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.

The system x′ = x + y, y′ = −x + 3y is a degenerate case, having only one straight line solution. Find the eigenvalues and eigenvectors of the system x' = x + y, y′ = −x+ 3y, then use pplane6 to depict the single straight line solution. Sketch several solution trajectories and note that they almost spiral, but the single straight line solution prevents full rotation about the equilibrium point at the origin. This degenerate case is a critical case. Its trace-determinant pair lies on a curve in the trace-determinant plane that separates systems that spiral from those that do not.

Sources, sinks, and saddles. If a system has two distinct, real eigenvalues, then it will have two independent straight line solutions, and all solutions can be written as a linear combination of these two straight line solutions, as in .