abscissa axis: this is achieved by means of the bisector of the first and third quadrants, whose points have equal abscissa and ordinate. The sequence is obtained by passing alternately from the graph of to the bisector along a line parallel to the abscissa axis and vice versa along a line parallel to the ordinate axis.
Use the graphical itertion method to explain the long term behaviour of the logistic map with respect to a given value of the parameter which is shown in the Feigenbaum diagram. What happens for ? How can you interpret graphically the situation? The Feigenbaun diagram shows a first bifurcation starting from : how does this bifurcation reflect into the graphic iteration? Looking at the second iterate map does enlighten the situation? The next bifurcation happens at : how does this bifurcation reflect into the graphic iteration? Looking at the fourth iterate map does enlighten the situation?