# Phase plane analysis: examples

## 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.