1.5. FIGURES 5
The methods used in the study of uncertainty in crisp differential equa-
tions usually falls into two areas. They are the so called “AI-based methods”,
also called “semiquantitative simulation”, and the Monte Carlo methods. In
the semiquantitative method the object is to give a qualitative description of
the behavior of all possible solutions. In the Monte Carlo technique the goal
is to construct all possible solutions. However, the set of all possible solutions
is infinite, so they compute some finite (discrete) approximation to the set of
all possible solutions. Both areas have their advantages and disadvantages
[17]. What we do in this book can be classified as the Monte Carlo method.
What then is new in this book is: (1) we argue in Chapter 4 that many
crisp continuous (dynamical) systems naturally become fuzzy through fuzzy
estimation (Chapter 3) of the uncertain initial conditions and parameters; (2)
we find an approximation to the band of uncertainty which is the trajectory of
the bases of the fuzzy number trajectories; and (3) we apply this to numerous
diverse applications in Chapters 7 - 26 using the readily available simulation
language Simulink [19].
1.5 Figures
The reader can see that there are three types of figures in this book. We now
explain why we used three different types of figures.
1.5.1 Maple
Some of the figures, graphs of certain fuzzy numbers, in the book are difficult
to obtain by standard methods (LaTeX) so they were created using a different
method. These graphs were done first in Maple [16] and then exported to
LaT eX2
. We did these figures first in Maple because of the “implicitplot”
command in Maple. Let us explain why this command was important in this
book. Suppose
X is a fuzzy estimator we want to graph. Usually in this book
we determine
X by first calculating its α-cuts. Let
X[α]=[x
1
(α),x
2
(α)].
So we get x = x
1
(α) describing the left side of the triangular shaped fuzzy
number
X and x = x
2
(α) describes the right side. On a graph we would
have the x-axis horizontal and the y-axis vertical. α is on the y-axis between
zero and one. Substituting y for α we need to graph x = x
i
(y), for i =1, 2.
But this is backwards, we usually have y a function of x. The “implicitplot”
command allows us to do the correct graph with x a function of y when we
have x = x
i
(y). All figures in Chapters 2 and 3 were done in Maple and then
exported to LaT eX2
.
1.5.2 LaTeX
Some other figures were easily constructed using the graphics in LaT eX2
.
These figures are Figures 11.1, 12.1, 13.1-13.3, 15.1, 17.1, 22.1 and 28.1.
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