Phase Plane Slides from Lectures 2/27 and 3/2:

1. 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.
2. 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.
3. 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).
4. 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.
5. 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.
6. 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.
7. 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).
8. 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.
9. 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).
10. 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.
11. 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.)