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Cogito

Differential Equations · Chapter 7 · Lesson 2

Practice Worksheet

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

NameDate

Figures — use these to answer the problems

Every trajectory drains to the origin, so this equilibrium is a stable node. Release from anywhere and see.
  • Released from(3, 2)
the rate1
starting first quantity3
starting second quantity2
Figure 1
Here trajectories come in along one axis and leave along the other. That is a saddle, and it is unstable.
  • Released from(0.5, 3)
the rate1
starting first quantity0.5
starting second quantity3
Figure 2

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. Trajectories circling the origin while approaching it. Enter 1 node, 2 spiral, 3 saddle.

    Answer

  2. What is plotted in a phase plane?

    • The two quantities against each other, with time as the direction of travel.
    • One quantity against time.

  3. Arrows pointing away from the origin. Is the equilibrium stable? 1 yes, 0 no.

    Answer

  4. Trajectories circling the origin while approaching it. Enter 1 node, 2 spiral, 3 saddle.

    Answer

  5. Trajectories circling the origin without approaching. Is the equilibrium stable in the strict sense? 1 yes, 0 no.

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. In a phase plane, what is plotted on the axes? Enter 1 for time and one quantity, 2 for the two quantities.

    Answer

  2. A saddle. Is it stable overall? 1 yes, 0 no.

    Answer

  3. At an equilibrium of a system, what is the length of the velocity vector?

    Answer

Stretch Yourself

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

4 problems

  1. Match each pattern to what it is called.

    Draw a line from each item on the left to its match on the right.

    • All arrows converge on the origin
    • In along one direction, out along another
    • Trajectories circle while approaching
    • A stable node
    • A saddle
    • A stable spiral

  2. A trajectory starting exactly at the origin of a linear system. How far does it move?

    Answer

  3. The Sink

    Arrows all pointing toward the origin. Is the equilibrium stable? 1 yes, 0 no.

    Answer

  4. The Saddle

    A saddle equilibrium. Along how many directions is it stable?

    Answer