The Logistic Map,
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 Plot||Timeseries Plot|
Do you think you know where the "cutoff" is? Link to the Bifurcation Diagram to see if you are right.