where 
are the dependent variables, t is
the independent variable time, and 
is the set of parameters for the system.
Sometimes the parameter dependence is denoted by a subscript as in 
.
A vector field is formally defined by a map 
that assigns a vector F(x) to each point 
in its domain A.
More generally, a vector
field on a manifold M is given by a map that assigns a vector to
each point in M.
For most of this chapter we
only need to work with 
.
A system governed by a time-independent vector field is called
autonomous ; otherwise it is called
nonautonomous . As we saw in section 3.4.3, any
nonautonomous vector field can be converted to an autonomous vector field of a
higher dimension.
The
flow , 
, of the vector field 
is analytically
defined by:
The position is the initial condition
or initial state . The initial condition is also written as
when it is specified at t = 0.
A solution curve ,
trajectory , or integral curve of
the flow is an individual solution
of the above differential equation based at
.
We often explicitly note the time-dependence of the position (the solution
curve to the above differential equation) by writing 
when we want to
indicate the position of the trajectory at a time t ;SPMgt; 0.
The collection of
all states of a dynamical system is called the phase space .
The term ``flow'' describing the
evolution of the system in phase space comes by analogy from the motion of
a real fluid flow. The flow 
is regarded as a function of the
initial condition and the single parameter time t. The
flow
tells us the position of the initial condition after a time
t. As illustrated in Figure 4.1(a), the position of the point on
the flow line through is carried or flows to the point .
Figure 4.1: (a) The flow of a solution curve. (b) The flow of a collection
of solution curves resulting in a continuous transformation of the manifold.
A geometric description of a flow says that it is a one-parameter family of diffeomorphisms of a manifold. That is, the flow lines of the flow, which are solution curves of the differential equation, provide a continuous transformation of the manifold into itself (Fig. 4.1(b)).
The flow is often written as 
to highlight its
dependence on the single parameter t. The flow is also known
as the evolution operator . Composition of the
evolution operator is defined in a natural way: starting at the state
at time s = 0, it first flows to the point 
and then onto the point 
.
The
evolution operator satisfies the group properties
which are taken as the defining relations of a flow for an abstract dynamical system . If the system is nonreversible we speak of a semiflow . A semiflow flows forward in time, but not backward.
An explicit example of an evolution operator is given by
solving the linear differential equation 
.
Here the vector field is found from the equivalent first-order system
( 
)
so that the vector field is
This vector field generates a flow that is a simple rotation
about the origin (see Fig. 3). The evolution operator is
given by the rotation matrix
