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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 xx versus t.
(d) Use a computer to plot accurately the curves requested in part (c).
1. — =
? 2. —
9.1 The Phase Plane: Linear Systems
c 5. dx
4. — =
6. - =
-- 4 -5
? 12. dX =
2 -5 L 2
In each of Problems 13 through 16 determine the critical point x = x0, and then classify its type and examine its stability by making the transformation x = x0 + u.
13 dx_ d 1
dt \1 -1
?4. dx=d -9 x+d
15. dx =
16. — = dt
a, p,y,S > 0
17. The equation of motion of a spring-mass system with damping (see Section 3.8) is
+ c---------------+ ku = 0,
where m, c, and k are positive. Write this second order equation as a system of two first order equations for x = u, y = du/dt. Show that x = 0, y = 0 is a critical point, and analyze the nature and stability of the critical point as a function of the parameters m, c, and k. A similar analysis can be applied to the electric circuit equation (see Section 3.8)
rd2I dl 1 r „
L—2 + R~.—+ 1 — 0.
dt2 dt C
Consider the system x' = Ax, and suppose that A has one zero eigenvalue.
(a) Show that x = 0 is a critical point, and that, in addition, every point on a certain straight line through the origin is also a critical point.
(b) Let rj = 0 and r2 = 0, and let and ?,(2) be corresponding eigenvectors. Show that the trajectories are as indicated in Figure 9.1.8. What is the direction of motion on the trajectories?
19. In this problem we indicate how to show that the trajectories are ellipses when the eigenvalues are pure imaginary. Consider the system
fx) = (a11 a12) lX
y a21 a22 y
(a) Show that the eigenvalues of the coefficient matrix are pure imaginary if and only if
a11 + a22 = °’ a11 a22 - a12a21 > °. (ii)
(b) The trajectories of the system (i) can be found by converting Eqs. (i) into the single equation
dy dy/dt a21 x + a22 y
dx dx/dt a11 x + a12 y
Chapter 9. Nonlinear Differential Equations and Stability
FIGURE 9.1.8 Nonisolated critical points; rj = 0, r2 = 0. Every point on the line through is a critical point.
Use the first of Eqs. (ii) to show that Eq. (iii) is exact.
(c) By integrating Eq. (iii) show that
a21 *2 + 2a22Xy - ai2Z2 = k, (iv)
where k is a constant. Use Eqs. (ii) to conclude that the graph of Eq. (iv) is always an
Hint: What is the discriminant of the quadratic form in Eq. (iv)?
20. Consider the linear system
dx/dt = a11 x + a12 y, dy/dt = a21 x + a22 y,
where a11, a22 are real constants. Let p = a11 + a22, q = a11a22 — a12a21, and A =
p2 — 4q. Show that the critical point (0, 0) is a
(a) Node if q > 0 and A > 0;
(b) Saddle point if q < 0;
(c) Spiral point if p = 0 and A < 0;
(d) Center if p = 0 and q > 0.
Hint: These conclusions can be obtained by studying the eigenvalues r1 and r2. It may also be helpful to establish, and then to use, the relations r1r2 = q and r1 + r2 = p.
21. Continuing Problem 20, show that the critical point (0, 0) is
(a) Asymptotically stable if q > 0 and p < 0;
(b) Stable if q > 0 and p = 0;
(c) Unstable if q < 0 or p > 0.
Notice that results (a), (b), and (c), together with the fact that q = 0, show that the critical point is asymptotically stable if, and only if, q > 0 and p < 0. The results of Problems 20 and 21 are summarized visually in Figure 9.1.9.
9.2 Autonomous Systems and Stability
FIGURE 9.1.9 Stability diagram.
9.2 Autonomous Systems and Stability
In this section we begin to draw together and to expand on the geometrical ideas introduced in Section 2.5 for certain first order equations and in Section 9.1 for second order linear homogeneous systems with constant coefficients. These ideas concern the qualitative study of differential equations and the concept of stability, an idea that will be defined precisely later in this section.
Autonomous Systems. We are concerned with systems of two simultaneous differential equations of the form
dx/dt = F(x, y), dy/dt = G(x, y). (1)
We assume that the functions F and G are continuous and have continuous partial derivatives in some domain D of the xy-plane. If (x0, y0) is a point in this domain, then by Theorem 7.1.1 there exists a unique solution x = 0(t), y = ^(t) of the system (1) satisfying the initial conditions
x (t0) = x0> y(t0) = y0 • (2)
The solution is defined in some time interval I that contains the point t^.
Frequently, we will write the initial value problem (1), (2) in the vector form