Appendix A
Theoretical foundation of the
Yoyo model
This appendix looks at the theoretical foundation on why such an intuition as the
systemic yoyo model of general systems holds for each and every system that is either
tangible or imaginable.
In particular, Section A.1 looks at the concept of blown-ups that commonly occur
in evolutions of systems. Section A.2 summarizes the mathematical properties of tran-
sitional changes that exist along with each evolution. Section A.4 provides a brand
new understanding of the quantitative infinity. And Section A.4 focuses on the study
of equal quantitative movements and equal quantitative effects.
A.1 BLOWN-UPS: MOMENTS OF EVOLUTIONS
When we study the nature and treat everything we see as a system (Klir, 2001), then
one fact we can easily see is that many systems in nature evolve in concert. When one
thing changes, many other seemingly unrelated things alter their states of existence
accordingly. That is why (OuYang, Chen, and Lin, 2009) proposes to look at the
evolution of a system or event of concern as a whole. That is, when developments and
changes naturally existing in the natural environment are seen as a whole, we have
the concept of whole evolutions. And, in whole evolutions, other than continuities, as
well studied in modern mathematics and science, what seems to be more important
and more common is discontinuity, with which transitional changes (or blown-ups)
occur. These blown-ups reflect not only the singular transitional characteristics of the
whole evolutions of nonlinear equations, but also the changes of old structures being
replaced by new structures. By borrowing the form of calculus, we can write the
concept of blown-ups as follows: For a given (mathematical or symbolic) model, that
truthfully describes the physical situation of our concern, if its solution u =u(t; t
0
, u
0
),
where t stands for time and u
0
the initial state of the system, satisfies
lim
tt
0
|
u
|
=+, (A.1)
and at the same time moment when t t
0
, the underlying physical system also goes
through a transitional change, then the solution u =u(t; t
0
, u
0
) is called a blown-up
solution and the relevant physical movement expresses a blown-up. For nonlinear
384 Appendix A: Theoretical foundation of the yoyo model
models in independent variables of time (t) and space (x, x and y,orx, y, and z),
the concept of blown-ups are defined similarly, where blow-ups in the model and the
underlying physical system can appear in time or in space or in both.
A.2 PROPERTIES OF TR ANSITION AL CH ANGES
To help us understand the mathematical characteristics of blown-ups, let us look at
the following constant-coefficient equation:
.
u
= a
0
+ a
1
u +···+a
n1
u
n1
+ u
n
= F, (A.2)
where u is the state variable, and a
0
, a
1
, ..., a
n1
are constants. Based on the
fundamental theorem of algebra, let us assume that equ. (A.2) can be written as
.
u
= F = (u u
1
)
p
1
···(u u
r
)
p
r
(u
2
+ b
1
u + c
1
)
q
1
···(u
2
+ b
m
u + c
m
)
q
m
, (A.3)
where p
i
and q
j
, i =1, 2, ..., r and j =1, 2, ..., m, are positive whole numbers, and
n =
-
r
i=1
p
i
+2
-
m
j=1
q
j
, =b
2
j
4c
j
< 0, j =1, 2, ..., m. Without loss of general-
ity, assume that u
1
u
2
···u
r
, then the blown-up properties of the solution of
equ. (A.3) are given in the following theorem.
Theorem A.1: The condition under which the solution of an initial value problem
of equ. (A.2) contains blown-ups is given by
1 When u
i
, i =1, 2, ..., r, does not exist, that is, F =0 does not have any real
solution; and
2IfF =0 does have real solutions u
i
, i =1, 2, ..., r, satisfying u
1
u
2
···u
r
,
(a) When n is an even number, if u > u
1
, then u contains blow-up(s);
(b) When n is an odd number, no matter whether u > u
1
or u < u
r
, there always
exist blown-ups.
A detailed proof of this theorem can be found in (Wu and Lin, 2002, p. 65–66)
and is omitted here. And for higher order nonlinear evolution systems, please consult
(Lin, 2008b).
A.3 QU ANTITATIVE INFINI TY
One of the features of the concept of blown-ups is the quantitative infinity , which
stands for indeterminacy mathematically. So, a natural question is how to compre-
hend this mathematical symbol . In applications, this symbol causes instabilities and
calculation spills that have stopped each and every working computer.
To address the previous question, let us look at the mapping relation of the
Riemann ball, which is well studied in complex functions (Figure A.1). This so-called
Riemann ball, a curved or curvature space, illustrates the relationship between the

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