Plot the two quantities against each other rather than against time. Each point is a possible state of the system.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Trajectories
A solution traces a curve through the phase plane. Time is no longer an axis; it is the direction of travel along the curve.
The direction field
The system assigns a velocity vector to every state. Drawing them fills the plane with arrows, and trajectories follow them.
Nodes
Arrows converging on the origin make a stable node, a sink. Arrows diverging make an unstable node, a source.
Saddles
Arrows coming in along one direction and leaving along another make a saddle. It is unstable, but stable along one line.
Spirals and centres
Trajectories circling the origin while approaching it form a stable spiral. Circling without approaching gives a centre.
The whole future in one picture
A phase portrait plots one variable against the other, with each curve a trajectory. Time is implicit, and the picture shows every possible history of the system at once.
Equilibria organise the plane
Points where both rates vanish are equilibria, and the trajectories arrange themselves around them. Locating the equilibria first is what makes a phase portrait interpretable.
The classification
Nodes, saddles, spirals and centres. Each has a characteristic pattern of trajectories, and identifying which one an equilibrium is describes the local behaviour completely.
Trajectories do not cross
Uniqueness forbids two trajectories meeting, since the crossing point would have two futures. That constraint disciplines any hand-drawn portrait and is a useful check.
Step 2: Try It Yourself
Tap and try it out.
- Released from(3, 2)
Every arrow points toward the origin, so trajectories settle there. The origin is stable.
- Released from(0.5, 3)
Arrows come in along one axis and leave along the other. The origin is a saddle, stable in one direction only.
Step 3: Watch an Example
One step at a time.
Watch Kofi Classify an Equilibrium
Kofi sees a phase portrait where every arrow points toward the origin.
- Step 1
He checks several points and finds the arrows all aim inward.
Step 4: Your Turn
Practice makes it stick.
The Sink
Problem 1 of 2
Arrows all pointing toward the origin. Is the equilibrium stable? 1 yes, 0 no.
The Saddle
Problem 2 of 2
A saddle equilibrium. Along how many directions is it stable?
Read the Portrait
1 of 8
Arrows pointing away from the origin. Is the equilibrium stable? 1 yes, 0 no.
2 of 8
Trajectories circling the origin while approaching it. Enter 1 node, 2 spiral, 3 saddle.
3 of 8
Trajectories circling the origin without approaching. Is the equilibrium stable in the strict sense? 1 yes, 0 no.
4 of 8
In a phase plane, what is plotted on the axes? Enter 1 for time and one quantity, 2 for the two quantities.
5 of 8
A saddle. Is it stable overall? 1 yes, 0 no.
6 of 8
At an equilibrium of a system, what is the length of the velocity vector?
7 of 8
Match each pattern to what it is called.
Tap a card on the left to start.
8 of 8
A trajectory starting exactly at the origin of a linear system. How far does it move?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Trajectories circling the origin while approaching it. Enter 1 node, 2 spiral, 3 saddle.
Question 2 of 2
What is plotted in a phase plane?
What You Learned
- 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.