elementary properties of nonlinear equations

Scientific Inquiry, vol. 10, no. 2, December, 2009, pp. 153 – 66

IIGSS Academic Publisher

ELEMENTARY PROPERTIES OF NONLINEAR EQUATIONS SHOUCHENG OUYANGa, YONG WUb, AND YI LINc a

Department of Scientific Research, Chengdu University of Information Technology, Chengdu, Sichuan 610041, PR China; b Bureau of National Land, Peilin District, Chongqing City, 408000, PR China; c Department of Mathematics, Slippery Rock University, Slippery Rock, PA 16057, USA; email: [email protected] (Received May 19, 2006; In final form December 6, 2007) Nonlinear problems of mathematics are essentially problems of interactions in physics. To meet the need of resolving physical problems, in this paper we discuss the elementary properties of nonlinear equations. Our purpose is to understand the mathematical properties of nonlinear equations and to find out what problems we should pay attention to in applications of these equations in solving physics problems. Keywords: nonlinearity, interaction, transitionality, evolution, blown-up

1. INTRODUCTION The circulation theorem, as studied in (OuYang and Lin, 2009), and the corresponding Euler equation or Navier-Stokes equation all involve nonlinearity, and are seen as world level difficult problems of mathematics. In the form of closed integrals, the circulation theorem shows that the nonlinearity of the acceleration circulation is a stirred rotation. In this paper, we will discuss the problem of variable acceleration in the form of calculus without imposing any condition on stability. The problem of mathematical nonlinearity was once a hot topic in the latter half of the 20th century. For a period of time, chaos theory, bifurcation, etc., on behalf of the nonlinear science, were wildly talked about in the community of theoretical physics. As a matter of fact, nonlinearity in physics problems is not a unfamiliar concept. For example, it can be traced back to Newton‟s second law of motion: f = ma.

(1.1)

If f, m, and a are all variables, it becomes a nonlinear equation. In history, Newton treated f as “godly” fixed and particlized m as the “quantity” of the object so that it also becomes fixed through the hand of a super natural being. Therefore, the acceleration a has to be a constant. That led to the establishment of the inertial system and the claim that modern science comes from the inertial system, force, and mass. As is known, practical accelerations are always varying with time. So, to guarantee that the acceleration is constant, one has to take the average of the practical, varying acceleration. That leads to the operational form of the law in modern science:

f

ma.

(1.2)

It implies that since the beginning of modern science, invariabilitites of materials have been described using averages and the generality of relatively stable and existing matters seen as the essential Scientific Inquiry: A Journal of International Institute for General Systems Studies, Inc. ISSN 1552-1222© 2009 IIGSS http://www.iigss.net/Scientific-Inquiry/mission.html

