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Chapter 9. Nonlinear Differential Equations and Stability
FIGURE 9.1.2 A saddle point; rx > 0, r2
< 0. (a) The phase plane. (b) Xj versus t.
by the eigenvector g(1) corresponding to the positive eigenvalue r1. The only solutions that approach the critical point at the origin are those that start precisely on the line determined by g(2). Figure 9.1.2b shows some typical graphs of x1 versus t. For certain initial conditions the positive exponential term is absent from the solution, so x1 0 as t to. For all other initial conditions the positive exponential term eventually dominates and causes x1 to become unbounded. The behavior of x2 is similar. The origin is called a saddle point in this case.
A specific example of a saddle point is in Example 1 of Section 7.5 whose trajectories are shown in Figure 7.5.2.
CASE 3 Equal Eigenvalues. We now suppose that r1 = r2 = r. We consider the case in which the eigenvalues are negative; if they are positive, the trajectories are similar but the direction of motion is reversed. There are two subcases, depending on whether the repeated eigenvalue has two independent eigenvectors or only one.
(a) Two independent eigenvectors. The general solution of Eq. (2) is
x = c1g(1)ert + c2g(2)ert, (8)
where g(1) and g(2)are the independent eigenvectors. The ratio x2/x1 is independent of t, but depends on the components of g(1) and g(2), and on the arbitrary constants c1 and c2. Thus every trajectory lies on a straight line through the origin, as shown in Figure 9.1.3a. Typical graphs of x1 or x2 versus t are shown in Figure 9.1.3b. The critical point is called a proper node, or sometimes a star point.
(b) One independent eigenvector. As shown in Section 7.8, the general solution of Eq. (2) in this case is
x = c1get + c2(gte( + ^ert), (9)
where g is the eigenvector and ^ is the generalized eigenvector associated with the repeated eigenvalue. For large t the dominant term in Eq. (9) is c2gtert. Thus, as t to, every trajectory approaches the origin tangent to the line through the eigenvector. This is true even if c2 = 0, for then the solution x = c1gert lies on this line. Similarly, for large negative t the term c2gtel is again the dominant one, so as t -to, each trajectory is asymptotic to a line parallel to g.
9.1 The Phase Plane: Linear Systems
FIGURE 9.1.3 A proper node, two independent eigenvectors; r1 = r2 < 0. (a) The phase plane. (b) x1 versus t.
The orientation of the trajectories depends on the relative positions of g and ^. One possible situation is shown in Figure 9.1.4a. To locate the trajectories it is helpful to write the solution (9) in the form
x = [(qg + C2^) + c2gt]ert = yert, (10)
where y = (c1g + ρ2φ) + c2gt. Observe that the vector y determines the direction of x, whereas the scalar quantity ert affects only the magnitude of x. Also note that, for fixed values of c1 and c2, the expression for y is a vector equation of the straight line through the point c1g + c2^ and parallel to g.
To sketch the trajectory corresponding to a given pair of values of c1 and c2, you can proceed in the following way. First, draw the line given by (c1g + c2^) + c2gt and note the direction of increasing t on this line. Two such lines are shown in Figure 9.1.4a, one for c2 > 0 and the other for c2 < 0. Next, note that the given trajectory passes through the point cjg + c2^ when t = 0. Further, as t increases, the direction of the vector x given by Eq. (10) follows the direction of increasing t on the line, but the magnitude of x rapidly decreases and approaches zero because of the decaying exponential factor ert. Finally, as t decreases toward to the direction of x is determined by points on the corresponding part of the line and the magnitude of x approaches infinity. In this way we obtain the heavy trajectories in Figure 9.1.4a. A few other trajectories are lightly sketched as well to help complete the diagram. Typical graphs of x1 versus t are shown in Figure 9.1.4b.
The other possible situation is shown in Figure 9.1.4c, where the relative orientation of g and ^ is reversed. As indicated in the figure, this results in a reversal in the orientation of the trajectories.
If r1 = r2 > 0, you can sketch the trajectories by following the same procedure. In this event the trajectories are traversed in the outward direction, and the orientation of the trajectories with respect to that of g and ^ is also reversed.
When a double eigenvalue has only a single independent eigenvector, the critical point is called an improper or degenerate node. A specific example of this case is Example 2 in Section 7.8; the trajectories are shown in Figure 7.8.2.