The Two Variable Function Constitutes a Necessary Condition of Chaotic Dynamical Systems

(1)

2017 International Conference on Mathematics, Modelling and Simulation Technologies and Applications (MMSTA 2017) ISBN: 978-1-60595-530-8

The Two Variable Function Constitutes a Necessary Condition of

Chaotic Dynamical Systems

Wan-bo YU

*

_{and Xiang-xiang WANG}

School of Information, Dalian University, Liaoning Dalian, 116622, China *Corresponding author

Keywords: Chaos, Dynamic system, Two variable function.

Abstract. This article works on the chaotic characteristic of a dynamic system which consists of two

variable function. It derives two necessary conditions for this chaotic system based on differential Taylor expansion. Which confirmed the sum of absolute value of the partial derivative on the two variable function in the periodic point of x,y is not greater than 1,and further presents the necessary conditions for product chaos which is described by recursive determinant. The recursive determinant calculation is simple, and it is suitable for a scale to measure as chaotic characteristics.

Introduction

Research on chaos, Lorenz (1963) discovered chaotic attractor [1],Li and Yorke (1975) give the discussion that cycle three means chaos [2], Devaney, based on chaotic traversal, gives a definition of chaos [3] and so on. These are groundbreaking research work. Chaos control based on Ott, Grebogi and Yorke (1990) [4], Pecora and Carroll (1990) put forward the concept and method of chaotic synchronization [5], These efforts have yielded important phased results. In the research process of chaos, A lot of chaos has come up. Oddly, these definitions are not consistent. Therefore, it is necessary to continue to excavate the mathematical mechanism and physical essence of chaos, and to search for a unified definition of chaos, In the research and practice of Physical Science, The Lyapunov index is often used as a sign of chaos [6], The Lyapunov index is defined by the multiplicative derivative, the index calculation is simple and can reflect the characteristics of chaotic systems in a certain extent. Although the Lyapunov index has played an important role in the research field of chaos, but the judgment of chaos also has some problems. For example the Lyapunov index greater than 0, but the system is not usually the sense of chaos; some systems satisfy some definitions of chaos, but the Lyapunov index is not greater than 0. Therefore, it is necessary to find some strict, accurate and simple methods and criteria for the determination of chaos. The research work of this paper is in this context. In previous study, using Taylor expansion research for one variable function, It is found that the necessary condition of the chaos of the one variable function is that the absolute value of the product of the absolute value of the derivative at the periodic point cannot be less than 1, and the author extends the conclusion to the dynamic system composed of two variables.

A dynamic system consisting of a simple function of two variables, such as equation (1),

1

2

( , ) ( , )

z f x y

z g x y

   

 ₍₁₎ Type (1) iteration once changed to

  

 

)) , ( ), , ( (

2 1

y x g y x f g z

y x g y x f f z

(2)

document [7-8], the document[9] found that the image function and sine function constitute the power system iteration can be obtained after the chaos attractor, and the probability of getting the attractor is very large, these attractors can as image feature for image recognition. Some images are flat and cannot produce chaotic attractors, After adjusting the gray image can also produce chaotic attractor. However, there are still a few images can not appear chaotic, looking for reasons. In view of these, the chaotic iterative generation mechanism is studied, and the generation conditions of chaotic attractors are very necessary and urgent in the present research.

The chaos in this article needs to satisfy the ergodicity in the Devaney chaos definition, That is, any two open sets in the domain U V, _{, there is a natural number}k_{, so} fk(U)



V _.

The Sum of Partial Derivatives at the Periodic Points of Two Variables Determines Its Chaotic Characteristics

It is found that the derivatives of chaotic functions at periodic points must satisfy certain conditions, as described in conclusion 1-3 below.

