# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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For p > 3 neither of the equilibrium solutions is stable, and the solutions of Eq. (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 2.9.4b 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)

(c)

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

u

n

2.9 First Order Difference Equations

121

(a) (b)

(c)

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

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

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.

The p-values at which the successive period doublings occur approach a limit that is approximately 3.57. For p > 3.57 the solutions possess some regularity, but no discernible detailed pattern for most values of p. For example, a solution for p = 3.65 is shown in Figure 2.9.6. It oscillates between approximately 0.3 and 0.9, but its fine structure is unpredictable. The term chaotic is used to describe this situation. One of the features of chaotic solutions is extreme sensitivity to the initial conditions. This is illustrated in Figure 2.9.7, where two solutions of Eq. (21) for p = 3.65 are shown. One solution is the same as that in Figure 2.9.6 and has the initial value u0 = 0.3, while the other solution has the initial value u0 = 0.305. For about 15 iterations the two solutions

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123

(a)

(b)

FIGURE 2.9.5 A solution of u j = pun(1 — un) for p = 3.5; period four. (a) un versus n; (b) a four-cycle.

un,<

0 a —

10 20 30 40 50 60 n

FIGURE 2.9.6 A solution of u j = pun(1 — un) for p = 3.65; a chaotic solution.

124

Chapter 2. First Order Differential Equations

10 20 30 40 50 60 n

FIGURE 2.9.7 Two solutionsof un+1 = pun(1 — un) for p = 3.65; u° = 0.3 and u° = 0.305.

remain close and are hard to distinguish from each other in the figure. After that, while they continue to wander about in approximately the same set of values, their graphs are quite dissimilar. It would certainly not be possible to use one of these solutions to estimate the value of the other for values of n larger than about 15.

It is only in the last several years that chaotic solutions of difference and differential equations have become widely known. Equation (20) was one of the first instances of mathematical chaos to be found and studied in detail, by Robert May13 in 1974. On the basis of his analysis of this equation as a model of the population of certain insect species, May suggested that if the growth rate p is too large, then it will be impossible to make effective long-range predictions about these insect populations. The occurrence of chaotic solutions in simple problems has stimulated an enormous amount of research in recent years, but many questions remain unanswered. It is increasingly clear, however, that chaotic solutions are much more common than suspected at first and may be a part of the investigation of a wide range of phenomena.

PROBLEMS In each of Problems 1 through 6 solve the given difference equation in terms of the initial value

! y°. Describe the behavior of the solution as n ^ ?.

n + 1

1 yn+1 = -°.9yn 2. yn+1 = n+2yn

In + 3 ,

3. yn+1 = y n+1 yn 4 yn+1 = (-1) + yn

5. yn+1 = °.5 yn + 6 6. yn+1 = -°.5 yn + 6

13R. M. May, “Biological Populations with Nonoverlapping Generations: Stable Points, Stable Cycles, and Chaos,”

Science 186 (1974), pp. 645-647; “Biological Populations Obeying Difference Equations: Stable Points, Stable Cycles, and Chaos,” Journal of Theoretical Biology 51 (1975), pp. 511-524.

2.9 First Order Difference Equations

125

7. Find the effective annual yield of a bank account that pays interest at a rate of 7%, compounded daily; that is, find the ratio of the difference between the final and initial balances divided by the initial balance.

8. An investor deposits $1000 in an account paying interest at a rate of 8% compounded monthly, and also makes additional deposits of $25 per month. Find the balance in the account after 3 years.

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