LOGISTIC MAP

The Logistic Map,

yn+1 = a yn (1 - yn)

is one of the most simple forms of a chaotic process. Basically, this map, like any one-dimensional map, is a rule for getting a number from a number. The parameter a is fixed, but if one studies the map for diffrent values of a (up to 4, else the unit interval is no longer invariant) it is found that a is the catalyst for chaos. Below, you will attempt to find for what values of a will the map be chaotic.

How will this be achieved? Below, the left hand plot is called a Cobweb Plot. These types of plots are very helpful in determining the asmyptotic (long-term) dynamics of a system. Upon inputing an inital condition, the plot will "map" the inital point to the mapped point. To do this, draw a straight line from the inital point on the graph to the diagonal. Then draw a straight line from the diagonal to the plot.

Why does this work? By drawing a straight line to the diagonal from the graph, you are in fact "resetting" the plot with an inital condition that is the result of your previous inital condition.

I invite you to utilize the below operati to make an estimate of what diffrent values of a will do to the map.

To see what the attractors for any value of a are, link to the Bifurcation Diagram

Cobweb PlotTimeseries Plot

Do you think you know where the "cutoff" is? Link to the Bifurcation Diagram to see if you are right.

Source Code