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b. Does the value of this limit change if the value of y(1) is modified,
while r is kept fixed?
Pb. 2.22 Find the iterates of the logistic equation for the following values of
r: 3.1, 3.236068, 3.3, 3.498561699, 3.566667, and 3.569946, assuming the follow-
ing three initial conditions:

y(1) = 0.2, y(1) = 0.5, y(1) = 0.7

In particular, specify for each case:
a. The period of the orbit for large N, and the values of each of the
iterates.
b. Whether the orbit is super-stable (i.e., the periodicity is present for
all values of N).

This section provided a quick glimpse of two types of nonlinear difference
equations, one of which may not necessarily converge to one value. We dis-
covered that a great number of classes of solutions may exist for different
values of the equation’s parameters. In Section 2.8 we generalize to 2-D. Sec-
tion 2.8 illustrates nonlinear difference equations in 2-D geometry. The study
of these equations has led in the last few decades to various mathematical
discoveries in the branches of mathematics called Symbolic Dynamical the-
ory, Fractal Geometry, and Chaos theory, which have far-reaching implica-
tions in many fields of engineering. The interested student/reader is
encouraged to consult the References section of this book for a deeper
understanding of this subject.

2.8 Fractals and Computer Art
In Section 2.4, we introduced a fractal type having a priori well-defined and
apparent spatial symmetries, namely, the Koch curve. In Section 2.7, we dis-
covered that a certain type of 1-D nonlinear difference equation may lead, for
a certain range of parameters, to a sequence that may have different orbits.
Section 2.8.1 explores examples of 2-D fractals, generated by coupled differ-
ence equations, whose solution morphology can also be quite distinct due
solely to a minor change in one of the parameters of the difference equations.
Section 2.8.2 illustrates another possible feature observed in some types of
fractals. We show how the 2-D orbit representing the solution of a particular
nonlinear difference equation can also be substantially changed through a
minor variation in the initial conditions of the equation.

© 2001 by CRC Press LLC