Phase Portraits, Equilibria and Stability · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Reading motion in the plane

Mathematics · Differential Equations · ages 21-22
Name ______________________   Date ____________

  1. The linear system x' = Ax has A = [[3, 0], [0, -2]]. What type of equilibrium is the origin?

    • Saddle
    • Stable node
    • Unstable spiral
    • Centre

  2. A linear system has eigenvalues 3 and minus 2 at the origin. What type of equilibrium is it?

    • Unstable node
    • Saddle
    • Stable spiral

  3. An equilibrium of a planar system is a point where both rates of change are zero.

    Circle one:   True   False

  4. A linear system has eigenvalues minus 2 and minus 5 at the origin. What type is it?

    • Stable node
    • Saddle
    • Unstable spiral

  5. A phase portrait shows closed loops around the origin. Every nearby trajectory stays nearby but never settles into the point. Is the origin a centre?

    Circle one:   True   False

  6. A linear system has eigenvalues 1 + 2i and 1 - 2i at the origin. What type of equilibrium is it?

    • Unstable spiral
    • Stable spiral
    • Centre
    • Saddle

  7. After linearising a predator prey model at an equilibrium you get a centre. What should you conclude?

    • The nonlinear portrait is certainly a centre
    • The centre may become a slow spiral in the full model
    • The equilibrium must be a saddle

  8. A linear system has eigenvalues 1 plus 2i and 1 minus 2i. What is the real part?

    Answer: ______________

  9. A student sees eigenvalues 2i and minus 2i and reports an unstable spiral. What is wrong?

    • A pure imaginary pair gives a node
    • Nothing is wrong
    • A pure imaginary pair gives a centre, not a spiral

  10. Linearisation of a nonlinear system at an equilibrium gives purely imaginary eigenvalues. Ana concludes the nonlinear system must show closed loops there. Is Ana right?

    Circle one:   True   False

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Reading motion in the plane W1-mt_ULuZk4lmGr-s1

  1. Saddle · The eigenvalues are 3 and -2: real with opposite signs. Trajectories approach along one axis and leave along the other, which is a saddle.
  2. Saddle · Opposite signs pull in along one direction and push out along the other, which is a saddle.
  3. True · That is the definition you learned: both rates vanish so the state can rest.
  4. Stable node · Two negative real eigenvalues pull every nearby trajectory straight in, which is a stable node.
  5. True · Closed loops that stay nearby mean stability, but since paths never settle into the point, it is not asymptotically stable. That is exactly a centre.
  6. Unstable spiral · The pair 1 + 2i and 1 - 2i is complex with positive real part 1. Rotation plus outward drift is an unstable spiral.
  7. The centre may become a slow spiral in the full model · A centre sits on the borderline, where the dropped nonlinear terms can change the picture.
  8. 1 · The pair shares the real part 1, which is what decides stability.
  9. A pure imaginary pair gives a centre, not a spiral · With zero real part there is rotation but no inward or outward drift, so the shape is a centre.
  10. False · Ana is wrong. A centre is a borderline case: tiny nonlinear terms can turn the loops into a slow inward or outward spiral, which the linear picture cannot show.
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