Conclusion 1 One power system is shown in equation (1) if the two function f(x,y) and g(x,y) define the domain as

   

0,1 0,1 , range is

 

0,1, Partial derivatives of x and y exist in the domain of definition. f(x,y) and g(x,y) constitute dynamical systems, If it's chaotic, Then the sum of partial derivatives of x and y is fixed at its fixed point (x0,y0) _, | fx(x0,y0)|| fy(x0,y0)| and

| ) , ( | | ) , (

|g_x x₀ y₀  g_y x₀ y₀ _{cannot be less than 1 at the same time.}

Let ( , )x y0 0 _{be a periodic point (fixed point) of equation (1),That is} f x y( , )0 0 x g x y0, ( , )0 0 y0, If at

0 0

( , )x y _, _the _partial _derivatives _of _x _and _y _are _obtained

1 | ) , ( | | ) , (

| f_x x₀ y₀  f_y x₀ y₀  _,|g_x(x₀,y₀)||g_y(x₀,y₀)|1_{, that, There must be a rectangular o-pen}

neighborhood u _with (x0,y0) _as _the _center, _For _all _the (x,y) _in _this

neighborhood, mxx0m, nyy0n ,Meet fx(x,y)  fy(x,y) 1 _and 1

| ) , ( | | ) , (

|g_x x y  g_y x y  _{,If there is no point in the iteration process into the}(x0,y0) centered open

neighborhood u_{,Then, according to the ergodicity of chaos, the function is not chaotic; If there is a}

) ,

(x0x y0y _{, enter into the open neighborhood}u_{,there is}|x||m,| |y||n|_{, According to}

the Taylor expansion equation of differential calculus, we can get the following equation (3):

0 0 0 0 0 0 0 0

( , ) ( , ) ( , ) ( , )

x y

f x x y y f x y f x k x y k y x f x k x y k y y

g x x y y g x y g x k x y k y x g x k x y k y y

 

                



 _{ } _{  } _ _ _{ } _{   } _ _{ } _{  }

 ₍₃₎

Among them, 0k 1_.

Because(x0,y0)_{is a fixed point, the equation (3) is equivalent to the equation (4),}

   

                  

y y k y x k x g x y k y x k x g y y y x x g

y y k y x k x f x y k y x k x f x y y x x f

y x

) ,

( )

, (

) ,

(

) ,

( )

, (

) ,

(

0 0

0

0 0

0

(4) Because f xx( 0 k x y, 0   k y) x f xy( 0 k x y, 0   k y) y

(3)

So, (x0x,y0y)_{after iteration the abscissa is still located in the neighborhood, similarly can}

prove the ordinate is also located in the neighborhood. There is also easy to prove that the sequence point position gradually close to the (x0,y0)_{, with the increase in the number of iterations, reaching}

to (x0,y0), so into the fixed point.

Conclusion 2 A power system, such as equation (1), if the two function f(x,y) and g(x,y) define the domain as

   

0,1 0,1 , range is

 

0,1,Partial derivatives of x and y exist in the domain of definition. f(x,y) and g(x,y) constitute dynamical systems, if the partial derivatives of x and y are satisfied ( , )x y0 0 and(x1,y1)at the second periodic point.

1

1 0 0 0 0

0 0 0

0       

(x ,y )| |f (x ,y )| , |g (x ,y )| |g (x ,y )| f

| x y x y _,

1 | ) , ( | | ) , ( | , 1 | ) , ( | | ) , (

| fx x1 y1  fy x1 y1  gx x1 y1  gy x1 y1  _, Then the dynamical system is not chaotic.

First, expand the two functions, as shown in equation (5):

1

0 0 0 0 1 1 1 1 1 1

( ( ) , ( , ) ) ( , ) ( , ) ( , )

x y

x

x y

y

f f x y x g x y y f x y f x k x y k y x f x k x y k y y

g f x y x g x y y g x y g x k x y k y x g x k x y k y y



 

                

 

                



 

   

  ₍₅₎

Among, x1 f(x0,y0), y1g(x0,y0) _{, because it's the two periodic point,} therefore,x0  f(x1,y1), y0 g(x1,y1)_{, equation (6) established}