154

SHOUCHENG OUYANG, YONG WU, AND YI LIN

property. That is, at the start, modern science has avoided special events. That leads to the chaos in our understanding of special events. That is why special events are seen as irregular, smallprobabilistic, producing the epistemological points of view of the relevant uncertainties and stochastics. The corresponding mechanic system, established by Newton and his followers, is considered the exact “determinism,” leading to the nearly one hundred years old debate between determinacy and indeterminacy without producing any constructive conclusion. Because of the doctrine of “chaos” in the later part of the 20th century, the phenomenon of inadequate numerical computation, met in the study of nonlinear dynamic equations due to the infinitesimal increments between large quantities, when seen as a physics problem, was considered as a fire started in Newton‟s backyard. That is, out of deterministic dynamic equations, one can also derive uncertainties, leading to the debate of a probabilistic universe. As for whether the “chaos” doctrine is correct or not, we have published papers and special volumes (Lin, OuYang, et al., 2001; OuYang, McNeil and Lin, 2002). So, we will omit all details here. As a matter of fact, the dynamic system with Newton as its representative is about the problem of invariability of materials, which is particularly shown in the stability of the initial value automorphism of mathematical physics equations. Evidently, in terms of the problem of invariability of materials, it is already meaningless to talk about certainty. Whether certain or not, it is about the problem of changes. It is only because of changes that materials have their origin and termination and events have their start and end. Initially, Newton studies rigid body‟s movements. So, it is not fair to throw the hat of determinism onto Newton, indeterminacy should be the study about what physics laws changing phenomena actually follow. We should not get Newton and others, such as Poincare and Einstein, involved because the quantitative analysis of his system cannot deal with the so-called smallprobability irregular information. That means that the school of stochastics not only did not understand the purpose of the Newtonian dynamic system, but also confused about the basic concepts and the aims of the problems of indeterminacy or stochastics. Average numbers were introduced as a general tool to describe the existing equilibrium problems. It would be a mistake from the start if average numbers were employed to investigate specifics. With such incorrect application of average numbers taking place first, criticizing determinism is not only indiscriminately blaming others but also pushing oneself into the determinism so that Newton‟s certainty of the present generality becomes that of the past generality. And, in particular, Newton‟s “future equal to the present” becomes “the future equal to the past” so that neither system has a future. After scathingly attacking Newton and his followers, the indeterminate stochastic system also cannot deal with true small probability information. This consequence indeed makes standbys do not know either to laugh or to cry. In the over one hundred years of debate between determinacy and indeterminacy, those involved did not even understand what they have been quarreling about. Evidently, to understand an event‟s cause, process of evolution, and the consequence, it is necessary for us to study the problem of change. If we still focus on problems of invariability, it will be surely the case that a difference exists between the past, the present, and the future. In this case, it is meaningless to continue to wrestle about either determinacy or indeterminacy. Therefore, the essence of the modern science, consisting of the dynamic system, established by Newton and his followers, and the later developed stochastic statistical system, is about how to deal with the quantitative methods of the existent generality without much substantial difference between the two systems. At the heights of epistemology and methodology, none of the two systems has revealed what specifics are? Change represents the general and fundamental problem regarding materials in the universe. In reality, there indeed exists generality with respect to specifics. It can also be said that specifics are the matter of generality. However, as an evolution problem on changes, specifics are exactly the generality of evolutions with the generality of the modern science being a specific detail. Hence, studying specifics is a call from the existing problems and also a need of science. The physics meaning and effects of specifics have been well-discussed in (OuYang, McNeil and Lin, 2002; OuYang, 1998) and can be summarized as follows: irregular information is a piece of evolutionary information about changes. In this paper, we will focus on the mathematical properties of nonlinear equations and the projected problems of physics.

Scientific Inquiry, vol. 10, no. 2, December, 2009

ELEMENTARY PROPERTIES OF NONLINEAR EQUATIONS

155

2. FUNDAMENTAL CHARACTERISTICS OF GENERAL NONLINEAR EQUATIONS Although the dynamic system of physics is originated from equ. (1.1), the practical process of development is based on equ. (1.2). Due to the custom of mathematics, the following model is also based on equ. (1.1): a

du dt

u

F,

(2.1)

where the mass is taken to be 1. The variable u is generally known as the variable of motion and the speed in particular. So, equ. (2.1) is Newton‟s second law in mathematical terms, known as the general mathematical model in the Lagrange language. Evidently, for the generality, F can take any form in variable u. For example, if F is a polynomial in u, then we can discuss the mathematical characteristics respectively for the second and third order polynomials.

2.1. The Second Order Polynomial In this case, we have pu q ,

u2

Fu

(2.2) where p and q are constant. From equs. (2.1) and (2.2), we have u

pu q .

u2

(2.3)

Figure 2-1 Continuity and transitional blown-ups

Evidently, if p and q are different, then equ. (2.3) has solutions of different forms. There are three scenarios here. 2.1.1. Assume that p 2

u

p 2

u

4q

0 . Then equ. (2.3) can be written as

2

.

