TABLE OF CONTENTS The system of nonlinear ODEs
written compactly as
where
and
and where the functions f i depend solely on y, and not explicitly on the independent variable t, is called an autonomous system. In the remainder of this chapter, we deal only with autonomous systems.
If the functions f i are Lipschitz continuous in some region 
of the real n-dimensional Euclidean space, then a unique solution of eq. (2.10.2), y( t), exists in 
for the initial value problem (cf. section 2.2).
A constant vector y s which satisfies
is called a steady state (also critical or equilibrium point) of the system (2.10.2). It should be evident that multiple steady states may exist if more than one value of y s satisfies eq. (2.10.3). Also, with the IC y(0) = y s, the solution is y( t) = y s for all t ? 0.
Given an IC, it is often convenient to express the solution y( t) as a curve in the n-dimensional phase space of y, and this curve is called a trajectory. Each steady state is a specific point in the phase space, and every point in the phase space is a potential IC y(0). Each IC thus generates a trajectory in the phase space, with the direction of increasing t indicated by an arrow.
