**LOGISTIC MAP
**

The Logistic Map,

####
y_{n+1} = a y_{n} (1 - y_{n})

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 Plot | Timeseries Plot |

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

Source Code