(2.4)

Integrating equ. (2.4) leads to

u

t

1 A0

p , 2

(2.5)

where A0 is a constant. So, from equ. (2.5), it follows that when A0 > 0, u varies with quantity t continuously, curve II in Figure 2-1. If A0 < 0, then the model contains a blown-up of transitional changes, curves I and III in Figure 2-1. Notice that blown-ups are not about jumping kinds of sudden changes appearing in problems of equilibrium states. Instead, they represent transitional changes in non-initial value systems, reflecting different properties of structures. Sudden changes are step-like variations in initial value systems, belong to the category of quantitative changes of the same properties, showing the difference between stable and accelerated changes. Scientific Inquiry, vol. 10, no. 2, December, 2009

156


2.1.2. When p 2 u

0 , equ. (2.3) can be written as

4q p 2

u

2

1 2 p 4

4q .

(2.6)

Integrating this equation leads to p2

p 2 p 2

u ln u

4q

.

2 p2

p2

4q t

4q

(2.7)

A0

2

So, from equ. (2.7) it follows that when the motion is limited to the bounded local region of u

p 2

p2

4q , we have 2

u

1 p2 2

4q th

1 p2 2

4q t

1 A0 2

p, 2

(2.8)

which is the smooth solution defined on the bounded area. What is interesting is that if we differentiate equ. (2.8) once, we can obtain the solitary wave solution, which has troubled the scientific community for over a century. As a matter of fact, such solutions are only special cases of wave motion systems and cannot represent the evolutions of nonlinearity. If the motion satisfies the overall unboundedness u

u

1 p2 2

4q cth

1 p2 2

4q t

1 A0 2

p, 2

p 2

p2

4q , then we have 2

(2.9)

which indicates that the movement can evolve into transitional blown-ups. When A0 > 0, a blown-up occurs at t

tb

A0

p2

4q .

Both equs. (2.8) and (2.9) imply that for the same nonlinear model, the solution takes different forms under different conditions. The commonality is that nonlinear equations possess the characteristics of discontinuity in the quantitative form with smooth and continuous solutions, which modern science has been pursuing after, as special cases. And, even for locally continuous and smooth solutions, their characteristics are not harmonic waves, either. Instead, these solutions are the well-known solitons. However, the general characteristic of nonlinearity is not following the rules of continuity systems. This end constitutes the essential cause of difficulties met in solving nonlinear equations. That is, nonlinear equations in general do not satisfy the initial value existence theorem of the traditional mathematical physics equations. 2.1.3. When p 2

4q

0 , equ. (2.3) can be written as


ELEMENTARY PROPERTIES OF NONLINEAR EQUATIONS 2

p 2

u

1 4q 4

p2 .

157

(2.10)

Integrating this equation leads to

1 4q 2

u Take 1 4q 2 blown-ups.

p2 t

1 4q 2

p 2 tan

A0

p2 t

(2.11)

a whole number, then equ. (2.11) describes periodic multiple

n , n

2

p. 2

A0

2.1.4. Examples of Applications For the convenience of applications and understanding, let us use the following population model, which once was known as a typical case of chaos and population explosion, as our example,

dx dt

x Let

x 1.

x2

> 0. Then

4 < 1, which implies

1 2 x 1 ln 1 4 2 x 1

1 4

c,

t

1 4

(2.12)

where c is a constant. With some manipulations, we obtain

1

x

1 4

et

c

2 1 e If we let

< 0, then

2 4 Taking 1 +

x

1

4

tan

1

1 4 t c

1

1 4

.

(2.13)

1 4

> 1. So, we have

2 x 1 4

1

t

c.

(2.14)

= 0 leads to

1 t

c

1 . 2

(2.15)

All the three cases in equs. (2.13), (2.14), and (2.15) imply that in the evolution with time, there exist blown-ups of transitional changes. Evidently, the population evolution or bacteria production, as described by the population model, is neither the chaos of the chaos doctrine nor the population explosion of the imaginary unconstrained development. Instead, it is about the problem of population decrease or extinction. The physics problem of nonlinearity reflects the characteristic of materials‟ changes that at extremes, the matter will surely evolve in the opposite direction, revealing transitions in materials‟ evolutions. That is one of the reasons why we established the theory of blown-ups.

