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.
Summarizing your findings. In a table use your results from Exercises 11–15 to fist conditions on the eigenvalue that predict when the equilibrium point at the origin is a nodal source, nodal sink, saddle, spiral source, spiral sink, and center.
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