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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.

a stable equilibrium – 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.

an unstable equilibrium – 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.

another saddle point

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.

an unstable spiral 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.

a center