3. EIGHT THEOREMS ON CUBIC POLYNOMIAL AND SECOND ORDER NONLINEAR MODELS Scientific Inquiry, vol. 10, no. 2, December, 2009

158


3.1. The general form of a cubic polynomial is

F

u3

pu q .

Substituting this general form into equ. (2.1) provides

u3

u

pu q .

(3.1)

The mathematical properties of equ. (3.1) include the following theorems. Theorem 3.1. If u1 is the root of F of multiplicity 3, then F true:

u

u1

A0

2t

2

u u1

3

so that the following holds

,

(3.2)

where A0 is a constant. Equ. (3.2) stands for a problem of second-order blown-ups. When this scenario is generalized to that of roots of multiplicity n, equ. (3.1) contains (n-1)-th order blown-ups. QED. Theorem 3.2. If F = 0 has a real and a pair of complex conjugate roots, then equ. (3.1) contains blown-ups. Proof. Let u1 be the only real root of F = 0. Then F can be written as

F

u u1 u 2

p1u q1 ,

(3.3)

where p1 , q1 are constants satisfying p12

u u1 u 2

u

4q1

0 . So, equ. (3.1) becomes

p1u q1 .

(3.4)

Using expansions of partial fractions leads to

1 u u1 u 2 p1u q1

A u u1

u2

Mu N , p1u q1

(3.5)

where 2

A

p1

2

1

2

, M

p1 2

u1 ,

u1 ,

2

1

2

, N

q1

1

2

,

2 1

p . 4

Substituting equ. (3.5) into equ. (3.3) and integrating the resultant expression produce

u

2

1 p1u1

q1

u u1

ln u

2

p1u q1

N

p1 M 2 p12 q1 4

u tan

1

q1

p1 2 p12 4

.

t

(3.6)

A0

Evidently, equ. (3.6) shows the fact that there appear multiple blown-ups for finite t-values. QED.



159

Theorem 3.3. If F = 0 has a real root u1 of multiplicity two and another root u 2 of multiplicity one, assume u1 > u 2 (the same results hold for the case of u 2 > u1 ). Equ. (3.4) contains blown-ups for the overall unbounded motion u ＞ u1 or u u > u2 . Proof. From the given conditions, F can be written as

F

2

u u1

u u2

and the corresponding equ. (3.1) as

u

u u1

2

u u2 .

(3.7)

By employing the expansion of partial fractions, we have

1 2

u u1

A u u2

u u1

B u u1

2

C , u u2

where

A

1 u1 u 2

1

, B

u1 u 2

2

, C

1 u1 u 2

. 2

Substituting this expression into equ. (3.7) and integrating the resultant expression provide

A u u1

C ln

u u2 u u1

A0 ,

t

(3.8)

where A0 is a constant. If u > u1 or u u > u 2 , that u is restricted to a local and bounded movement, equ. (3.8) can be written as

A u u1

C ln

u u2 u1 u

t

A0 .

(3.10)

Note that the difference between equs. (3.10) and (3.8) is that when u1 > u , the denominators of the second terms are different. Substituting the expressions of A and C into equ. (3.10) produces

u

u2

u1e

1 e

So, when t

0,

u 2 u1 u1 u

u 2 u1 u1 u

e u1

e u1

u2

u2

2

2

t A0

.

t A0

, equ. (3.9) does not contain any blown-up. QED.


(3.11)

160


Theorem 3.4. If F = 0 has three different real roots u1 , u 2 and u3 satisfying u1 > u 2 > u3 , then if u > u1 or u < u3 , then equ. (3.1) contains blown-ups; if u 2 > u > u3 or u1 > u > u 2 , then equ. (3.1) does not contain any blown-up. Proof. From the assumption, equ. (3.1) can be written as

u

u u1 u u2 u u3 .

