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國 立 交 通 大 學

應用數學系

碩 士 論 文

橢圓函數之理論與運用

The Theory and Applications of the Elliptic

Functions

研 究 生 : 劉家豪

指導教授 : 李榮耀 教授

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橢圓函數之理論與運用

The Theory and Applications of the Elliptic Functions

研 究 生 : 劉家豪 Student :Ja-Hou Liu

指導教授 : 李榮耀 Advisor : Jong-Eao Lee

國 立 交 通 大 學

應 用 數 學 系

碩 士 論 文

A Thesis

Submitted to Department of Applied Mathematics

College of Science

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of

Master

in

Applied Mathematics

January 2010

Hsinchu, Taiwan, Republic of China

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橢圓函數之

理論與運用

研 究 生 : 劉家豪 指導教授 : 李榮耀

國 立 交 通 大 學

應 用 數 學 系

摘 要

本篇文章主要是研究古典橢圓函數的理論及其在微分方程式上的運用和分析。 在 第 一 章 裡 我們 定 義了 橢 圓 函 數 並分 析 其性 質 , 接 著 介紹 了 Weierstrass 和 Jacobian 這兩個代表性的橢圓函數。 在第二章裡提供了一些分析相位圖的技巧與方法。

在第三章裡利用 Jacobian 函數解 Sine-Gordon equation 所描繪的理想平面鐘擺運 動。接著使用第二章所提供的技巧與方法分析非理想狀態下的平面鐘擺運動。

在第四章裡提供了五個物理問題之數學模式用微分方程式來描繪並運用 Jacodian 函數來求解。

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The Theory and Applications of the Elliptic

Functions

Student:Ja-Hou Liu

Advisor:Jong-Eao Lee

Department of Applied Mathematics

National Chiao Tung University

Abstract

In this paper, we study the classical elliptic functions and the applications to the differential equations.

In chapterⅠ, we define the elliptic functions and analyze it’s properties. And then, we introduce Weierstrass functions and Jacobian functions, the two typical elliptic functions.

In chapterⅡ, we analyze phase portraits.

In chapterⅢ, we study the Sine-Gordon equation that describes the ideal

pendulum motion and use Jacobian functions to represent the solutions. We then use the methods in chapterⅡ to analyze pendulum motion with friction.

In chapterⅣ, we provide other five physical models described by differential equations and solve them by Jacobian functions.

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誌 謝

本篇論文的完成,首先要感謝我的指導老師-李榮耀教授。老師在

研究的初期扮演著引導的角色,帶著我一步一步的研究理論內容的深

處,在後期則是扮演了敦促與激勵的角色,激發學生在已有的基礎上

更加的往前深入與探索。若不是老師ㄧ步步的指引就不會有本篇論文

的產生。

在口試期間,十分感謝兩位口試委員的幫助,尤其在論文的用詞

上給予了十分寶貝的意見,使本篇的論文更佳的完善。

當然一定要謝謝與我一同生活的同伴們與在新竹一個個關心我的

家,在這段時間給予無限的支持與幫助。謝謝羅晟天天陪我討論物理

的理論、詹博全在英文文法及用字上提供寶貴的意見、鍾宗穎在許多

程式以及文字編排上的幫助、周節在圖表上的協助、邱俊傑連續好幾

個禮拜早上五點陪我在寒冷的冬天吃早餐,要感謝的人實在述說不盡。

最後要感謝我的父母親,養育之恩難言,也因為他們的堅持,我

才會註冊進到校園裡完成碩士的學業,才會有今天論文的產生。

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Contents:

中文摘要………Ⅰ

英文摘要………Ⅱ

誌謝…..…………... Ⅲ

Contents………..Ⅳ

Chapter 1 Elliptic Functions

1.1 General theorem and properties of elliptic functions

1.1.1 Induction……….1

1.1.2 Doubly-Periodic functions……… .1

1.1.3 Elliptic functions……….. ..1

1.1.4 Period-parallelograms……….2

1.1.5 Some properties of elliptic functions………..2

1.1.6 The order of an elliptic function……….…….3

1.1.7 The Weierstrass elliptic function………...3

1.1.8 Some properties of Weierstrass elliptic function………....….5

1.1.9 The Riemann-Zeta function ζ(z)……….……….6

1.1.10 Some properties of The function ζ(z)……….………7

1.1.11 The sigma function σ(z)…………...……..………7

1.1.12 Some properties of The functionσ(z)…….………7

1.1.13 Expression of elliptic function……….………..7

1.2 Theta functions

1.2.1 Theta functions……….……….8

1.2.2 The four types of Theta functions……….………...….8

1.2.3 Some properties of Theta functions….………9

1.3 Jacobian function

1.3.1 Jacobian function………...12

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1.3.2 Some relation between Jacobian function…….……….13

1.3.3 Some properties of Jacobian functions…….………15

1.3.4 The Jacobian elliptic function and the trigonometric

functions………20

1.3.5 General form of the elliptic integral of the first kind….…...23

1.3.6 Some graphs of Jacobian functions……….……..25

Chapter 2 Linearzied, Nullclines, Hamiltonian system and Dissipative

system

2.1 Linearzied………...41

2.2 Nullclines………... 42

2.3 Hamiltonian system…….………...42

2.4 Dissipative system…….……….44

2.5 Discussion……….……….…...………..44

Chapter 3 Pendulum

3.0 Some induction of Pendulum

3.0.1 Physic aspect……….……….45

3.0.2 Mathematic aspect………..………...……… 46

3.1 Ideal Pendulum

3.1.0 Ideal Pendulum……….……….….47

3.1.1 Apply Linearzied to analysis Ideal pendulum……..….……..48

3.1.2 Apply the Hamiltonian system to analysis Ideal pendulum….49

3.1.3 Apply the Jacobian elliptic function to solve the Ideal pendulum

motion………..…….49

3.1.4 The graph of the Ideal pendulum motion………...52

3.2 Pendulum motion with friction

3.2.0 Pendulum motion with friction………...55

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3.2.2 Apply the nullclines to analysis pendulum with friction….… 56

Chapter 4 Physical Applications of Elliptic functions

4.1 Whirling Chain………....….59

4.2 Duffing‟s Equation………....…...62

4.3 Orbit motion

4.3.1 Orbits under aμ /r

4

Law of Attraction………...65

4.3.2 Orbits under aμ /r

5

Law of Attraction………...…...73

4.3.3 Relativistic Planetary Orbits………...…………...…...78

Chapter5 Conclusion

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Chapter 1 Elliptic Functions

1.1.1 Introduction:

In this chapter, we study the classical elliptic functions and it mainly follows the references [3, 4, 5, 10]. Some of the tables follow another reference [6]. Moreover, the relation between one period functions and double-period functions are according to the references [4, 7]. Figures inside are drawn by the program Mathematica.

1.1.2 Doubly-Periodic functions:

Let 12 be any two numbers (real or complex) whose ratio is not purely real. A function which satisfies the equations

z 2  f z

f  1  fz22 f z ,

and no further period lies between 0 and 1, and 0 and 2 respectively,

for all values of z for which f z exist, is called a doubly-periodic function of z, with periods 21 22

1.1.3 Elliptic functions:

1. The singularity (singular point)

If f(z) is not analytic at zz0 then we call z0 is the singularity of f(z).

For a singular point z0 if there exists a neighborhood N

 

z0 of z0 such that the function f(z) is analytic in N

 

z0 /z0 then z0 is called a isolated singularity of f(z).

