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In-Class Exercise

Pb. 2.20 Using the difference equation given by Eq. (2.38):
a. Write down a routine to compute 2 . As an initial guess, take the
initial value to be successively: 1, 1.5, 2; even consider 5, 10, and
20. What is the limit of each of the obtained sequences?
b. How many iterations are required to obtain 2 accurate to four
digits for each of the above initial conditions?
c. Would any of the above properties be different for a different choice
of A.

Now, having established that the above sequence goes to a limit, let us
prove that this limit is indeed A . To prove the above assertion, let this limit
be denoted by y ; that is, for large k, both y(k) and y(k + 1) ⇒ y , and the
lim lim
above difference equation goes in the limit to:

1  A 
y = y + (2.39)
lim  lim 
2  y lim 
Solving this equation, we obtain:

y = A (2.40)
lim

It should be noted that the above derivation is meaningful only when a
limit exists and is in the domain of definition of the sequence (in this case, the
real numbers). In Section 2.7.2, we encounter a sequence where, for some val-
ues of the parameters, there is no limit.

2.7.2 The Logistic Equation
Section 2.7.1 illustrated the case in which the solution of a nonlinear differ-
ence equation converges to a single limit for large values of the iteration
index. In this chapter subsection, we consider the case in which a succession
of iterates (called orbits) bifurcate, yielding orbits of period length 2, 4, 8, 16,
ad infinitum, ending in what is called a “chaotic” orbit of infinite period
length. We illustrate the prototype for this class of difference equations by
exploring the logistic difference equation.
The logistic equation was introduced by Verhulst to model the growth of
populations limited by finite resources (the name logistic was coined by the
French army under Napoleon when this equation was used for the planning
of “logement” of troops in camps). In more modern settings of ecology, the

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