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> 6. =
dx dt dx dt
> !2. -γ = dt
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.
dx /1 1
13 dt = 1 -1 U -
14. = dt
15. = dt
16. = dt
ΰ, β, y, S > 0
17. The equation of motion of a spring-mass system with damping (see Section 3.8) is
d2u 12 dt2
+ 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)
d2I dI 1 r L2 + R~,+ ~p, I 0. dt2 dt C
18. 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 r1 = 0 and r2 = 0, and let g(1) and g(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?
In this problem we indicate how to show that the trajectories are ellipses when the eigenvalues are pure imaginary. Consider the system
(a) Show that the eigenvalues of the coefficient matrix are pure imaginary if and only if
a11 + a22 = 0,
a11 a22 a12a21 > °.
(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 au x + a^ y
Chapter 9. Nonlinear Differential Equations and Stability
FIGURE 9.1.8 Nonisolated critical points; r1 = 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 ^ + 2a22ΥΣ - ai2^ = 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 = ajj x + aj2 y, dy/dt = a21 x + a22 y,
where ajj, a22 are real constants. Let p = ajj + a22, q = ajja22 aj2a2j, and Δ =
p2 4q. Show that the critical point (0, 0) is a
(a) Node if q > 0 and Δ > 0;
(b) Saddle point if q < 0;
(c) Spiral point if p = 0 and Δ < 0;
(d) Center if p = 0 and q > 0.
Hint: These conclusions can be obtained by studying the eigenvalues rj and r2. It may also be helpful to establish, and then to use, the relations rjr2 = q and rj + 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.