Linear systems. Exercise are designed to help classify all potential behavior of planar, a...

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.

Center. If the real part of the complex eigenvalue is zero, there can be no growth or decay of solution trajectories. Enter the system x' = −x + y, y' = −2x + y, and note that pplane6 reports a nearly closed orbit in its message window. Note that pplane6 has difficulty with tliis type of equilibrium point, commonly called a center. It should report that the trajectories are purely periodic. This happens because this system is a critical case, separating linear systems with a spiral sink from systems with a spiral source. The slightest numerical error, even an error of 1 × 10−16, makes the real part of the complex eigenvalue non-zero, indicating a spiral source or sink.