(3.12)

So, we get

A ln u u1

ln u u3

u 2 u3 ,B u1 u 2

where A

B ln u u3

t

A0 , C

(3.13)

u1 u3 , and A is a constant. C 0 u1 u 2

If u > u1 or u < u3 , equ. (3.13) can be written as

ln

u u1

u 2 u3 u1 u 2

u u2

u u3

1 t C

u1 u3 u1 u 2

A0

.

(3.14) If the overall motion is unbounded, then equ. (3.14) contains blown-ups and is multi-valued. If u 2 > u > u3 or u1 > u > u 2 , that is, when the motion is local and bounded, then no blown-up will appear. That is, Theorem 3.4 holds true. QED.

3.2. Second-Order Nonlinear Models For generality, let us use the following second order, one variable nonlinear model as our example. u

d 2u dt 2

u3

bu 2

d,

cu

(3.15)

where b, c, and d are constants. Using the method of lowering orders, we can rewrite equ. (3.15) as the following system of first order, two-variable equations:

u v v u 3

.

bu 2

cu

(3.16)

d

Integrating the second equation in equ. (3.16) produces

v

1 4 u 4

b 3 u 3

c 2 u 2

where h0 is a constant. Let u

v

du

h0 ,

E , we have



u

E

1 2 u 4

v

b2 u c 2 ,

b1u c1 u 2

161

(3.17)

where b1 , c1 , b2 , and c 2 are also constants. So, we have the following results. Theorem 3.5. If E = 0 has a root of multiplicity 4, then equ. (3.17) contains blown-ups. The proof is similar to that of Theorem 3.1 and is omitted. QED. Theorem 3.6. If E = 0 has two different roots of multiplicity 2, equ. (3.17) can be rewritten as

1 2 u 4

E

2 c1 ,

b1u

(3.18)

and, when b12 4c1 < 0, the model contains blown-ups. When b12 4c1 > 0, the overall unbounded motion becomes a problem of blown-ups; for any local, unbounded motion, it is a problem without any blown-up. For the proof of this result, see equs. (2.10) and (2.6) for details. QED. Theorem 3.7. If E = 0 has only one root of multi-multiplicity, then equ. (3.17) contains blown-ups. Proof. For simplicity, let the only root be 0. Then, equ. (3.17) can be written as u u2

u

b2 u c2 .

(3.19)

1 . Then, the previous equation becomes u

Take the transformation w

.

dw b2 2c 2

c2 w

(1) If c 2

0 and b22

1 4c 22 4c 2

1

b2 2c 2

1 2 b2 4c 2

(2) If c 2 > 0 and b22

c2 w

c2 t

A0

,

(3.21)

4c 2

where A0 is a constant. Since w

ln

b22

4c2 > 0, integrating equ. (3.20) produces w

sin

2

(3.20)

dt

, equ. (3.20) contains blown-ups, which are periodic.

0, u

4c2 < 0, integrating equ. (3.20) produces b2 2c 2

c2 w


b2 2c 2

4c 22 b22 4c 22

c2 t

A0 .

(3.22)

162


Similar to previous discussions, equ. (3.22) contains blown-ups. That is, Theorem 3.7 holds true. QED.

3.3. The Nonlinearity Problem of the nth-Order Polynomials For the nonlinearity problem of the nth-order polynomials, even though we cannot obtain the precise solution, we can still analyze the mathematical properties and the physical significance reflected by the model. In the field of real numbers, according to the theorem of unique factorization of polynomials, the autonomous system of equ. (2.1) can be written as

u

a1u  a n 1u n

a0

1

un ,

(3.23)

which can be factored into

u

F

u u1

p1

 u ur

where pi , i 1,2,, r , and q j , j r

m

n

pi

2

i 1

pr

u2

q1

 u2

bm u cm

qm

,

(3.24)

1,2,, m , are positive whole numbers, and r

qj, n j 1

b1u c1

m

pi i 1

2

b 2j

qj,

4c j

0,

j

1,2,, m .

j 1

Without loss of generality, if u1 ≥ u 2 ≥ … ≥ u r , then the mathematical properties of equ. (3.23) are given in the following result. Theorem 3.8. (1) When F = 0 does not have the real number roots ui , i 1,2,, r , then the model equ. (3.23) contains blown-ups; (2) If F = 0 does have the real roots ui , i 1,2,, r , then (i) When n is even, if u > u1 , the model (3.23) contains blown-ups. If u u1 or u 0.

u2

b1u c1 ≥

qm

.