Moreover, for a isolated singularity of f(z) if there exist an analytic function C

N ( z ) :

g ( z )  such that g(z) f(z) on N

 

z0 /z0 then the point z0 is called a removable singularity.

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2. Pole

If z0 is isolated singularity and min kN such that z-z0kf(z) is analytic at

0

z then z0 is a pole of function f(z)

3. Elliptic function

A doubly-periodic function which is analytic (except at poles), and which has no singularities other than poles in the finite part of the plane, is called an elliptic

function.

Remark 1:

A function defined in real is defined in one dimension. It means we can see all the function if there is one certain period in it. Furthermore, a function defined in complex number is defined in two dimension and a “good” function defined in complex number should have two period that is doubly-periodic function.

1.1.4 Period-parallelograms:

Suppose that in the plane of the variable z for a elliptic function with two primitive periods 1 and 2 is completely determined in any one of the parallelograms with

vertices at z0, z021, z02122, z0 22, where z0C . For proper choice of z0, the poles of this elliptic function will not reside on the boundary of any of these parallelograms. Such parallelograms are called the cells.

1.1.5 Some properties of elliptic functions:

1. The number of poles of an elliptic function in any cell is finite. 2. The number of zeros of an elliptic function in any cell is finite.

3. The sum of the residues of an elliptic function at its poles in any cell is zero. 4. Liouvilles theorem

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1.1.6 The order of an elliptic function:

1. The order of an elliptic function

If f z be an elliptic function and c be any constant, the number of roots of equation

 z

f =c which lie in any cell depends only on f z , and not on c ; this number is called the order of the elliptic function.

2. Some properties of the order of an elliptic function

a. The number of the elliptic function f z is equal to the number of poles of f z in the cell.

b. The order of an elliptic function is2.

Remark2:

The elliptic functions with order two are classified as two kinds; the Weierstrassian elliptic functions, which have a single double pole in a cell, and the Jacibian elliptic functions, which have two simple poles in a cell.

The importance of the elliptic functions with order two is as indicated by the fact that any elliptic function can be in terms of either of these type. We study the elliptic functions of order two in the next section.

1.1.7 The Weierstrass elliptic function:

The Weierstrass elliptic function

   

           Z n m, 2 2 1 2 2 1 / 2 2n 2m 1 -2n -2m -z 1 z 1 (z)     where

/

denotes that the sum excludes the term when m  n  0and 1,2satisfy

the condition that the ratio is not purely real.

For brevity, we write m,nin place of 2m1+2n2,

so that

      Z n m, 2 -n m, 2 -n m, / 2 z- -z 1 (z) (1.1.1)

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Remark3:

1. If 21 and 22 are periods whose ratio is real, then it is not double period for a noconstant elliptic function.(In reference [2])

a. If b a 2 2 2 1   

, where a and b are relatively prime integers, then there exists integers m and n such that mb na 1

Let   21 22. Then  is a period and we have the following

  b na mb b b a n m n m 1 1 1 2 1 1                          ,

So 1  b and 2  a . Thus 1 and 2 integer multiples of  .

b. If   

2

1 , is an irrational number. Given   0, there exist integers p and q such

that 2 2 1 2 1 2 2q -2p or q -p q -p                

but then 2p1-2q2 would be a period of arbitrary small modulus, which is impossible.

2. The weierstrassian function is a elliptic function. a. For 1         ( z ) 2n 2m 1 -2n -2m -z 1 z 1 2n 2m 1 -2n -2m -2 z 1 ) 2 (z 1 ) 2 (z Z n m, 2 2 1 2 2 1 / 2 Z n m, 2 2 1 2 2 1 1 / 2 1 1                        

            

By the same way

(z 22)( z )

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b. The singular points of (z) are m,n and for any m,n take k 2 then (z -m,n)k( z ) is analytic on z m,n.

c. For any finite plane the number of poles are finite.

From a. b. and c. (z) is an elliptic function.

1.1.8 Some properties of Weierstrass elliptic function:

1. (z) is an even elliptic function of z. 2.

m,n

-3 Z n m, / / -z -2 (z)   

is an odd elliptic function of order three with poles at {m,n }and zeros (2n 1)w1,(2m 1)w 2; moreover,

(z 2w1)(z  2w 2)( z ) . 3. The differential equation satisfied by (z)

((z))2  4(z)3 -g2(z) -g3 (1.1.2) where m,n-6 Z n m, / 3 -4 n m, Z n m, / 2 60 g 140 g 

 

  

and the constantg2, g3 are called invariants of(z) ; moreover,

)) (w -(z) ))( (w -(z) ))( (w -(z) 4( ) e -(z) )( e -(z) )( e -(z) 4( (z)) ( 3 2 1 3 2 1 2             (1.1.3) Where (w1)e1 (w2)e2 (w3)e3 with w 3 -w1-w2. and ei  ej for i j . 4. The intergral formula for (z)

z

4t -g t-g

2dt 1 -3 2 3 (z)

   (1.1.4) 5. The addition-theorem for the function (z)

If u v w  0,then

( u ) (( v )-( w ) )-( v ) (( u )-( w ) )( w ) (( u )-( v ) ) 0 (1.1.5)

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That is 0 1 ) (w (w) 1 ) (v (v) 1 ) (u (u)          

and (u) ,(v) ,(w) ,are all unequal.

Remark 4:

By (1.1.2)and (1.1.3)

If (u)  e1,(v)  e2,(w) e3 and u vw  0

then e1,e2,e3,are the roots of the equation 4t -g2t-g3 0

3 That is e1e2 e3 4 g e e e e e e1 22 33 1  2 4 g e e e1 2 3  3

7. Another form of the addition-theorem

-(y) -(z) (y) -(z) 4 1 y) (z 2              (z) -(y) (1.1.6) -2 ( z ) ( z ) ( z ) 4 1 ( 2 z ) 2             unless 2z is a period. (1.1.7)

1.1.9 The Riemann-Zeta function ζ(z):

The function z defined by the equation

  - ( z ) dz z d    with   } 0 z 1 -z { lim 0 -z    . (1.1.8)

The limit condition in (1.1.8) is to assure that   z has simple pole at z  0. and so  

               / n m, 2 n m, n m, n m, z 1 -z 1 z 1 z 

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1.1.10 Some properties of The function ζ(z):

 z

 is odd with a simple poles m,n, but is not doubly periodic. In fact,   z is quasi-periodic elliptic function,

    z 2 12  z 2 12 2 z 2 z               (1.1.9) where 1   1 , 2  21 with i 2 1 - 2 1 2 1     

1.1.11 The sigma function σ(z):

The function (z) defined by the equation

 z (z) log dz d    with 1 z (z)

lim

0 z          (1.1.10) The limit condition in (1.1.10) is to assure that (z) has simple zero atz 0.

So                               n m, 2 2 n m, n m, / n m, 2 z z exp z -1 z (z) 

1.1.12 Some properties of The functionσ(z):

The function (z) is an odd entire function with simple zeros at all the points m,n,

and is quasi-periodicity, (z) )} (z -exp{2 ) 2 (z (z) )} (z -exp{2 ) 2 (z 2 2 2 1 1 1                 (1.1.11)

1.1.13 Expression of elliptic function:

1. Any elliptic function can express in terms of (z) and (z)/

2. Any elliptic function can express in terms of linear combination of Zeta-functions and their derivates.

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3. Any elliptic function can express in terms of quotient of Sigma-functions.

1.2.1 Theta functions:

Let τ be a (constant) complex number whose imaginary part is positive; and write i

e

q  , so that q 1

Consider the function z,q, defined by the series  

    -n 2 n i z n n e q ( - 1 ) q z, 2  , (1.2.1) qua function of the variable z.

