A simple linear map 
of the
real line R to itself is given by f(x) = m x. Unlike the quadratic map,
this linear map can have stretching, but no folding .
The graphical
analysis shown in Figure 2.8 quickly convinces us that for 
Figure 2.8: Graphical iteration of the linear map.
only three possible asymptotic states exist, namely:
The period one points of a map (points that map to themselves after one iteration) are also called fixed points . If m ;SPMlt; 1, then the origin is an attracting fixed point or sink since nearby points tend to 0 (see Figure 2.8(a)). If m ;SPMgt; 1, then the origin is still a fixed point. However, because points near the origin always tend away from it, the origin is called a repelling fixed point or source (see Figure 2.8(b)). Lastly, if m = 1, then all initial conditions lead immediately to a period one orbit defined by y = x. All the periodic orbits that lie on this line have neutral stability.
The story for the more complicated function 
is not
much different. For this parabolic map a simple graphical
analysis shows that as 
,
In this case all initial conditions tend to either 
or 0,
except for the point x = 1, which is a repelling fixed point since all
nearby orbits move away from 1. The special initial
condition 
is said to be eventually fixed because, although it is not a fixed point itself, it goes exactly to a
fixed point in a finite number of iterations. The sticking
solutions of the bouncing ball system
are examples of orbits that could be called eventually periodic since they
arrive at a periodic orbit in a finite time.
Graphical analysis also allows us to see why certain fixed points are locally attracting and others repelling. As Figure 2.9 illustrates, the local stability of a fixed point is determined by the slope of the curve passing through the fixed point.
Figure 2.9: The local stability of a fixed point is determined by the slope
of f at 
.
If the absolute value of the slope is less than one--or equivalently, if the absolute value of the derivative at the fixed point is less than one--then the fixed point is locally attracting. Alternatively, if the absolute value of the derivative at the fixed point is greater then one, then the fixed point is repelling.
An orbit of a map is periodic if it repeats itself after a finite number of
iterations. For instance, a
point on a
period two orbit has the property that 
, and a period three point satisfies 
, that is, it repeats
itself after three iterations. In general a period n point repeats itself
after n iterations and is a solution to the equation
In other words, a period n point is a fixed point of the nth composite
function of f . Accordingly,
the stability of this fixed point and of the corresponding
period n orbit is determined by the derivative of 
.
Our discussion about the fixed points of a map is summarized in the following two definitions concerning fixed points, periodic points, and their stability [2]. A more rigorous account of periodic orbits and their stability in presented in section 4.5.
Definition.
Let . The point 
is a
fixed point for f if 
. The point is a periodic
point of period n for f if 
but
for 0 ;SPMlt; i ;SPMlt; n. The point is eventually
periodic if 
, but
is not itself periodic.
Definition.
A periodic point of period n is attracting if
. The prime denotes differentiation with respect to x.
The periodic point is repelling
if 
. The point is neutral
if 
.
We have just shown that the dynamics of the linear map and the parabolic map are
easy to understand.
By combining these two maps we arrive at the quadratic map, which
exhibits complex dynamics. The
quadratic map is then, in a way, the simplest
map exhibiting nontrivial nonlinear
behavior.
