## Example 1 - a stable equilibrium - a sink

Consider the system \(\frac{dx}{dt}=-x, \frac{dy}{dt}=-4y\)

Plotted with a large number of initial conditions, we see that all
solutions converge to \((0,0)\), which is a stable equilibrium point
for the system - a *sink*.

## Example 2 - an unstable equilibrium - a saddle

Consider the system \(\frac{dx}{dt}=-x, \frac{dy}{dt}=4y\)

Again plotted with a large number of initial conditions, we see
that all solutions apart from \(y=0\) flee the point \((0,0)\) which
therefore is an *unstable* equilibrium point for the system -
a *saddle*.

## Example 3 - another saddle point

Consider the system \(\frac{dx}{dt}=2x, \frac{dy}{dt}=2x-y\)

Again most solutions flee the point \((0,0)\), which therefore is
an *unstable* equilibrium for the system.

## Example 4 - an unstable spiral source

Consider the system \(\frac{dx}{dt}=x+2y, \frac{dy}{dt}=-2x+y\)

Solutions flee the point \((0,0)\) in a spiral mode. Again
\((0,0)\) is an *unstable* equilibrium for the system - a *source*.

## Example 5 - a center

Consider the system \(\frac{dx}{dt}=-x-y, \frac{dy}{dt}=4x+y\)

Solutions circle around the point \((0,0)\), which is a *center*
for the system.