2

1 1 1 1 1 1 0 0 1 1 0 1 1 1 0 1 1 0 1 1 1

( ( , ) , ( , ) ) ( , ) ( , )

x y

x

x y

y

f f x y x g x y y x f x k x y k y x f x k x y k y y

g f x y x g x y y y g x k x y k y x g x k x y k y y



 

                

 

                



 

   

   ₍₆₎

So, at the point of the periodic point (fixed point) and the two periodic point, compared with the absolute value of x,y, The absolute value of x1,y1 is decreasing, Compared with the absolute

value of x1,y1, the absolute value of x2,y2decreases further, so if

1 | ) , ( | | ) , ( | , 1 | ) , ( | | ) , (

| fx x0 y0  fy x0 y0  gx x0 y0  gy x0 y0  _,

At the same time, the establishment of |fx(x1,y1)|| fy(x1,y1)|1, |gx(x1,y1)||gy(x1,y1)|1,

Then the function must not be chaotic, And conclusion 1, conclusion 2 similar, can get the following conclusion 3.

Conclusion 3 The two variable continuous derivative function forms the dynamical system, such

as equation (1), if the N point (N is a positive integer) in the N cycle is satisfied (7)

N i

y x g y x g y

x f y x

fx( i, i)| | y( i, i)| 1, | x( i, i)| | y( i, i)| 1, 1,2,3 ,

|           ₍₇₎

So the power system is not chaotic.

(4)

The Determinant of the Partial Derivative at the Periodic Point of the Two Variable Function Determines Its Chaotic Characteristics.

The partial derivatives of X and y are bounded in the defined domain of the two functions in the hypothetical equation (1), then a main result of this paper can be obtained by deduction, as shown below.

First, expand the function into a equation (8),

                                                                             1 1 ) ( ) , ( ) , ( ) , ( ) , ( ) ( ) , ( ) , ( ) , ( ) , ( 2 2 2 0 0 0 0 0 0 0 0 2 2 1 0 0 0 0 0 0 0 0 y y x x y x y x y y x g x y x g y x g y y x x g y x y y x f x y x f y x f y y x x f   (8) in fact, by using this expansion form, we can also draw conclusions(1)











 

















 







                                                                                                                                                                                      ) ( ) ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) ( ) ( ) , ( ) , ( ) , ( ) ( ) , ( ) , ( ) , ( ) ( ) , ( ) , ( ) ) , ( , ) , ( ( )) , ( ), , ( ( ) ( ) ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) ( ) ( ) , ( ) , ( ) , ( ) ( ) , ( ) , ( ) , ( ) ( ) , ( ) , ( ) ) , ( , ) , ( ( )) , ( ), , ( ( 2 1 2 1 2 2 2 1 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 2 1 2 1 2 2 2 1 0 0 0 0 1 1 2 2 1 0 0 0 0 1 1 1 2 1 2 1 2 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 2 1 2 1 2 2 2 1 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 2 1 2 1 2 2 2 1 0 0 0 0 1 1 2 2 1 0 0 0 0 1 1 1 2 1 2 1 2 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 2 2 y x y x y x g y x g y y x g y x g y x f y x g x y x g y x g y x f y x g y y x y x y y x g x y x g y x g y x y y x f x y x f y x g y y x y y x g x y x g y y y x g x y x f g y y x x g y y x x f g y x y x y x f y x f y y x g y x f y x f y x f x y x g y x f y x f y x f x y x y x y y x g x y x g y x f y x y y x f x y x f y x f x y x y y x f x y x f x y y x g x y x f f y y x x g y y x x f f y x y y y x x y x x y x y y x x y y x y x y y y x x y x x y x y y x x x y x                                                       (9)

Using conclusion 1, the method of analysis (9) in conclusion 2 can get the result: if the equation (1)

is chaos, Then

| ) , ( ) , ( ) , ( ) , ( | | ) , ( ) , ( ) , ( ) , (

| f_x x₁ y₁ f_x x₀ y₀  f_y x₁ y₁ g_x x₀ y₀  f_x x₁ y₁ f_y x₀ y₀  f_y x₁ y₁ g_y x₀ y₀ _{and cannot}

be less than 1, at the same time.