It is evident that  

    1 n n n cos2nz q (-1) 2 1 q z, 2  Sand that z,qz,q; Further   n 2n 2niz -n n e q q (-1) q , z 2

     

         -n iz 1 n 2 1) (n 1 n 2iz -1 -e q (-1) e -q 2 , And so z,q-q-1e-2izz,q

1.2.2 The four types of Theta functions:

It is customary to write 4z,q in place of z,q;the other three type of Theta-functions are defined as follows:

 

              0 n ) 2 1 (n n 4 i 4 1 iz 1 ,q 2 (-1) q sin(2n 1)z 2 1 z -ie q z, 2     (1.2.2)   ,q 2 q cos(2n 1)z 2 1 z q z, 0 n ) 2 1 (n 1 2 2          

      (1.2.3)  

            1 n n 4 3 ,q 1 2 q cos2nz 2 1 z q z, 2    (1.2.4) z,q 1 (-1) q cos2nz 2 n 1 n n 4

     (1.2.5)

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Writing down the series at length, we have

z,q 2q sinz -2q sin3z 2q 4 sin5z -...

2.5 4 9 4 1 1    (1.2.6)

z,q 2q cosz 2q cos3z 2q 4 cos5z ...

2.5 4 9 4 1 2      (1.2.7)

z,q 1 2qcos2z 2q4cos4z 2q9cos6z ...

3     

 (1.2.8)

z,q 1-2qcos2z 2q4cos4z -2q9cos6z ...

4   

 (1.2.9)

For brevity,

1. The parameter q will usually not be specified, so that 1 z ,...will be written for

z,q,...

1

2. When it is desired to exhibit the dependence of a Theta-function on the parameter  , it will be written z|.

3.1 0 ,2   0 ,3 0 ,4 0 will be replaced by 1,2,3,4and 1will denote the

result of making z equal to zero in the derivate of 1(z) .

1.2.3 Some properties of Theta functions:

1. 1z, qis an odd function of z and that the other Theta-functions are even functions of z.

2. The relations between the squares of the Theta-functions

      2 3 2 1 2 2 2 4 2 4 2 2 z  z  z - (z)   (1.2.10)       2 2 2 1 2 3 2 4 2 4 2 3 z  z  z - ( z )   (1.2.11)       2 3 2 2 2 2 2 3 2 4 2 1 z  z  z - ( z )   (1.2.12)       2 2 2 2 2 3 2 3 2 4 2 4 z  z  z - ( z )   (1.2.13) Form equation (1.2.13), let z=0 we get fallowing equation.

4 3 4 2 4 4      . (1.2.14)

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3. The addition-formulae for the Theta-functions

3z y 3 z-y32 y32( z )-12( y )12( z ) (1.2.15) 4. Jacobis expressions for Theta-functions as infinite priducts

 



    1 n 4n 2n 2n 4 1 1 z,q 2q sinz 1-q 1-2q cos2z q  (1.2.16)  



     1 n 4n 2n 2n 4 1 2 z,q 2q cosz 1-q 1 2q cos2z q  (1.2.17)  



     1 n 2 -4n 2n 2n 3 z,q 1-q 1 2q cos2z q  (1.2.18)  



    1 n 2 -4n 2n 2n 4 z,q 1-q 1-2q cos2z q  (1.2.19) 5. The differential equation satisfied by Theta-function

               z i 4 -z z 3 2 3 (1.2.20) 6. A relation between Theta-functions of zero argument

 0 2 0 3(0) 4(0)

1   

   (1.2.21) 7. Sigma-function can express in terms of Theta-functions

so any elliptic function can express in terms of Theta-functions 8. Landen‟s type of transformation

                       2 | 0 | 0 | 0 2 | 2z | z | z 4 4 3 4 4 3 (1.2.22) 9. The differential equation satisfied by quotients of Theta-functions

a.            z z z z z z dz d 4 3 4 2 2 4 4 1               (1.2.23) b.            z z z z -z z dz d 4 3 4 1 2 3 4 2               (1.2.24) c.            z z z z -z z dz d 4 2 4 1 2 2 4 3               (1.2.25)

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From (1.2.23) We write    z z 4 1  

  and use the result of relations between the squares of the Theta-functions. We see that ( - )( - ) dz d 2 2 2 2 3 2 3 2 2 2               (1.2.26) Write 2 2 3 1 2 3 2 3/ u z k /

y       and k is called modulus We get equation

) (1-y )(1-k y ) du

dy

( 2  2 2 2 (1.2.27)

This differential equation has the particular solution

-2

3 4 -2 3 1 2 3 u u y        (1.2.28) Rmark5:

1. Let k be called the complementary modulus such that k2k2 1, that is

2

k -1 k

2. The number u will be called the argument and the number 2

k

m  be called the

parameter of the functions.

3. The complementary parameter is the number m1 1-m, that is 2

1 k

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1.3.1 Jacobian function:

From (1.2.26) and (1.2.27) we know

-2

3 4 -2 3 1 2 3 u u y        is a particular solution of differential equation ) (1-y )(1-k y ) du dy ( 2  2 2 2

We have the integral representation of y is

dt ) t k -)(1 t -(1 1 u 2 2 2 y(u) 0

 (1.3.1)

so we defined y = sn (u, k) or simply y = sn (u) ,when it is unnecessary to emphasize the modulus k Clearly, dt ) t k -)(1 t -(1 1 k) (x, sn 2 2 2 x 0 1

-

Jacobian functions defined as follow sn (u)=

2

3 4 2 2 3 1 3 u u       (1.3.2) cn (u)=

2

3 4 2 2 3 2 4 u u       (1.3.3) dn (u)=

2

3 4 3 2 3 3 4 u u       (1.3.4) From (1.2.24) (1.2.25)

We get the following integral equations

If dt ) t k k )( t -(1 1 u 2 2 2 2 1 y(u)  

then y(u)  cn (u, k)

and dt ) t k k )( t -(1 1 k) (x, cn 2 2 2 2 1 x 1 -  

(1.3.5) If dt ) k -)(t t -(1 1 u 2 2 2 1 y(u) 

then y(u) dn (u, k)

and dt ) k -)(t t -(1 1 k) (x, dn 2 2 2 1 x 1 - 

(1.3.6)

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This integrals (1.3.1), (1.3.5), (1.3.6) are called the elliptic integral of the first

kind.

Glaishers nation for quotients.

