Download (direct link):
(b) Show that each system is almost linear.
(c) Let r2 = x2 + j2, and note that xdx/dt + j dj/dt = r dr/dt. For system (ii) show that dr/dt < 0 and that r — 0 as t — to; hence the critical point is asymptotically stable. For system (i) show that the solution of the initial value problem for r with r = r0 at t = 0 becomes unbounded as t — 1 /2r^, 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. Showthatthe 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 rj = — 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 ± -./kT, 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 â and dâ/dt as functions of t. However, if we reverse the roles of t and â, we can regard t as a function of â, and consequently also think of dâ/dt as a function of â. Then derive the following sequence of equations:
(c) By using the identities cos â = 1 — 2 sin2 (â /2) and cos a = 1 — 2 sin2(a/2), followed by the change of variable sin (â/2) = k sin ô 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
—~2 + c(x) — dt2 dt
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 6 with 6 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.
(a) Write the Lienard equation as a system of two first order equations by introducing the variable j = dx/dt.
(b) Show that (0, 0) is a critical point and that the system is almost linear in the neighborhood of (0, 0).
(c) Show that if c(0) > 0 and g(0) > 0, then the critical point is asymptotically stable, and that if c(0) < 0 or g (0) < 0, then the critical point is unstable.
Hint: Use Taylor series to approximate c and g in the neighborhood of x = 0.
9.4 Competing Species
In this section and the next we explore the application of phase plane analysis to some problems in population dynamics. These problems involve two interacting populations and are extensions of those discussed in Section 2.5, which dealt with a single population. While the equations discussed here are extremely simple, compared to the very complex relationships that exist in nature, it is still possible to acquire some insight into ecological principles from a study of these model problems.
Suppose that in some closed environment there are two similar species competing for a limited food supply; for example, two species of fish in a pond that do not prey on each other, but do compete for the available food. Let x and j be the populations of the two species at time t. As discussed in Section 2.5, we assume that the population of each of the species, in the absence of the other, is governed by a logistic equation. Thus
dx/dt = x(e1 - î1x), (1a)
dj/dt = j(e2 - o2j), (1b)
respectively, where e1 and e2 are the growth rates of the two populations, and å1/ñò1 and å2/ñò2 are their saturation levels. However, when both species are present, each will impinge on the available food supply for the other. In effect, they reduce the growth rates and saturation populations of each other. The simplest expression for reducing the growth rate of species x due to the presence of species j is to replace the growth rate factor e1 - î 1 x in Eq. (1a) by e1 - î1 x - a1 j, where a1 is a measure of the degree to