These products are denoted as determinant forms, as shown below,

) , ( ) , ( ) , ( ) , ( 1 1 1 1 0 0 0 0 1 y x f y x f y x g y x f a x y x x       , ( , ) ( , ) ) , ( ) , ( 1 1 1 1 0 0 0 0 1 y x f y x f y x g y x f b x y y y       , ( , ) ( , ) ) , ( ) , ( 1 1 1 1 0 0 0 0 1 y x g y x g y x g y x f c x y x x       , ) , ( ) , ( ) , ( ) , ( 1 1 1 1 0 0 0 0 1 y x g y x g y x g y x f d x y y y      

That is, if the dynamical system (1) is chaotic, then the two periodic points of the dynamical system cannot satisfy the |a1||c1|1, |b1||d1|1, simultaneously.

(5)

) , ( ) ,

( 2 2 2 2

1 1

2

y x f y x f

c a

a

x

y 

 

, ( 2, 2) ( 2, 2)

1 1

2

y x f y x f

d b

b

x

y 

 

, ( 2, 2) ( 2, 2)

1 1

2

y x g y x g

c a

c

x

y 

  

,

) , ( ) ,

( 2 2 2 2

1 1

2

y x g y x g

d b

d

x

y 

  

,

) , ( ) , (

1 1

n n x n n y

n n

n

y x f y x f

c a

a _ _



  

, ( , ) ( , )

1 1

n n x n n y

n n

n

y x f y x f

d b

b _ _



  

, ( , ) ( , )

1 1

n n x n n y

n n

n

y x g y x g

c a

c

 



  

,

) , ( ) , (

1 1

n n x n n y

n n

n

y x g y x g

d b

d  _ _  _ 

, (10) It can be concluded that three periodic point cannot satisfy, |a2||c2|1, |b2||d2|1,... , at the

same time. It cannot satisfy |an||cn|1, |bn||dn|1. at the n+1 periodic point because it is a recursive relation, so this method is convenient to calculate. This determinant can be used as a criterion to judge whether the dynamical system (1) is chaos or not.

Compared with the Lyapunov index, the determinant representation method has many characteristics, for example, the method does not need to calculate the characteristic value, do not need to take logarithm, you can save time; this method can accurately exclude whether a system is ergodic chaotic.

Summary and Prospect

The article makes a research on the function of two variables based on differential dynamical system, Differential Taylor expansion gives a necessary condition for ergodic chaos. A new determinant composed of partial derivatives is used in the necessary condition, has many advantages. At least the determinant can express a chaotic degree of the system. In addition, we can give the following guess: Conjecture and conjecture: The determinant in equation (10) is not less than 1 at the same time, then the system is chaos, or in other words, So long as each determinant(ai,bi,ci,di) in equation (10) is not less than 1 at the same time, then it can be called a chaos.

The following work extends the conclusion to the dynamic system composed of three dimensional functions, and studies whether the above assumptions are correct.

Acknowledgement

This research was financially supported by the Liaoning Natural Science Foundation of China (201602034)

Reference

[1] Lorenz E.N. 1963 J. Atmos. Sci. 20 130.

[2] Li T.Y., Yorke J.A. 1975 Amer. Math. Monthly. 82 985. [3] Liu H., Wang L.D., Chu Z.Y. 2009 Nonlinear Analysis 71 6144. [4] Ott E., Grebogi C., Yorke J.A. 1990 Phys. Rev. Lett. 64 1196. [5] Pecora L.M., Carroll T.L. 1990 Phys. Rev. Lett. 64 821. [6] Li X, Liu X.B. 2014 Chaos, Solitons & Fractals 68 40.

[7] Yu W.B., Zhao B. 2014 Acta Phys. Sin. 63, 120502 [Yu W.B., Zhao B. 2014 Acta Phys Sin 63 120502].