A short and convenient notation has been invented by Glaisher to express reciprocals and quotients of the Jacobian elliptic functions

ns (u) 1/sn (u) nc (u) 1/cn (u) nd ( u )1 / d n ( u ) (1.3.7) sc(u) = sn(u) /cn(u) sd(u) =sn(u) /dn(u) cd(u) =cn(u) /dn(u)

cs(u) = cn(u) /sn(u) ds(u) =dn(u) /sn(u) dc u=dn(u) / cn(u) (1.3.8)

We have the following results

                         u nd 1 2 1 -2 2 2 1 -2 u nc 1 2 1 -2 2 2 2 1 -2 u ns 2 1 -2 2 2 1 -2 1 u dc 2 1 -2 2 2 1 -2 1 u cd 2 1 -2 2 2 1 -2 u ds 2 1 -2 2 2 1 -2 2 u sd 0 2 1 -2 2 2 1 -2 2 u cs 2 1 -2 2 2 1 -2 u sc 0 2 1 -2 2 2 1 -2 dt. ) t k -(1 1) -(t dt ) k t k ( 1) -(t dt ) k -(t -1) (t dt t) k -(t 1) -(t dt ) t k -(1 ) t -(1 dt ) k (t ) k -(t dt ) t k (1 ) t k -(1 dt ) k (t 1) (t dt ) t k (1 ) t (1 u

1.3.2 Some relation between Jacobian function:

1. s n ( u ) c n ( u )d n ( u ) du d  (1.3. 9) 2. sn2(u) cn2(u) 1 (1.3.10) 3. k2sn2(u) dn2( u )1 (1.3.11) 4. c n ( 0 )d n ( 0 )1 (1.3.12)

Differentiate the equation sn2(u) cn2(u) 1 and use relation equation (1.3.9) we get equation (1.3.13)

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5. -sn (u) dn (u) du (u) cn d  (1.3.13)

From equation k2sn2(u) dn2(u) 1and equation (1.3.9) we have equation (1.3.14)

6. -k sn (u) cn (u) du (u) dn d 2 (1.3.14) Moreover (u)) sn -(u))(1 sn k -(1 (u) dn (u) cn (u) sn du d 2 2 2   (1.3.15)

cn (u) -sn (u) dn (u) (k -k cn (u))(1 -cn ( u ) ) du d 2 2 2 2    (1.3.16) (u)) dn -)(1 k -(u) (dn (u) cn (u) sn -k (u) dn du d 2 2 2 2 (1.3.17) And (u) dn (u) cn (u) sn du d

 , cn (u) -sn (u) dn (u) du d  , (1.3.18) (u) cn (u) sn m -(u) dn du d  (1.3.19) (u) ns (u) ds -(u) cs du d  ns (u) -cs(u) cs( u ) du d  (1.3.20) (u) ns (u) cs -(u) ds du d  (1.3.21) (u) nc (u) dc (u) sc du d  nc (u) sc (u) dc ( u ) du d  (1.3.22) (u) nc (u) sc m (u) dc du d 1  (1.3.23) (u) nd (u) sd (u) sd du d  cd (u) -m sd (u) nd ( u ) du d 1  (1.3.24) (u) cd (u) sd m (u) nd du d  (1.3.25)

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7. Relations between the Squares of the Jacobian Functions 1 (u) cn (u) sn2  2  dn2 (u) msn2( u )1 (1.3.26) 1 2 2 m (u) cn m -(u) dn  (1.3.27) 1 (u) cs -(u) ns2 2  2 2 1 m ( u ) cs -(u) ds  (1.3.28) m (u) ds -(u) ns2 2  (1.3.29) 1 (u) cd -(u) sd m 2 2  nd2 (u) -msd2( u )1 (1.3.30) 1 (u) nd m -(u) cd m 2 1 2  (1.3.31) 1 (u) sc -(u) nc2 2  dc2(u) -m1sc2 ( u )1 (1.3.32) m (u) nc m -(u) dc2 1 2  (1.3.33)

with the aid of these identities the square of any function can be expressed in terms of the square of any other. In particular

(u) ds m 1 (u) cs 1 1 (u) sn 2 2 2     (1.3.34) m -( u ) ds m (u) sc 1 1 (u) cn 2 1 2 2   (1.3.35) ( u ) cd m -1 m (u) sd m 1 1 (u) dn 2 1 2 2   (1.3.36)

1.3.3 Some properties of Jacobian functions:

1. sn(u) is an odd function of u cn(u) is an even function of u dn(u) is an even function of u

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Double and Half Arguments ( u ) ( u ) d n sn (u) cn (u) dn (u) 2sn(u)cn (u) sn k -1 dn(u) cn(u) (u) sn 2 (2u) sn 2 2 2 4 2   (1.3.37) ( u ) ( u ) d n sn (u) cn (u) dn (u) sn -(u) cn (u) sn k -1 (u) dn (u) sn -(u) cn (2u) cn 2 2 2 2 2 2 4 2 2 2 2    (1.3.38) 1 ) -( u ) ( u ) ( d n cn -(u) dn 1) -(u) (u)(dn cn (u) dn (u) sn k -1 (u) cn (u) sn k -(u) dn (2u) dn 2 2 2 2 2 2 4 2 2 2 2 2   (1.3.39) ( u ) cn (u) dn (u) sn (2u) cn 1 (2u) cn -1 2 2 2   (1.3.40) ( u ) dn (u) cn (u) sn k (2u) dn 1 (2u) dn -1 2 2 2 2   (1.3.41) ( 2 u ) d n 1 ( 2 u ) c n -1 u) 2 1 ( sn2   (1.3.42) (2u) dn 1 (2u) cn (u) dn u) 2 1 ( cn2    (1.3.43) ( u ) d n 1 ( u ) c n k ( u ) d n k u) 2 1 ( dn 2 2 2      (1.3.44)

3. The addition-theorem for Jacobian function . ( v ) sn (u) sn k -1 (u) dn (u) cn (v) sn (v) dn (v) cn (u) sn v) sn(u 2 2 2    (1.3.45) ( v ) sn (u) sn k -1 (v) sn (u) sn -(v) cn (u) cn v) cn(u 2 2 2   (1.3.46) ( v ) sn (u) sn k -1 (v) cn (u) cn (v) sn (u) sn k -(v) dn (u) dn v) dn(u 2 2 2 2   (1.3.47) (v) (u)sn sn k -1 (v) sn -(u) sn v) -(u sn v) (u sn 2 2 2 2 2   (1.3.48) (v) (u)sn sn k -1 (u) dn (v) cn sn(v) (v) dn (u) cn (u) sn v) -(u cn v) (u sn   2 22 (1.3.49) (v) (u)sn sn k -1 (u) cn (v) dn sn(v) (v) cn (u) dn (u) sn v) -(u dn v) (u sn   2 22 (1.3.50) (v) (u)sn sn k -1 (u) dn (v) sn -(u) cn v) -(u cn v) (u cn 2 2 2 2 2 2   (1.3.51)

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(v) (u)sn sn k -1 (v) sn sn(u) k (v) cn(v)dn (u) dn (u) cn v) -(u dn v) (u cn 2 2 2 2     (1.3.52) (v) (u)sn sn k -1 (v) sn (u) cn k -(u) dn v) -(u dn v) (u dn 2 2 2 2 2 2 2   (1.3.53) 4. The constant K , K

a. Symbol K is a function of k such that sn (K, k) = 1 In other words, 

1 0 2 1 -2 2 2 1 -2 d t ) t k -(1 ) t -(1 (k) K (1.3.54) and sn K = 1, cn K = 0 , dn K = k b. Symbol Kis a function of k   

1  0 2 1 -2 2 2 1 -2 dt ) t k -(1 ) t -(1 ) k ( K (1.3.55) Remark5: 1. K (m)  K(1-m)  K(m) 2.  2 1 K(0)  K( 0 ) 

3. Another form of K and K

2 0 2 1 -2 2 d )) ( sin k -(1 (k) K    (1.3.56)  

2  0 2 1 -2 2 d )) ( sin k -(1 (k) K    (1.3.57)

5. The periodic properties of the Jacobian elliptic functions a. associated with K ( u ) d n 2 K ) (u dn (u) cn 4K) (u cn (u) sn 4K) (u sn       (1.3.58)

