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(c) How do the amplitude and period of the pendulums motion depend on the initial position A? Draw a graph to show each of these relationships. Can you say anything about the limiting value of the period as A ^ 0?
(d) Let A 4 and plot x versus t. Explain why this graph differs from those in parts (a)
and (b). For what value of A does the transition take place?
? 22. Consider again the pendulum equations (see Problem 21)
dx/dt y, dy/dt 4 sinx.
If the pendulum is set in motion from its downward equilibrium position with angular velocity v, then the initial conditions are x(0) 0, y(0) v.
(a) Plot x versus t for v 2 and also for v 5. Explain the differing motions of the pendulum that these two graphs represent.
(b) There is a critical value of v, which we denote by vc, such that one type of motion occurs for v < v and the other for v > v . Estimate the value of v .
c c c
? 23. This problem extends Problem 22 to a damped pendulum. The equations of motion are
dx/dt y, dy/dt 4 sin x y y,
where y is the damping coefficient, with the initial conditions x(0) 0, y(0) v.
(a) For y 1/4 plot x versus t for v 2 and for v 5. Explain these plots in terms of the motions of the pendulum that they represent. Also explain how they relate to the corresponding graphs in Problem 22(a).
(b) Estimate the critical value vc of the initial velocity where the transition from one type of motion to the other occurs.
(c) Repeat part (b) for other values of y and determine how vc depends on y .
24. Theorem 9.3.2 provides no information about the stability of a critical point of an almost linear system if that point is a center of the corresponding linear system. That this must be the case is illustrated by the systems
dx/dt y + x (x2 + y2),
2 2 (i)
dy/dt x + y(x + y2)
dx/dt y x (x2 + y2),
2 2 (ii)
dy/dt x y(x + y2).
(a) Show that (0, 0) is a critical point of each system and, furthermore, is a center of the corresponding linear system.
(b) Show that each system is almost linear.
(c) Let /2 x2 + y2, and note that xdx/dt + ydy/dt /d//dt. For system (ii) show that d//dt < 0 and that / ^ 0 as t ^ <x>; hence the critical point is asymptotically stable. For system (i) show that the solution of the initial value problem for / with / /0 at t 0 becomes unbounded as t ^ 1/2/0, and hence the critical point is unstable.
Chapter 9. Nonlinear Differential Equations and Stability
25. In this problem we show how small changes in the coefficients of a system of linear equations can affect a critical point that is a center. Consider the system
where |e| is arbitrarily small. Show that the eigenvalues are e ± i. Thus no matter how small |e | = 0 is, the center becomes a spiral point. If e < 0, the spiral point is asymptotically stable; if e > 0, the spiral point is unstable.
26. In this problem we show how small changes in the coefficients of a system of linear equations can affect the nature of a critical point when the eigenvalues are equal. Consider the system
Show that the eigenvalues are r1 = 1, r2 = 1 so that the critical point (0, 0) is an asymptotically stable node. Now consider the system
where |e| is arbitrarily small. Show that if e > 0, then the eigenvalues are 1 ± i*fe, so that the asymptotically stable node becomes an asymptotically stable spiral point. If e < 0, then the roots are 1 ± ^/\?\, and the critical point remains an asymptotically stable node.
? 27. In this problem we derive a formula for the natural period of an undamped nonlinear pendulum [c = 0 in Eq. (10) of Section 9.2]. Suppose that the bob is pulled through a positive angle a and then released with zero velocity.
(a) We usually think of 9 and d9/dt as functions of t. However, if we reverse the roles of t and 9, we can regard t as a function of 9, and consequently also think of d9/dt as a function of 9. Then derive the following sequence of equations:
(c) By using the identities cos 9 = 1 2 sin2 (9/2) and cos a = 1 2 sin2 (a/2), followed by the change of variable sin (9/2) = k sin 0 with k = sin (a/2), show that
Show that the eigenvalues are ±i so that (0, 0) is a center. Now consider the system
Why was the negative square root chosen in the last equation? (b) If T is the natural period of oscillation, derive the formula
9.4 Competing Species
The integral is called the elliptic integral of the first kind. Note that the period depends on the ratio L/g and also the initial displacement a through k = sin(a/2).
(d) By evaluating the integral in the expression for T obtain values for T that you can compare with the graphical estimates you obtained in Problem 21.
28. A generalization of the damped pendulum equation discussed in the text, or a damped spring-mass system, is the Lienard equation
d2 x dx
-JT + c(x)
2 + c(x) + g(x) = 0.
If c(x) is a constant and g(x) = kx, then this equation has the form of the linear pendulum equation [replace sin 9 with 9 in Eq. (12) of Section 9.2]; otherwise the damping force c(x) dx/dt and restoring force g(x) are nonlinear. Assume that c is continuously differentiable, g is twice continuously differentiable, and g(0) = 0.