Linear systems. Exercise are designed to help classify all potential behavior of planar, autonomous, linear systems. It is essential that you master this material before attempting the analysis of non-linear systems. Be sure to turn off milIclines in these exercises.
Straight line solutions. If λ, v is an eigenvalue-eigenvector pair of matrix A, then we know that X(t) = eλt V is a solution. Moreover, if λ and v are real, than this solution is a line in the phase plane Because solutions of linear systems cannot cross in the phase plane, straight line solutions prevent any sort of rotation of solution trajectories about the equilibrium point at the origin.
In Exercise, find the eigenvoines and eigenvectors with the nig and null commands, as demonsnated in Example 4 of Chapter 12. Yon may find foroat rat helpful Then enter the system into pplanaG, and draw the straight line solutions. For example, if one eigenvector happens to bey = [l, −2]T, use the Keyboard input window to start straight line solutions at (1, –2) and (–1, 2). Perform a similar task for the other eigenvector. Finally, the straight line solutions in those exercises divide the phase plane into four regions. Use your mouse to start several solution trajectories in each region.
Spiral source. As we saw in equation (13.12), if the real part of a complex eigenvalue is negative, the equilibrium point will be a spiral sink. What happens if the real part of the complex eigenvalue is positive? Enter the system x' = 2.1x + 5.0y, y' = — 1.0x − 1.9y in pplane6 and note that solutions spiral outward with the passage of time. Select Solutions→Find an equilibrium point, find the equilibrium point at the origin, note the real part of the complex eigenvalue, then note the classification of the equilibrium point in the PPLANE6 Equilibrium point data window. If your computer is too fast to differentiate an outward spiral from an inward spiral, you might try Options→Solution direction→Forward.
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