At the very least, it seems that the state should remain fixed at
, if it is reached. A point
is called an *equilibrium point* (or *fixed
point*) of the vector field
if and only if
. This does not, however, characterize
how trajectories behave in the vicinity of . Let
denote some initial state, and let refer to the state
obtained at time after integrating the vector field from
.

See Figure 15.1. An equilibrium point
is
called *Lyapunov stable* if for any open
neighborhood^{15.1} of there exists
another open neighborhood of such that
implies that
for all . If
, then
some intuition can be obtained by using an equivalent definition that
is expressed in terms of the Euclidean metric. An equilibrium point
is called *Lyapunov stable* if, for any , there exists some
such that
implies that
. This means that
we can choose a ball around with a radius as small as
desired, and all future states will be trapped within this ball, as
long as they start within a potentially smaller ball of radius
. If a single can be chosen independently of every
and , then the equilibrium point is called *uniform* Lyapunov
stable.

Steven M LaValle 2020-08-14