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b. associated with K iK ( u ) d n ) K 4i 4K dn(u (u) -dn ) K 2i 2K dn(u (u) cn ) K 4i 4K cn(u (u) cn ) K 2i 2K cn(u (u) sn ) K 4i 4K sn(u (u) sn -) K 2i 2K sn(u                         (1.3.59) c. associated with iK ( u ) d n ) K 4i (u dn (u) -dn ) K 2i (u dn (u) cn ) K 4i (u cn (u) -cn ) K 2i (u cn (u) sn ) K 4i (u sn (u) sn ) K 2i (u sn                   (1.3.60) Table 1

Special Values of Argument

u sn (u) cn (u) dn (u)

0 0 1 1 K 2 1 2 1 2 1 1 ) m (1 1  2 1 2 1 1 4 1 ) m (1 m  4 1 1 m K 1 0 2 1 1 m 2K 0 -1 1 K i 2 1  4 1 -im 4 1 2 1 2 1 m ) m (1 2 1 2 1 ) m (1 K i     K 2i  0 -1 -1 K i K   2 1 -m 2 1 1/m) i(m -0 K 2i 2K   0 1 -1

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Remark6:

For any Jacobian elliptic functions pq (u) pq (u4K ,k ) pq (u 4iK,k) pq (u, k)

Table 2 Periods

sn (u) cn (u) dn (u) Zeros 0, 2K K, 3K K iK,K 3iK Poles iK,2K iK iK,2K iK iK,3iK Periods 4K, 2iK 4K, 2K 2iK 2K, 4iK Table 3 Zeros, Poles and Periods

6. Jacobi´s imaginary transformation

sn (iu , k)= i sc (u, k) cn (iu , k) = nc (u, k) dn (iu , k) = dc (u , k) (1.3.61)

If z xiy, the addition theorems the give with k) , (x sn s1  , s2 sn (y ,k), k) , (x cn c1  , c2  cn (y ,k), k) , (x dn d1  , d2  dn (y ,k) 2 2 2 1 2 2 2 2 2 1 1 2 1 s s k c c s d c i d s k) , sn(z    (1.3.62) 2 2 2 1 2 2 2 2 1 1 1 2 1 s s k c d s d s i -c c k) , cn(z   (1.3.63) 2 2 2 1 2 2 2 2 1 1 2 2 2 1 s s k c s c s k i -d c d k) , dn(z   (1.3.64) Periods K 4i ,

2K  cs (u) sc (u) dn (u) nd (u) K

2i ,

4K  ns (u) dc (u) sn (u) cd (u) K

2i 2K ,

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Argument sn cn dn

u sn (u) cn (u) dn (u)

-u -sn (u) cn (u) dn (u)

K

u  cd (u) -ksd (u) knd (u)

K

-u - cd (u) ksd (u) knd (u)

u

-K cd (u) ksd (u) knd (u)

2K

u  - sn (u) - cn (u) dn (u)

2K

-u - sn (u) - cn (u) dn (u)

u

-2K sn (u) - cn (u) dn (u)

K i u   k-1 ns (u) (u) ds k i - -1 -ics(u) K 2i

u   sn (u) - cn (u) - dn (u)

K i K u    k-1 dc (u) (u) nc k k i -  -1 iksc (u) K 2i 2K

u    - sn (u) cn (u) - dn (u)

Table 4 Chang of Argument

1.3.4 The Jacobian elliptic function and the trigonometric

functions:

1. Observation:

Recall the integral representation of sn (u) in (1.3.1),

 sn (u) 0 2 2 2 dt ) t k -)(1 t -(1 1 u (1.3.65) a. When 0 k  , (1.3.65) becomes

 sin (u) 0 2 x -1 dt u (1.3.66) That is,

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s n ( u )degenerate to sin (u) as k  0. b. When 1 k  , (1.3.65) becomes   d -1 1 u (u) tanh 0 2

 (1.3.67) That is, (u)

sn degenerate to tanh (u) as k 1. Similarly,

(u)

cn degenerate to cos (u) as k  0, d n ( u ) degenerate to 1 as k  0,

and

(u)

cn degenerate to sech (u) as k 1, d n ( u ) degenerate to sech (u) as k 1, 2. Exact:

Changing the variable by t sin (), the integral (1.3.65) is reduced to

Legendre’s form,    ,k) sn (sin ,k) (1-k sin ) d F( 2 1 -0 2 2 1

-

  (1.3.68) Then, expanding the integrand in ascending powers of 2

k and integrating term by term, we find that

... ) cos sin -( k 4 1 k) , (sin sn-1    2     , (1.3.69) which is equivalent to ... ) x -1 x -(x) (sin k 4 1 (x) sin k) (x, sn u  -1  -1  2 -1 2  . (1.3.70) where x  sin

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This series can now be inverted to expand sn (u) in powers of 2 k , thus:

k (x (1-x )-sin (x)) O(k )] 4 1 sin[u x k) (u, sn    2 2 -1  4

k (x (1-x )-sin (x))cos(u) O(k ) 4

1

sin(u)  2 2 -1  4

k (sin(u)cos (u) -u)cos(u) O(k ) 4

1

sin(u)  2  4

 (1.3.71)

Moreover,

1. When the parameter 2

k

m  is small that its square may be neglected, the

following approximations may be used to calculate the elliptic functions in terms of circular functions. m c o s ( u ) ( u-s i n ( u ) c o s (u ) ) 4 1 -( u ) s i n ) m ( u s n  , (1.3.72) ms i n ( u ) ( u-s i n ( u ) c o s (u ) ) 4 1 ( u ) c o s ) m ( u c n   , (1.3.73) ms i n ( u ) 2 1 -1 ) m ( u d n  2 , (1.3.74)

2. When the parameter 2

k

m  is so near unity that the square of the complementary

parameter 2

1 1-m 1-k

m   may be neglected, the following approximations may be used to calculate the elliptic functions in terms of hyperbolic functions.

m sech (u)(sinh(u )cosh(u) -u) 4 1 (u) tanh ) m (u sn   1 2 , (1.3.75)

m tanh (u)sech (u)(sinh(u )cosh(u) -u) 4 1 -(u) sech ) m (u cn  1 , (1.3.76)

m tanh (u)sech (u)(sinh(u )cosh(u) u) 4 1 (u) sech ) m (u dn   1  , (1.3.77)

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1.3.5 General form of the elliptic integral of the first kind:

1. integral of the Jacodian function sn (u)

By changing the variable in the integral (1.3.1) using the substitution t s b , we calculate that 1) x (0 dt )} t k -)(1 t -{(1 u k) (x, sn x 0 2 1 -2 2 2 1 -

a {(a -s )(b -s )} ds bx 0 2 1 -2 2 2 2

 (1.3.78) where 0 b  b k a It now follows that

dt )} t -)(b t -{(a 1 ) a b , b x ( sn a 1 x 0 2 2 2 2 1

-

 (1.3.79) where 0 x  b  a

2. integral of the Jacodian function cn (u) In the same way

By changing the variable in the integral (1.3.5) using the substitution, b

s

t  ,k b (a2 b2)

After some manipulation, we arrive at the formula

b) x (0 dt )} t -)(b t {(a 1 ] ) b (a b , b x [ cn ) b (a 1 b x 2 2 2 2 2 2 1 -2 2      

(1.3.80)

3. More results below:

a, b x 0 , dt ) t -)(b t -(a 1 a ] a b , b x [ cd b x 2 2 2 2 1 -   

(1.3.81) b, x 0 , dt 1 ) b (a ] ) b (a b , ab x ) b (a [ sd x 0 2 2 2 2 2 2 2 2 2 2 1 -      

(1.3.82)

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x, a b , dt ) b -)(t a -(t 1 a ] a b , b x [ dc x a 2 2 2 2 1 -  

