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 trace-determinant plane. It is a nice exercise to classify linear systems based on their position in the trace-determinant plane. Consider the matrix

a) Show that the characteristic polynomial of the matrix A is p(λ) = λ2 — Tλ + D, where T = a + d is the trace of A and D = det(A) = ad − bc is the determinant of A.

b) We know that the characteristic polynomial factors as p(λ) = (λ − λ1)(λ − λ2), where λ1 and λ2 are the eigenvalues. Use this and the result of part (a) to show that the product of the eigenvalues is equal to the determinant of matrix A. Note: This is a useful fact. For example, if the determinant is negative, then you must have one positive and one negative eigenvalue, indicating a saddle equilibrium point. Also, show that the sum of the eigenvalues equals the trace of matrix A.

c) Show that the eigenvalues of matrix A are given by the formula

Note that there are three possible scenarios. If T2 − AD<0, then there are two complex eigenvalues. If T2 − 4D > 0, there are two real eigenvalues. Finally, if T2 − 4D = 0, then there is one repeated eigenvalue of algebraic multiplicity two.

d) Draw a pair of axes on a piece of poster board. Label the vertical axis D and the horizontal axis T. Sketch the graph of T2 − 4D = 0 on your poster board. The axes and the parabola defined by T2 − 4D = 0 divide the trace-determinant plane into six distinct regions, as shown in Figure 13.17.

Figure 13.21. The trace-determinant plane.

e) You can classify any matrix A by its location in the trace-determinant plane. For example, if

then T = 3 and D = 8, so the point (T, D) is located in the first quadrant. Furthermore, (3)2 − 4(8)<0, placing the point (3,8) above the parabola T2 − 4D = 0. Finally, if you substitute T = 3 and D = 8 into the formula , then you get eigenvalues that are complex with a positive real part, making the equilibrium point of the system x' = Ax a spiral source. Use pplana6 to generate a phase portrait of this particular system and attach the printout to the poster board at the point (3, 8).

f) Linear systems possess a small number of distinctive phase portraits. Each of these is graphically different from the others, but each corresponds to the pair of eigenvalues and their multiplicities. For each case, use pplane6 to construct a phase portrait, and attach a printout at its appropriate point (T, D) in your poster board trace-determinant plane. Hint: There are degenerate cases on the axes and the parabola. For example, you can find degenerate cases on the parabola in the first quadrant that separate nodal sources from spiral sources. There are also a number of interesting degenerate cases at the origin of the trace-determinant plane. One final note: We have intentionally used the words “small number of distinctive cases” so as to spur argument amongst our readers when working on this activity. What do you think is the correct number?

Nonlinear Systems. We analyze the behavior of a nonlinear system near an equilibrium point with a linear approximation.