Phase Portraits

The whole future of a system, drawn in one picture.

12 problems · about 24 minutes · F-IF.B.4, N-VM.C.11

What this lesson teaches

The student reads a phase portrait and classifies the behaviour near an equilibrium.

A phase portrait plots the two quantities against each other, with trajectories following the arrows.
Converging arrows make a stable node, diverging ones an unstable node.
A saddle is in along one direction and out along another; a spiral circles while approaching.

Straightforward practice. Get the method working first.

5 problems

Trajectories circling the origin while approaching it. Enter 1 node, 2 spiral, 3 saddle.
Answer 2
Why A stable spiral.
What is plotted in a phase plane?
Answer The two quantities against each other, with time as the direction of travel.
Why The state, not the timeline.
Arrows pointing away from the origin. Is the equilibrium stable? 1 yes, 0 no.
Answer 0
Why Everything escapes.
Trajectories circling the origin while approaching it. Enter 1 node, 2 spiral, 3 saddle.
Answer 2
Why Circling plus approaching.
Trajectories circling the origin without approaching. Is the equilibrium stable in the strict sense? 1 yes, 0 no.
Answer 0
Why A centre neither attracts nor repels.

The same ideas with more to keep track of.

3 problems

In a phase plane, what is plotted on the axes? Enter 1 for time and one quantity, 2 for the two quantities.
Answer 2
Why Time is not an axis.
A saddle. Is it stable overall? 1 yes, 0 no.
Answer 0
Why Most directions escape.
At an equilibrium of a system, what is the length of the velocity vector?
Answer 0
Why Nothing is changing.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Match each pattern to what it is called.
Answer All arrows converge on the origin → A stable node; In along one direction, out along another → A saddle; Trajectories circle while approaching → A stable spiral
Why Circling is what distinguishes a spiral from a node.
A trajectory starting exactly at the origin of a linear system. How far does it move?
Answer 0
Why It is an equilibrium.
The Sink: Arrows all pointing toward the origin. Is the equilibrium stable? 1 yes, 0 no.
Answer 1
Why Yes.
The Saddle: A saddle equilibrium. Along how many directions is it stable?
Answer 1
Why 1.