# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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Now consider the other equilibrium solution un = (p — 1)/p. To study solutions in the neighborhood of this point, we write

u- = ^ + v„. (25)

p

where we assume that vn is small. By substituting from Eq. (25) in Eq. (21) and simplifying the resulting equation, we eventually obtain

v-+1 = (2 — P)v- — Pv-. (26)

Since vn is small, we again neglect the quadratic term in comparison with the linear terms and thereby obtain the linear equation

vn+1 = (2 — P)vn ¦

(27)

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Chapter 2. First Order Differential Equations

Referring to Eq. (9) once more, we find that vn ^ 0as n ^ to for 12 — pi < 1, that is, for 1 < p < 3. Therefore we conclude that, for this range of values of p, the equilibrium solution un = (p — 1)/p is asymptotically stable.

Figure 2.9.1 contains the graphs of solutions ofEq. (21) for p = 0.8, p = 1.5, and p = 2.8, respectively. Observe that the solution converges to zero for p = 0.8 and to the nonzero equilibrium solution for p = 1.5 and p = 2.8. The convergence is monotone for p = 0.8 and p = 1.5 and is oscillatory for p = 2.8. While the graphs shown are for particular initial conditions, the graphs for other initial conditions are similar.

Another way of displaying the solution of a difference equation is shown in Figure

2.9.2. In each part of this figure the graphs of the parabola y = px(1 — x) and of the straight line y = x are shown. The equilibrium solutions correspond to the points of intersection of these two curves. The piecewise linear graph consisting of successive vertical and horizontal line segments, sometimes called a stairstep diagram, represents the solution sequence. The sequence starts at the point u0 on the x-axis. The vertical line segment drawn upward to the parabola at u0 corresponds to the calculation of pu0(1 — u0) = u1. This value is then transferred from the y-axis to the x-axis; this step is represented by the horizontal line segment from the parabola to the line y = x. Then the process is repeated over and over again. Clearly, the sequence converges to the origin in Figure 2.9.2a and to the nonzero equilibrium solution in the other two cases.

To summarize our results so far: The difference equation (21) has two equilibrium solutions, un = 0and un = (p — 1)/p; the former is stable for 0 < p < 1, and the latter is stable for 1 < p < 3. This can be depicted as shown in Figure 2.9.3. The parameter p is plotted on the horizontal axis and u on the vertical axis. The equilibrium solutions u = 0 and u = (p — 1)/p are shown. The intervals in which each one is stable are indicated by the heavy portions of the curves. Note that the two curves intersect at p = 1, where there is an exchange of stability from one equilibrium solution to the other.

For p > 3 neither of the equilibrium solutions is stable, and the solutions ofEq. (21) exhibit increasing complexity as p increases. For p somewhat greater than 3 the sequence un rapidly approaches a steady oscillation of period 2; that is, un oscillates back and forth between two distinct values. For p = 3.2 a solution is shown in Figure 2.9.4. For n greater than about 20, the solution alternates between the values 0.5130 and

0.7995. The graph is drawn for the particular initial condition u0 = 0.3, but it is similar for all other initial values between 0 and 1. Figure 2.9.46 also shows the same steady oscillation as a rectangular path that is traversed repeatedly in the clockwise direction.

0.8

0.6

0.4

0.2

(a)

( b)

(ñ)

FIGURE 2.9.1 Solutions of un+l = pun(1 — un): (a) p = 0.8; (b) p = 1.5; (c) p = 2.

u

n

(a) (b)

FIGURE 2.9.2 Iterates of u+ = pun(1 - un). (a) p = 0.8; (b) p = 1.5; (c) p = 2.8.

u

1 - u

II

p

1

1

0.5 Stable

u = 0 /\ 1 1 _

-0.5 /1 ^ 2 3 p

----- ---^Unstable

- /

FIGURE 2.9.3 Exchange of stability for un+1 = pun(1 — un).

122

Chapter 2. First Order Differential Equations

(a)

FIGURE 2.9.4 A solution of u 1 = pun(1 — un) for p = 3.2; period two. (a) un versus n; (b) a two-cycle.

At about p = 3.449 each state in the oscillation of period two separates into two distinct states, and the solution becomes periodic with period four; see Figure 2.9.5, which shows a solution of period four for p = 3.5. As p increases further, periodic solutions of period 8, 16, ... appear. The appearance of a new solution at a certain parameter value is called a bifurcation.

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