(1.3.83) x, a b , dt ) b -)(t a -(t 1 a ] a b , a x [ ns x 2 2 2 2 1 -  

 (1.3.84) a, x b , dt ) b -)(t t -(a 1 a ] a ) b -(a , b x [ nd x b 2 2 2 2 2 2 1 -  

(1.3.85) a, x b , dt ) b -)(t t -(a 1 a ] a ) b -(a , a x [ dn a x 2 2 2 2 2 2 1 -  

(1.3.86) x, a , dt ) b )(t a -(t 1 ] ) b (a b , a x [ nc x a 2 2 2 2 2 2 1 -   

(1.3.87) x, a , dt ) b )(t a -(t 1 ) b (a ] ) b (a b , ) b (a x [ ds x 2 2 2 2 2 2 2 2 2 2 1 -     

 (1.3.88)

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, x, 0 a, b 0 , dt ) b )(t a (t 1 a ] a ) b -(a , b x [ sc x 0 2 2 2 2 2 2 1 -     

(1.3.89) x, 0 a, b 0 , dt ) b )(t a (t 1 a ] a ) b -(a , a x [ cs x 2 2 2 2 2 2 1 -     

 (1.3.90)

1.3.6 Some graphs of Jacobian functions:

1. Jacobian function sn u a. sn (u, k)

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b. sn(u,1) 15  10  5 5 10 15 u 1.0 0.5 0.5 1.0 sn Figure 1.2 c. ) 3 2 sn(u,  15  10  5 5 10 15 u  1.0  0.5 0.5 1.0 sn Figure 1.3 d. ) 2 1 sn(u,  15  10 5 5 10 15 u  1.0  0.5 0.5 1.0 sn Figure 1.4

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Remark 7: .

sn(u) is an odd periodic function of u for u is real. Moreover the period is larger when k is larger.

We only consider that the Jacobian function sn u is defined in real number when we sketch the graphs above. However, the Jacobian function sn (u) is a function from complex number to complex number. Because this function is from two dimension to two dimension, it means that we have to analyze it in a four dimension space. It is difficult for us to do this.

Therefore, we use the method below to analyze the Jacobian function sn (u) defined in complex number.

First, we define two new functions Re(u), Im(u). Re(u) is a function that we take the real part of the Jacobian function sn u and Im(u) is another function we take the

imaginary part of the Jacobian function sn (u). That is Re(u) = Re{sn(u)} and Im(u)=Im{sn(u)}. Example : 1. ) 0.660252 -0.203738 i 2 1 , 4i (3 sn   R e ( 34 i ) 0 . 6 6 0 2 5 2 I m ( 34 i ) 0 . 2 0 3 7 3 8 2. sn(6 -2i) -1.47598 +0.0512946 i Re(6 -2i)  -1.47598 Im(6 -2i)  0.0512946

Second, we can use two three-dimensional figures of Re(u) and Im(u) to represent the behavior of the Jacobian function sn (u). It is obvious that the Jacobian function sn (u) is a doubly-periodic function of u and there is a smallest unit parallelogram that

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can be repeated to form the entire graph.

In the end, we want to examine whether the graphs of Re(u) and Im(u) are right when the domain of the Jacobian function sn (u) is restricted in real number. It is coincident that the range of the Jacobian function sn (u) defined in real umber is also real.

Now, in order to observe the graph or Re(u) clearly, we take the value of x-axis ( real part of u ) from -5 to 5 and the value of y-axis ( imaginary part of u ) from 0 to 5. It is easy to discover that the intersection of the plane y = 0 and the graph of Re(u) is the figure of the Jacobian function sn u defined in real number.

In comparison, we find that the three-dimensional figure of the intersection and the two-dimensional graph of the Jacobian function sn (u) defined in real number are the same. In the other hand, the value of Im(u) is zero in the three-dimensional figure when y = 0.

We can use the same way to observe the Jacobian function cn(u) and the Jacobian function dn(u) .

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Figure 1.5

The figure Im(u) represents the imaginary part of Jacobian function ) 2 1

sn(u, for

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Figure 1.6

The figure Re(u) represent the real part of Jacobian function ) 2 1

sn(u, for complex

number u .

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Figure 1.8

The graph of Re(u) for x-axis from -5 to 5 and y-axis from 0 to 5

.  4 2 2 4 u  1.0  0.5 0.5 1.0 sn Figure 1.11 The jacobian function sn(u).

Figure 1.9

We can see the intersection line from this direction.

Figure 1.10

The imaginary part of function sn(u).

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2. Jacobian function cn(u) a. cn(u, k) Figure 1.12 b. cn(u,1)  4  2 0 2 4 u 0.2 0.4 0.6 0.8 1.0 cn Figure 1.13 c. ) 3 2 cn(u,  15  10  5 5 10 15 u  1.0  0.5 0.5 1.0 cn Figure 1.14

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d. ) 2 1 cn(u,  15  10  5 5 10 15 u  1.0  0.5 0.5 1.0 cn Figure 1.15 Remark 8:

c n ( u )is an even periodic function of u for u is real. Moreover the period is larger

when k is larger.

Figure 1.16

The imaginary part of the Jacobian function       2 1 u, cn

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Figure 1.17

The real part of the Jacobian function       2 1 u, cn Figure 1.18

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Figure 1.19 4 2 2 4 u  1.0  0.5 0.5 1.0 cn u Figure 1.22

The jacobian function cn(u)

The real part of the Jacobian function       2 1 u, cn for

x-axis from -5 to 5 and y-axis from 0 to 5

Figure 1.20

We can see the intersection line from this direction.

Figure 1.21

The imaginary part of function cn(u).

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3. Jacobian function dn u a.dn(u, k) Figure 1.23 b.dn(u,1)  4  2 0 2 4 u 0.2 0.4 0.6 0.8 1.0 dn Figure 1.24 Remark 9

:

d n ( u , 1 ) c n ( u , 1 ) By definition dt ) k -)(t t -(1 1 V dt ) t k k )( t -(1 1 u 2 2 2 1 k) dn(v, 2 2 2 2 1 k) cn(u,     

k2k2 1

For k=1 then k0 and

) 0 -)(t t -(1 1 dt )t t -(1 1 dt ) t 1 )(0 t -(1 1 u 2 2 2 1 dn(u,1) 2 2 1 cn(u,1) 2 2 2 2 1 cn(u,1)

    dt

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c. ) 3 2 dn(u,  15  10  5 5 10 15 u 0.7 0.8 0.9 1.0 dn Figure 1.25 d. ) 2 1 dn(u,  15  10  5 5 10 15 u 0.80 0.85 0.90 0.95 1.00 dn Figure 1.26 Remark 10:

d n ( u )is an even periodic function of u for u is real. Moreover the period is larger

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Figure 1.27

The imaginary part of the Jacobian function       2 1 u, dn Figure 1.28

The real part of the Jacobian function       2 1 u, dn

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Figure 1.29

Figure 1.30

The real part of the Jacobian function       2 1 u, dn for

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4 2 2 4 u 1.00 dn Figure 1.33

The jacobian function dn(u)

4. Jacobian function sc u

Figure 1.34

Figure 1.32

The imaginary part of dn(u).

Figure 1.31

We can see the intersection line from this direction.

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Chapter 2 Linearzied , Nullclines , Hamiltonian

system and Dissipative system

In this chapter, according to the reference [1], we analyze phase portraits by some ways.

