# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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CASE 1 Real Unequal Eigenvalues of the Same Sign. The general solution of Eq. (2) is

x = c1g(1)er1t + c2g(2)erd, (5)

where r1 and r2 are either both positive or both negative. Suppose first that q < r2 < 0, and that the eigenvectors g(1) and g(2) are as shown in Figure 9.1.1a. It follows from Eq. (5) that x ^ 0 as t regardless of the values of q and c2; in other words, all solutions approach the critical point at the origin as t ^ro. If the solution starts at an

9.1 The Phase Plane: Linear Systems

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initial point on the line through ?(1), then c2 = 0. Consequently, the solution remains on the line through ?(1) for all t, and approaches the origin as t ^<x>. Similarly, if the initial point is on the line through ?(2), then the solution approaches the origin along that line. In the general situation, it is helpful to rewrite Eq. (5) in the form

x = er2t [c1g(1)e(r1 -r2)t + c2g(2)j. (6)

Observe that ^ — r2 < 0. Therefore, as long as c2 = 0, the term q?(1) exp[(r1 — r2)t] is negligible compared to C2^(2) for t sufficiently large. Thus, as t ^ <x>, the trajectory not only approaches the origin, it also tends toward the line through ?(2). Hence all solutions approach the critical point tangent to ?(2) except for those solutions that start exactly on the line through g(1). Several trajectories are sketched in Figure 9.1.1a. Some typical graphs of ^ versus t are shown in Figure 9.1.1 b, illustrating that all solutions exhibit exponential decay in time. The behavior of ^ versus t is similar. This type of critical point is called a node or a nodal sink.

FIGURE 9.1.1 A node; r1 < r2 < 0. (a) The phase plane. (b) x1 versus t.

If rx and r2 are both positive, and 0 < r2 < rv then the trajectories have the same pattern as in Figure 9.1.1a, but the direction of motion is away from, rather than toward, the critical point at the origin. In this case xx and X2 grow exponentially as functions of t. Again the critical point is called a node, or (often) a nodal source.

An example of a node occurs in Example 2 of Section 7.5, and its trajectories are shown in Figure 7.5.4.

CASE 2 Real Eigenvalues of Opposite Sign. The general solution of Eq. (2) is

x = c^Vd + , (7)

where ri > 0 and ^ < 0. Suppose that the eigenvectors ?(1) and ?(2) are as shown in Figure 9.1.2a. If the solution starts at an initial point on the line through ?(1), then it follows that C2 = 0. Consequently, the solution remains on the line through ?(1) for all t, and since ri > 0, ||x|| ^ ?as t ^ ?. If the solution starts at an initial point on the line through ?(2), then the situation is similar except that ||x|| ^ 0 as t ^ ? because r2 < 0. Solutions starting at other initial points follow trajectories such as those shown in Figure 9.1.2a. The positive exponential is the dominant term in Eq. (7) for large t, so eventually all these solutions approach infinity asymptotic to the line determined

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Chapter 9. Nonlinear Differential Equations and Stability

(a) (b)

FIGURE 9.1.2 A saddle point; r1 > 0, r2 < 0. (a) The phase plane. (b) x1 versus t.

by the eigenvector ?(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 ?(2). Figure 9.1.2bshows some typical graphs of Xj versus t. For certain initial conditions the positive exponential term is absent from the solution, so Xj ^ 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 q = q = 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 = q^V + c?{2)ert, (8)

where ?(1) and ?(2)are the independent eigenvectors. The ratio X2/xx is independent of t, but depends on the components of ?(1) and ?(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 = q^ + c^e + ^), (9)

where ? is the eigenvector and ^ is the generalized eigenvector associated with the repeated eigenvalue. For large t the dominant term in Eq. (9) is C2^tert.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 = q^e^ lies on this line. Similarly, for large negative t the term c2^teft is again the dominant one, so as t ^ -to, each trajectory is asymptotic to a line parallel to ?.

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