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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 ? 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 = [(c1? + c2^) + c2?t]ert = yert, (10)
where y = (q? + c2^) + c2^t. 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 q and C2, the expression for y is a vector equation of the straight line through the point q? + C2^ and parallel to ?.
To sketch the trajectory corresponding to a given pair of values of q and C2, you can proceed in the following way. First, draw the line given by (q? + C2^) + c2%t 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 q? + 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 ? 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 ? 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.
Chapter 9. Nonlinear Differential Equations and Stability
x2 ‘ Increasing X \ ^ s' ^^Increasingt
c2 < 0 / + C2n^>"' 'c2 > 0 s'*' t X
x2 c2 > 0 n Increasing t C2^ -1 ,
c2 < 0 Increasing t + C2^ Xl <C2,
FIGURE 9.1.4 An improper node, one independent eigenvector; r1 = r2 < 0. (a) The phase plane. (b) x1 versus t. (c) The phase plane.
CASE 4 Complex Eigenvalues. Suppose that the eigenvalues are X ± ii, where X and i are real, X = 0, and i > 0. It is possible to write down the general solution in terms of the eigenvalues and eigenvectors, as shown in Section 7.6. However, we proceed in a different way.
Systems having the eigenvalues X ± ii are typified by
x=( X ^ x (11)
or, in scalar form,
Xi = Xx! + jxx2,
X*2 = —flXi + Xx2.
9.1 The Phase Plane: Linear Systems
We introduce the polar coordinates r, Q given by
r2 = X2 + a| , tan Q = x2/ x1.
By differentiating these equations we obtain
rr' = x1 x1 + x2 x2, (sec2 Q)Q' = (x1 x2 — x2 x1 )/x\. (13)
Substituting from Eqs. (12) in the first of Eqs. (13), we find that
r = kr, (14)
r = cekt, (15)
where c is a constant. Similarly, substituting from Eqs. (12) in the second of Eqs. (13), and using the fact that sec2 Q = r2/xj!, we have
Q' = -fi. (16)
Q = —fit + Qo, (17)
where Q0 is the value of Q when t = 0.
Equations (15) and (17) are parametric equations in polar coordinates of the trajectories of the system (11). Since i > 0, it follows from Eq. (17) that Q decreases as t increases, so the direction of motion on a trajectory is clockwise. As t ^ to, we see from Eq. (15) that r ^ 0 if k < 0 and r ^to if k > 0. Thus the trajectories are spirals, which approach or recede from the origin depending on the sign of k. Both possibilities are shown in Figure 9.1.5, along with some typical graphs of x1 versus t. The critical point is called a spiral point in this case. Frequently, the terms spiral sink and spiral source, respectively, are used to refer to spiral points whose trajectories approach, or depart from, the critical point.
More generally, it is possible to show that for any system with complex eigenvalues k ± ii, where k = 0, the trajectories are always spirals. They are directed inward or outward, respectively, depending on whether k is negative or positive. They may be elongated and skewed with respect to the coordinate axes, and the direction of motion may be either clockwise or counterclockwise. While a detailed analysis is moderately difficult, it is easy to obtain a general idea of the orientation of the trajectories directly from the differential equations. Suppose that