State of the System: A real number between -1 and 1.
Rule of Evolution: 
, where 
.
You can check that for these values of a, 
stays within the interval
[-1,1] when 
belongs to this interval.
Motivations: One of the simplest non-linear system. Related maps are sometimes used for population growth (see e.g. J. Hofbauer and K. Sigmund, The theory of Evolution of Dynamical Systems, Cambridge University Press, 1988). See also chapter 11 of Chance and Chaos.
Main Results:
If a<0.75, the system has one fixed point. When a is
increases slightly above 0.75, a cycle of length 2 appears.
This qualitative change is called a bifurcation.
This change is illustrated in Figs. 
and .
Figure: 20 iterations with a=0.5 and x(0)=0.4
Figure: 20 iterations with a=0.9 and x(0)=0.4
If a is
increased slightly above 1.25, a cycle four appears.
The lengths of the cycles keep being doubled when a
is further increased until a reaches the value 1.401155...
The successive bifurcations are sketched in Fig.
.
Figure: Asymptotic values of the quadratic map as afunction of a
Between this value of a and 2, many interesting things happen (fell free to
make experiments!). At a=2, we have a typical example of sensitive
dependence on the initial conditions. The situation is illustrated in
fig. .
Between this value of a and 2, many interesting things happen (fell free to make experiments!). At a=2, we have a typical example of sensitive dependence on the initial conditions. Nevertheless, a probabilistic description is possible (see next section)
Important Concepts: Non-linearity, bifurcations, sensitive dependence on the initial conditions.
Exercise : Find a value of a between 1.25 and 1.4 for which you have a cycle of length 8.
Exercise : Repeat the exercise suggested in the Introduction (for
a=2). Describe as precisely as you can the different evolutions
obtained by changing 
by 0.001.
