A system of ordinary differential equations can often be written in the form [2][1]
where 
is the independent variable (normally time) and
Equation (1) consists of 
first-order differential equations.
If the original set of equations is of higher-order, they can often be
transformed to Eq. (1) by redefining variables and expanding their
number.
(It may be recalled that in classical mechanics the 
second-order
differential equations arising from a Lagrangian formulation are
equivalent to the 
first-order equations from the Hamiltonian
formulation.)
The system of equations in (1) is an example of a flow,
which gives rise to a continuous evolution of ``field lines'' in
the -dimensional space (sometimes called the ``phase space'')
[2].
We say that the flow is conservative if the volume in phase space
remains constant with time. If the volume decreases with time, there is a
damping effect in the dynamics, and we refer to the resulting flow as
dissipative.
(A frictional force in a mechanical system gives rise to dissipation, as
does a resistor in an electrical system.)
Also, it is sometimes convenient to write Eq. (1) in column vector form
where
Chaotic effects can arise from Eq. (1) if two conditions are
satisfied. First, at least one of the functions 
must contain a nonlinear term (e.g., 
, 
, 
).
Second, there is a theorem [1]
which states that chaos only occurs in
Eq. (1) if 
.
For example, the famous Lorenz equations are given by
[2][1]
where 
, 
, and 
are all constants. Note the nonlinear
terms
and 
in the second and third equations, respectively, of
Eq. (5). For 
, 
, and 
, one finds
that the system of equations in (5) gives rise to the so-called
Lorenz attractor [2][1]
(a ``strange attractor''), which in 1963 was historically
the first evidence of chaos in a dissipative system.
(Shortly before 1900, Henri Poincaré discovered chaos in a conservative
system.)
On the other hand, dynamical systems are often modelled as maps, e.g., [3][2]
where we define the column vectors
The symbol 
labels the 
iteration of the map.
Comparing Eqs. (3), (4), (6),
and (7), we see that the
iteration index can, in some cases, be considered a label for the time
(taken in discrete ``jumps'').
Indeed, a map can be generated from a flow by taking:
where 
is the ``time delay'' [2].
In the right hand part of Eq. (8) we indicate the iterations
corresponding
to the discrete flow times in the left hand side.
In other examples, maps can
arise from flows by taking various types of Poincaré sections, in which
one or more variables are fixed [2].
However, in many cases maps have been constructed as pure dynamical
systems, without recourse to an underlying system of flows. Also, just as
with flows, one can determine whether the map is conservative or
dissipative [3][2].
As before, in order to obtain chaos for a map the equations (6)
must contain at least one nonlinear term. However, for a map chaos can
arise even for a one-dimensional system [3],
which should be contrasted with the
corresponding criterion for a flow. This is a very important point since,
in order to study chaotic effects one need only consider one- or
two-dimensional maps, whereas for a flow at least three dimensions is
mandatory. Moreover, for a given dimensionality , the map equations
(6) are much easier to use than the flow equations (3).
For a mapping, we study the dynamics by simply iterating, whereas for a
flow we must solve a system of differential equations, which in general is
nontrivial.
Thus, for the remainder of this chapter we will focus on chaos in several
simple one- and two-dimensional maps.
