**
**

**Tools that you need**
You will need the following this week (look them up in the online help):

**Do** (you can also use Table or NestList)

You will also need to learn how to plot lists of numbers using:

**ListPlot**

In addition, you need to recall that animation is
very simple in mathematica. Simply generate a series of
frames (e.g. using a ``Do'' loop) and then double click on one of the
frames. This automatically animates the set of frames.

**The new physics - Chaos**

Chaos, though discussed extensively for a couple of
centuries (e.g. Boltzmann and Maxwell discussed ``molecular
chaos''), it has really come into its own since the
widespread use of computers. An early surprise is that
even quite simple looking systems can have chaos, whereas
it was originally thought that chaos only
occured in systems with billions of molecules. In this
worksheet you will study perhaps the simplest
system which shows chaos, namely the ``mapping''

(1) |

This mapping models, for example, how a population density,

(2) |

where the Lyapunov exponent, , is positive.

**Problem.**

(i) Write a Mathematica code to iterate the mapping
(Eq. 1). Plot the steady state behavior of the
mapping as a function of the parameter for
**.
**

(ii) In the regime in which ``looks chaotic'' in your graph, obtain an estimate of the Lyapunov exponent using Eq. (2).