(3.25)

b j u c j has a minimum value

So, we have

u ≥

where

0

0

q1 1 1

q2 2



qm m

0

u

1 b1 2

2

> 0,

(3.26)

. If u is a monotonic increasing function, equ. (3.26) implies



u0 1 u0

1 b1 2

u≥

,

163

(3.27)

0t

where u 0 is the initial value. So, the evolution of u contains transitional blown-ups. That is, result (1) in Theorem 3.8 holds. (2) If F = 0 has real roots ui , i 1,2,, r , then we have m

(i) When n is even, since when q

q j is even, p

2

r

pi is also even. So, for u > u1 or u 0. That is, u is monotonically increasing, which contradicts the assumption that u u1 , there are blown-ups. Take u1 ≥ u 2 ≥ … ≥ u r . Then, equ. (3.24) can be written as u ≥

where

1

q1 1

1

p u u1 ,

q2 2



(3.28)

qm m

. Solving the inequality (3.28) leads to

1

u≥ u 1

p 1

A0

,

p 1

1

(3.29)

t

where A0 is a constant, determined by the initial value. Evidently, equ. (3.29) contains transitional blown-ups. (ii) when n is odd, since q is even, p is odd. So, when u > u1 , we have

u > 0.

(3.30)

So, u is monotonically increasing. And, we have

u ≥

1

u u1

p

> 0.

Solving this inequality and taking the (+) branch produce

1

u≥ u 1

p 1

p 1

, 1

t

A0

which implies that u contains blown-ups. When u < u r , we have u < 0. So, u is monotonically decreasing. We have

u ≤

1

u u1

p

< 0.

Solving this inequality and taking the ( ) branch produce Scientific Inquiry, vol. 10, no. 2, December, 2009

(3.31)

164


u≤ u r

1 p 1

p 1

. 1t

(3.32)

A0

Hence, no matter whether u is a decreasing or increasing function, it contains blown-ups of evolutionary transition. That is, Theorem 3.8 (2) holds true. So, Theorem 3.8 is shown. QED.

4. DISCUSSION The proofs and statements of our theorems indicate that the traditional linearity system treats invariance as the generality. That is why we have that “from the initial values, we know everything about the future” (Laplace) and that “modern science cannot tell the difference between the past, the present, and the future” (Einstein). On the other hand, nonlinearity reveals changes and blown-ups of transitional changes as the generality with invariance as special cases of evolutions. That is, a complete flip-over in our comprehension has occurred, where the traditional “generality” becomes a special case so that from knowing the initial value, there is no way for one to foretell the future. And, the traditional special case – singularities – is now the generality of evolution, which can point to the difference between the past, the present, and the future. As hinted by the V. Bjerknes‟ Circulation Theorem, we generalized the idea of varying accelerations to the (n-1)-th order. Changes in accelerations are a physics problem on variable accelerations, while nonlinearity belongs to non-inertial system. This end implies explicitly that nonlinearity cannot be linearized and the essence of linearization is to erase evolution. So, all the laws of physics obtained from using linearization need to be reconsidered. At the same time when the well-posedness theory of the fundamental mathematical physics equations limits the numerical instability, it also loses its validity to the philosophical law – qualitative changes occur on accumulations of quantitative changes, which was initially derived on the achievements of modern science. Numerical instabilities become transitions in the events of curvature spaces. That is, in practice, before a numerical instability occurs, the corresponding event has already gone through a transitional change. Combining with the post-event effect of numbers, it can be seen that the modern scientific law “qualitative changes occur on accumulations of quantitative changes” is not a law of materialism. As an additional remark, let us note that in quantitative formal analysis, it is possible to introduce general quantities of physics. However, we should know that each force F does not occupy a material dimension. If we follow the conclusion that forces are originated in the gradient unevenness in materials‟ structures, then only materials can occupy material dimensions. So, one should be able to take m 1 and F = 1. In this case, equ. (2.1) can be rewritten as u (u ) . Then, analysis and results as above can be obtained. What‟s important to this end is that the analysis and results can be employed to study materials‟ evolutions. Or in other words, acting forces are not the ultimate cause for changes in movement. Changes in acting forces should be caused by changes in the materials‟ structures, which can be naturally included in the evolution science. What needs to be clear is that the problem about changes in materials belongs to a non-inertial system. Although the community of science has accustomed to that without assumptions there will not be any science, each science is after all about finding causes. The reason why we have asked questions as what is force? Where is force located? What is energy? Where is energy stored? What is time? And where is time? etc., is due to the fact that the spirit of finding causes in modern science where materials and forces are separate has been lost. To be specific, both force F and acceleration a dwell in materials so that they do not occupy any material dimension except that m does occupy a material dimension. Therefore, as the materialistic, monistic epistemological point of view, we should not isolate either F or a away from the carrying material. In ancient China, the world and nature were seen from the angle of materials‟ structures. That is why there existed only the monistic materialism. That might be why modern science did not appear in China. However, without modern science is not the same as without science. By looking at the meaning of exploring science, modern science can only be a school of thought, which is divided into Scientific Inquiry, vol. 10, no. 2, December, 2009