2.1 Linearzied:

1. linearized system

For a general form of a nonlinear system

x, y f dt dx  x, y g dt dy 

The linearized system at the equilibrium point

x0,y0

is

 

 

                                      v u y x y g y x x g y x y f y x x f dt dv dt du 0 0, 0 0, 0 0, 0 0, Where u x-x0 v y-y0 And J=

 

 

                   y x y g y x x g y x y f y x x f 0 0, 0 0, 0 0, 0 0,

is called the Jacobian matrix of the system at

x0,y0

2. Classerify equilibrium points by linearized system

a. If all eigenvalues of J are negative real numbers than

x0,y0

is sink.

If all eigenvalues of J are complex numbers with negative real parts than

x0,y0

is spiral sink

b. If all eigenvalues of J are positive real numbers than

x0,y0

is source

If all eigenvalues of J are complex numbers with positive real parts than

x0,y0

is spiral source

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2.2 Nullclines:

1. x-nullcline y-nullcline For the system

f(x, y) dt dx  g(x, y) dt dy 

The x-nullcline is the set of points (x,y) wheref(x, y)is zero that is, the level curve

where f(x, y)is zero. The y-nullcline is the set of points whereg(x, y) is zero.

Example:

2. Some properties of nullclines

a. Along the x-nullcline, the x-component of the vector field is zero, and consequently the vector field is vertical.

Along the y-nullcline, the y-component of the vector field is zero, and consequently the vector field is horizontal.

b. The intersections of the nullclines are the equilibrium points

c. The regions separated by nullclines offer information of vector field

2.3 Hamiltonian system:

1. Conserved quantity

A real-valued function H(x,y) of the two variables x and y is a conserved quantity for a system of differential equations if it is constant along all solution curves of

system. That is, if (x(t),y(t)) is a solution of the system, then H(x(t),y(t))is constant. In other words,

H/( x ( t ) ,y ( t ) )0 for (x(t),y(t)) is a solution of the system

2. Hamiltonian system

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real-valued function H(x,y) such that x H -dt dy y H dt dx      

for all x and y. The function H is called the Hamiltonian function for the system.

3. Relation between conserved quantity and Hamiltonian system Letting (x(t),y(t)) be any solution of the system then

0 ) d x d H )(-y H ( ) dy dH )( x H ( ) dt dy )( y H ( ) dt dx )( x H ( y(t)) H(x(t), dt d             

So Hamiltonian system is conserved.

4. Equilibrium points of Hamiltonian system

Suppose

x0,y0

is our equilibrium point for the Hamiltonian system

x H -dt dy y H dt dx      

The Jacobian matrix at this equilibrium point is given by

                      x y H -x H -y H y x H 2 2 2 2 2 2

where each of these partial derivatives is evaluated at

x0,y0

. Since x y H y x H 2 2       

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          -where 2 2 2 2 2 x H y H y x H             

The characteristic polynomial of this matrix is  -- -- 2-2-

And the eigenvalues are    2 

.Thus we see that there are only three possibilities for the eigenvalues:

a. If 2  0 both eigenvalues are real and have opposite signs.

b. If 2  0 both eigenvalues are imaginary with real part equal to zero.

c. If 2  0 then 0 is the only eigenvalue

In case a. we know the equilibrium point must be a saddle

5. Solution of Hamiltonian system

Solution curves of the system lie along the level curves of H . Sketching the phase portrait for Hamiltonian system is the same as sketching the level sets of the

Hamiltonian function.

2.4 Dissipative system:

A function L(x,y) is called a Lyapunov function for a system of differential equation if, for every solution (x(t),y(t)) that is not an equilibrium solution of the system ,

L ( x ( t ) ,y ( t ) ) 0 dt

d

For all t with strict inequality except for set of t‟s.

2.5 Discussion:

There are four methods to analyze the solution of nonlinear systems. By linearized we have some information near equilibrium points. By nullclines we get the trend in the whole phrase plane. In the special case we have the properties of Hamiltonian system and Dissipative system.

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Chapter 3 Pendulum

We study the motions of Pendulum in the following paragraphs. The references for this part are [1, 8, 10].

3.0.1 Physic aspect:

Consider a pendulum made of a light rod of length l with a ball at one end of mass

m .The position of the bob at time t is given by an angle U(t),which we choose to

measure in the counterclockwise direction with 0 corresponding to the downward vertical axis (see Figure2.1)

Figure3.1

A pendulum with rod length l and angle θ

The speed of the bob is the length of the velocity vector , which is l U (t) . The

component of the acceleration that points along the direction of the motion of the bob is

(t) U

l  We take the force due to friction to be proportional to the velocity , so this force is

(t) U bl

-  where b>0 is a parameter that corresponds to the coefficient of damping .

Using Newton‟s second law,F  ma we obtain the equation of motion

-blU(t) -mg sinU(t) mlU( t )

Which is often written as

s i n U ( t )0 l g ( t ) U m b ( t ) U     (3.0.1)

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Figure 3.2

3.0.2 Mathematic aspect:

Consider the Sine-Gordon equation is

Uxx (x, y)-Uyy(x, y) s i n [ U ( x ,y ) ] 0 (s-G) (3.0.2) Let tx-y Then U( t ) x t ( t ) U y) (x, Ux      - U( t ) y t ( t ) U y) (x, Uy       Uxx (x, y) 2U( t ) Uyy(x, y) 2U( t )

So we can rewrite equation (3.0.2) 2U( t )-2U( t )s i n [ U ( t ) ]0

Let h2 -w2 1

We get U(t) sin[U(t)] 0 (3.0.3) Compare with equation (3.0.1). This is a equation describing ideal pendulum

Multiple U(t)

U(t)U(t) U(t)sin[U(t )]0 (3.0.4) Integrated by t

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2 2 E c o s U ( t ) -( t ) U 2 1   where E -1 2  is a constant Add 1 2 2 1 E E 1 c o s U ( t ) ] -[1 (t) U 2 1      (3.0.5) We can see U (t) 2 1 2

as kinetic energy and [1-cosU(t)] as potential energy and this system has total energy E1

   2  2 u 0.5 1.0 1.5 2.0 P .E Figure 3.3

The relation between potential energy P.E and angle u

From (3.0.5) 2 1 E c o s U ( t ) ] -[1 (t) U 2 1    U(t)  2E1-2 [ 1-c o s U ( t ) ] (3.0.6) 1 c o s U ( t ) ] -2 [ 1 -2E (t) U 1   Integrated We get 

U ( t ) 0 1 d ] c o s -2 [ 1 -2E 1 t   (3.0.7)

3.1.0 Ideal Pendulum:

A system of pendulum with no friction is called ideal pendulum.

When no friction is present, the coefficient b vanish. We get the equation 0 ) ( sin g (t) U  U t    

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For convenience we suppose g 1 

We can rewrite this equation U(t) sin U(t) 0 as first-order system in the usual manner by letting the variable v represent the angular velocity U(t) .The corresponding

system is V dt dU   dt dV - sinU

Equilibrium points of this system are (nπ,0) for nZ

3.1.1 Apply Linearzied to analysis Ideal pendulum:

For system V dt dU   dt dV - sinU

1.The linearized system at the equilibrium point n,0 for n is odd integer

                         v u 0 1 1 0 dt dv dt du

The Jacobian matrix of the system is J =       0 1 1 0

and its eigenvalues are 1

There are saddle points at the equilibrium point n,0 for n is odd integer 2. The linearized system at the equilibrium point n,0 for n is even integer

                         v u 0 1 -1 0 dt dv dt du

The Jacobian matrix of the system is J =       0 1 -1 0

and its eigenvalues are 1

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3.1.2 Apply the Hamiltonian system to analysis Ideal

pendulum:

Consider U(t)  sin[U(t)]  0

Let H(U,V)= -cosU 2 1

v

2 Because V H V dt dU     U H --cosU dt dV    

it is Hamilonian system with Hamiltonian function H(U,V)= -cosU 2

1

v

2

Figure 3.4

Level curve of H(U,V)

3.1.3 Apply the Jacobian elliptic function to solve the Ideal

pendulum motion:

We want to solve (3.0.5) by Jacobian elliptic functions

Consider 

U(t) 0 1 d ] cos -2[1 -2E 1 t   with 1 E2  E1 i.e.

