This app allows you to explore the behaviour of the logistic map

x_{new} = r x (1 - x),
including the ability to compare the evolution form different initial conditions.

Use the slider to choose a value for the growth parameter r, and an initial
size for each population. Click the Next buttons to
move to a new generation. The value for the new population (a normalised value from 0 to 1) is shown and also plotted.

Study the behaviour of the

populations for various values of the
growth parameter

r, and for different initial conditions.

## Some things to consider

Two aspects of this population process are particularly important:

- How does the specific value of r effect the long term behaviour
of x?
- How does the series of x values depend on the initial
population value?

To explore these, try various values for

r, and test the sensitivity
to the initial value by looking at the

convergence or

divergence of the two populations over time.

Look at the behaviour for the following ranges of r and make
a table to record what you find: (you can type directly into the table below)

r | Results |

0 – 1 | |

1 – 2 | |

2 – 3 | |

3 – 3.45 | |

3.5 | |

3.56 | |

3.74 | |

3.83 | |

4 | |

When the population ends on a constant, fixed value, can you calculate what that value must be?
(Hint: think about quadratic equations.)

A

bifurcation occurs when a solution to an
equation suddenly splits into two, such as you will have seen in the logistic
map when the value of

r crosses 3 (if you didn't
notice, run the above app with

r = 2.95 and then

r = 3.05). Repeated bifurcations lead to cycles with
4 values, then 8, 16 etc, before things get really crazy around

r = 3.57.

Chaos is the mathematical term that describes
when a system is infinitely sensitive to its
starting conditions, i.e. even the tiniest difference in starting point
leads to a completely different outcome. The mathematics of chaos is behind the
somewhat over-dramatised
Butterfly Effect that says the
tiny change due to the beating of a butterfly's wings in Brazil can set off
a tornado in Texas.

For values of r close to 4 it seems the logistic
map could be chaotic, but how to tell for sure? Well, there is a famous
result that says if a map has a 3-cycle, that is a solution where it keeps repeating
the same three values over and over again, then it is chaotic.

In the next section you can visualise the long term population dynamics for a range
of r values using a bifurcation diagram.
This is a diagram that shows the long time behaviour of the map for each
r — if it is a fixed value, then a single point is plotted,
if it is a 2-cycle, the both points are shown, and so on. The first few progressive
bifurcations are clearly seen, as is the chaotic region, including its
contained islands of stability.

See if you can find a value of r
that leads to a 3-cycle. Test your value with the app above.