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two branches. One is the numerical initial value stability of the inertial systemic mechanics, and the other stable quantitative statistical inferences. Their effectiveness exists only when faced with existence problems of relative stability without involving evolution problems of non-equilibrium. As for the practical significance of the transitional changes, let to by nonlinear equations, we have discussed in details in (OuYang, McNeil and Lin, 2002; OuYang, Wu and Lin, 1998). To summarize, in the studies of the traditional mathematical physics equations, numerical instabilities are limited by the requirement of well-posedness. This method is also applied in various tests of numerical models, leading to the appearance of the great number of different stable computational schemes. However, the mathematical characteristic of nonlinearity is exactly numerical instability. So, most of the methods of limiting numerical instability cannot be applied to nonlinear equations. Next, although quantitative analysis can reveal transitions in evolution, in practical applications, since quantities cannot deal with ∞, we see the fact that quantities themselves cannot deal with transitions. In evolutions of physical matters, transitions occur before the relevant quantities become instable or approach ∞. No doubt, this is a weakness of the quantitative analysis. To meet this challenge, we in practice employ the direction of rotation and construct our structuralized quantitative analysis (OuYang, McNeil and Lin, 2002) outside the methodology of modern science. Its practical significance is that the observational error in directions is relatively small and can be obtained prior to the event data. But, we must be careful that information digitization is not the same as quantification. Or, the realization that events are not quantities is the truly major reform in epistemology. In other words, the 300 some years old science where events are replaced by numbers will be laughed at in the scientific history.

References Lin, Y., S. C. OuYang, et al., (2001) “On Fundamental Problems of „Chaos‟ Doctrine”. International Journal of Applied Mathematics, vol.5, no. 1, 37 – 64. OuYang, S. C., (1998) “Explosive growth of general nonlinear evolution equation and related problems”, Applied Mathematics and Mechanics 19, 165 – 173. OuYang, S. C., and Lin, Y., (2009) “V. Bjerknes Circulation Theorem, Universal Gravitation and Rossby‟s Waves”. Scientific Inquiry, vol. 10, no. 2, pp. 137 – 152. OuYang, S. C., McNeil, D. H., and Lin, Y., (2002) Entering the Era of Irregularity. Meteorological Press, Beijing. OuYang, S. C., Wu, Y., and Lin, Y., (1998) “The Discontinuity Problem and “Chaos” of Lorenz‟s Model”. Kybernetes, vol. 27, no. 6/7, pp. 621 – 635.