1.1 PHASE PORTRAITS

In this section you will learn to

- Plot direction field diagrams
- Plot phase plane diagrams
- Plot graphs of x and y against t.

1.1.1 Systems of differential equations

This course is about predicting the long term behaviour of systems which can be modelled by differential equations. There are many different types of differential equations but in this section we will only study autonomous systems of first order equations.

A pair of autonomous first order differential equations have the general form

If the system is linear then the functions are linear and the system is of the form

Any second order differential equation of the form

A description or the method and examples of converting a second order differential equation into a pair
of first order differerential equations can be found in the handbook.

1.1.2 Direction field diagram

Consider the harmonic oscillator with damping given by the equations

Click on field plot to open the Maple work sheet for producing the diagram.

1.1.3 Drawing the phase portrait

The next step is to turn this field diagram into a phase plane diagram by drawing trajectories through a selection of initial points scattered over the x-y plane.
Click on phase plot to open the Maple worksheet for producing a phase plot

1.1.4 Interpreting the phase portrait

Look at the phase portrait below.

Suppose you start at the point A. As t increases x increases and y decreases until x reaches a positive maximum at . Then x and y both decrease until y reaches a negative minimum. Finally x decreases and y increases as the point approaches .This behaviour can be confirmed by looking at the time series plots of x and y against t. Click on x-t plot and y-t plot to open the maple worksheets to show you how to do this.

Can you describe what happens if you start at the point B?

Can you find the equilibrium point?

Which way are the arrows pointing? Do you think the point is stable or unstable?

Can you find the asymptote? In this diagram there is only one.

Write your answers in the space provided in the handbook.