- Simple Harmonic Motion: x-x' phase
plane for the second order equation x'' + x = 0, with 1 trajectory
shown. Trajectories = concentric circles, centered at
(0,0). (0,0) is a Stable Equilibrium.

- Damped Harmonic Motion: x-x' phase plane
for the second order equation x'' + 2x + 4 = 0, with 1 trajectory
shown. Trajectories = spirals converging to (0,0). (0,0) is
a Stable Equilibrium.

- Opposite Sign Real Eigenvalues: x-y phase
plane for the 2x2 system x' = x-y; y' = -9x+y, with two trajectories
shown. (0,0) is an (Unstable) Saddle Equilibrium: stable in the
direction (1,3), and unstable in the direction (1,-3).

- Positive Eigenvalues: x-y phase plane for the 2x2 system x' = 2x + 2y; y' = x + 3y, with 2 trajectories shown. (0,0) is an Unstable Equilibrium.
- Negative Eigenvalues: x-y phase plane for the 2x2 system x' = -2x + y; y' = x - 2y, with several trajectories shown. (0,0) is a Stable Equilibrium.
- Complex Eigenvalues (Spirals): x-y phase
plane for the 2x2 system x' = 3x - y; y' = x + 3y. Eigenvalues
are 3+i and 3-i. Since 3>0, the trajectories spiral away from
the origin, and (0,0) is an Unstable Equilibrium.

- Complex Eigenvalues (Better Spirals): x-y
phase plane for the 2x2 system x' = -x + 20y ; y' = -20x-y.
Eigenvalues have much larger imaginary parts (beta) than real parts
(alpha).

- Complex Eigenvalues (Ellipses): x-y phase plane for the 2x2 system x' = 4x - 5y; y' =5x-4y. Eigenvalues are 3i and -3i. Since the real part alpha is 0, the trajectories are ellipses. (0,0) is a Neutrally Stable Equilibrium.
- Repeated Real Eigenvalues: x-y phase plane for the 2x2 system x' = 2x - y; y' = 4x+6y. The only eigenvalue is 4, and the only eigenvector (up to scalar multiples) is (1, -2).
- Nonlinear System: x-y phase plane
for the 2x2 nonlinear system x' = 2y; y' = x+y -y^3. There
are 3 equilibrium points: (0,0) is a saddle, and (-1,0) and
(1,0) are unstable repelling spirals.

- Nonlinear System (predator-prey): x-y
phase plane for the 2x2 nonlinear system x' = 2x-xy; y' = -3y +
xy. There are 2 equilibrium points: (0,0) is a saddle, and (3,2)
is a center (neutrally stable).

Additional Samples of Phase Plane Portraits. (courtesy of Prof. Goodearl.)

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