  U(t) 0 2 d 2cos 2E 1 t   (3.1.1)

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a. If 0  E1  2 , i.e. -1 E2 1

then there is  such that -cos  E2

  U ( t ) 0 d 2 c o s 2 c o s -1 t        d 2 2 s i n -1 2 2 2 s i n -1 2 -1 2 2 U ( t ) 0

             = 2 1    d 2 sin -2 sin 1 2 2 U(t) 0

Let 0< k = 2 sin  <1 z = k 2 sin  = dz z k -1 2k z k -k 1 2 1 2 2 2 2 2 k 2 u(t) sin 0

= dz ) z k -)(1 z -(1 1 2 2 2 k 2 u(t) sin 0

According to Jacobian function 2 U ( t ) s i n k 1 k) sn(t, 

So U(t) 2sin -1(k sn(t, k)) where k= 2 sin  (3.1.2) b. If E1  2 , i.e. E2 1

  U(t) 0 d 2cos 2 1 t  

  U ( t ) 0 d c o s 1 1 2 1   

 U ( t ) 0 d c o s 1 1 2 1  

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=

U(t) 0 2 d 2 2sin -2 1 2 1   =

U(t) 0 2 d 2 sin -1 1 2 1   Let 2 sin x   =

2 U(t) sin 0 2 2 dx x -1 2 x -1 1 2 1 =

2 U(t) sin 0 1-x2 dx 1 = 2 U(t) sin -1 2 U(t) sin 1 ln 2 1  (3.1.3) So tanh(t) 2 U(t)

sin  i.e U(t) 2sin -1tanh(t) (3.1.4)

c. If E1  2 i.e. E2 1

  U ( t ) 0 2 d 2 c o s 2E 1 t  

  U ( t ) 0 2 2 d ) 2 4 s i n -(2 2E 1  

   U ( t ) 0 2 2 2 d ) 2 s i n 2 2E 4 -1 2 2E 1   Compare with

dx sin x k -1 1 2 Let 2 x 2 2E 2 k 2    

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2 U ( t ) 0 2 2 dx x sin k -1 k Let ysin x 

2 U(t) sin 0 2 2 2 dy y -1 1 y k -1 1 k So 2 U ( t ) s i n k) , k t sn(  i.e. ,k) k t sn( sin 2 1 U(t)  -1 where 2 2E 2 k 2   (3.1.5)

3.1.4 The graph of the Ideal pendulum motion:

1.    2  2 u 0.5 1.0 1.5 2.0 2.5 3.0 P .E Figure 3.5

The relation between potential energy P.E and angle u with total energy E1 3

   2  2 u  2  1 1 2 v Figure 3.6

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 2    2 u 0.5 1.0 1.5 2.0 P .E Figure 3.7

The relation between potential energy P.E and angle u with total energyE1 2

   2  2 u  2  1 1 2 v Figure 3.8

The relation between vector v and angle u with E1  2

   2  2 u 0.5 1.0 1.5 2.0 P .E Figure 3.9

The relation between potential energy P.E and angle u with total energy E1  0.5

   2  2 u  1.0  0.5 0.5 1.0 v Figure 3.10

The relation between vector v and angle u with E1 0.5

2. From (3.0.2)

U( t )s i n [ U ( t ) ]0 with t  hx  wy and h2 -w2 1 Let h 2 w  3

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a. Graph of the ideal pendulum motion with E1 1 (0  E1  2)

Figure 3.11

b . Graph of the ideal pendulum motion with E1 2

Figure 3.12

c .Graph of the ideal pendulum motion with E1  3 (2  E1)

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3.2.0 Pendulum motion with friction:

Recall that the second-order equation governing the motion of the pendulum is 0 sin[U(t)] l g (t) U m b (t) U    

Where b is the coefficient of damping m is the mass of the pendulum bob, g is the acceleration of gravity, and l is the length of the pendulum arm.

For convenies we let B m

b

 and 1 l g

 .And rewrite this equation as first-order system

in the usual manner by letting the variable v represent the angular velocity U(). The corresponding system is

V dt dU   dt dV -BV- sinU

3.2.1 Apply Linearzied to analysis pendulum with friction:

For system V dt dU  -BV- sinU

1. The linearized system at the equilibrium point n,0 for n is odd integer

                         v u B 1 1 0 dt dv dt du

The Jacobian matrix of the system is J =       B 1 1 0

and its eigenvalues are

2 4 B B -  2   

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2. The linearized system at the equilibrium point n,0 for n is even integer                          v u B 1 -1 0 dt dv dt du

The Jacobian matrix of the system is J =       B 1 -1 0

and its eigenvalues are 1

There are center points at the equilibrium point n,0 for n is even integer.

3.2.2 Apply the nullclines to analysis pendulum with friction:

For system V dt dU   dt dV -BV- sinU U-nullcline is U,V;V  0 If V>0 then 0 dt dU

 This means vector filed “right”

If V<0 then 0

dt dU

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V-nullcline is U, V;BV sinU 0

If BV sinU 0 then 0

dt dV

 This means vector filed “up”

If BV sinU  0 then 0

dt dV

 This means vector filed “down”

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It obvious

1. There are saddle points at the equilibrium point n,0 for n is odd integer.

2. There are center or spiral sink or spiral source points at the equilibrium point n,0 for n is even integer.

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Chapter 4 Physical Applications of Elliptic functions

In the paragraphs below, we study five physical models‟ differential systems. An ideal whirling chain, and Duffing‟s Equation. The other three describe the motions of orbit planets. These parts mainly follow the reference [4].

4.1 Whirling Chain:

Consider a uniform length l of rope or chain, whose ends are fixed at point 0 and A and which is set rotating about the axis OA with constant angular velocity .

For ideal case

1. Gravity will be neglected

2. It will be assumed that the chain always lies in a plane through the axis of rotation. We shall take O to be the origin of axes Ox, Oy, the x-axis lying along OA, and y-axis lying in the plane of the chain at same instant t (Fig.). Consider the motion of an element ds  PQ of the chain, where P and Q have coordinates (x, y), (x+dx, y+dy) respectively.

The forces acting on this element are tensions T and T+dT at its ends P and Q, and their lines of action are the tangents to the chain at these points; let these tangents make angles ,  d respectively with the x-axis.

Figure 4.1

Resolving the forces tangentially and normally, we obtain components (dT, Td ) respectively. The element moves around a circle of radius y with angular velocity  and its acceleration is accordingly 2y directed in the negative sense parallel to the y-axis.

We can now write down the tangential and normal components of the equation of motion thus; ) sin( y ds -dT   2  , Td - ds2y cos() (4.1.1) Where is the mass per unit length of the chain.

數據

Table 4  Chang of Argument
Figure 1.19        4  2 2 4 u  1.00.50.51.0 cn u             Figure 1.22
Figure 3.4  Level curve of H(U,V)

參考文獻

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