Download (direct link):
In general, when the eigenvalues are pure imaginary, it is possible to show (see Problem 19) that the trajectories are ellipses centered at the origin. A typical situation is shown in Figure 9.1.7, which also shows some typical graphs of x1 versus t.
FIGURE 9.1.7 A center; r1 = ië, r2 = —i/ë. (a) The phase plane. (b) x1 versus t.
By reflecting on these five cases and by examining the corresponding figures, we can make several observations:
1. After a long time each individual trajectory exhibits one of only three types of behavior. As t — to, each trajectory either approaches infinity, approaches the critical point x = 0, or repeatedly traverses a closed curve, corresponding to a periodic solution, that surrounds the critical point.
2. Viewed as a whole, the pattern of trajectories in each case is relatively simple. To be more specific, through each point (x0, y0) in the phase plane there is only one trajectory; thus the trajectories do not cross each other. Do not be misled by the figures, in which it sometimes appears that many trajectories pass through the critical point x = 0. In fact, the only solution passing through the origin is the equilibrium solution x = 0. The other solutions that appear to pass through the origin actually only approach this point as t — to or t — —to.
3. In each case the set of all trajectories is such that one of three situations occurs.
(a) All trajectories approach the critical point x = 0 as t — to. This is the case if the eigenvalues are real and negative or complex with negative real part. The origin is either a nodal or a spiral sink.
(b) All trajectories remain bounded but do not approach the origin as t — to. This is the case if the eigenvalues are pure imaginary. The origin is a center.
Chapter 9. Nonlinear Differential Equations and Stability
(c) Some trajectories, and possibly all trajectories except x = 0, tend to infinity as t ^ to. This is the case if at least one of the eigenvalues is positive or if the eigenvalues have positive real part. The origin is either a nodal source, a spiral source, or a saddle point.
The situations described in 3(a), (b), and (c) above illustrate the concepts of asymptotic stability, stability, and instability, respectively, of the equilibrium solution x = 0 of the system (2). The precise definitions of these terms are given in Section 9.2, but their basic meaning should be clear from the geometrical discussion in this section. The information that we have obtained about the system (2) is summarized in Table 9.1.1. Also see Problems 20 and 21.
TABLE 9.1.1 Stability Properties of Linear Systems x' = Ax with det(A — rI) = 0 and det A = 0
Eigenvalues Type of Critical Point Stability
r1 > r2 > 0 Node Unstable
r11 < r22 < 0 Node Asymptotically stable
r2 < 0 < r1 Saddle point Unstable
r1 = r2 > 0 Proper or improper node Unstable
r1 = r2 < 0 Proper or improper node Asymptotically stable
r1 , r2 = ê ± ip Spiral point
ê > 0 Unstable
ê < 0 Asymptotically stable
^ = ip, Ã2 = ---ip Center Stable
The analysis in this section applies only to second order systems x' = Ax whose solutions are represented geometrically as curves in the phase plane. A similar, though more complicated, analysis can be carried out for an nth order system, with an n x n coefficient matrix A, whose solutions are curves in an n-dimensional phase space. The cases that can occur in higher order systems are essentially combinations of those we have seen in two dimensions. For instance, in a third order system with a threedimensional phase space, one possibility is that solutions in a certain plane may spiral toward the origin, while other solutions may tend to infinity along a line transverse to this plane. This would be the case if the coefficient matrix has two complex eigenvalues with negative real part and one positive real eigenvalue. However, because of their complexity, we will not discuss systems of higher than second order.
For each of the systems in Problems 1 through 12:
(a) Find the eigenvalues and eigenvectors.
(b) Classify the critical point (0, 0) as to type and determine whether it is stable, asymptotically stable, or unstable.
(c) Sketch several trajectories in the phase plane and also sketch some typical graphs of Xj versus t.
(d) Use a computer to plot accurately the curves requested in part (c).
dx /3 —2\ dx /5 — Ë
1 dt = (2 —2)x > 2. dt = (3 Jx
9.1 The Phase Plane: Linear Systems
5. — =
dx dt dx dt dx dt dx dt