由本身系統之增項及代換達成之Lorenz系統超渾沌, 渾沌控制及渾沌同步

國立交通大學

機械工程學系

工學院精密與自動化工程學程

碩 士 論 文

由本身系統之增項及代換達成之 Lorenz 系統超渾沌, 渾沌

控制及渾沌同步

Hyperchaos and Chaos Control and Synchronization of

Lorenz System Obtained by Addition and Replacing of

Its Own Chaos and Regular Functions of Time

研究生:林森生

Student: Sen-Sheng Lin

指導教授:戈正銘

Advisor: Zheng-Ming Ge

國立交通大學

機械工程學系

碩士論文

由本身系統之增項及代換達成之 Lorenz 系統超渾沌, 渾沌

控制及渾沌同步

Hyperchaos and Chaos Control and Synchronization of

Lorenz System Obtained by Addition and Replacing of

Its Own Chaos and Regular Functions of Time

研究生:林森生 Student: Sen-Sheng Lin 指導教授:戈正銘 Advisor: Zheng-Ming Ge

A Thesis

Submitted to Institute of Mechanical Engineering College of Engineering

National Chiao Tung University In Partial Fulfillment of the Requirement

For the Degree of Master of Science In

Mechanical Engineering November 2007

Hsinchu, Taiwan, Republic of China

中華民國九十七年六月

由本身系統之增項及代換達成之 Lorenz 系統超渾沌,渾沌控

制及渾沌同步

研究生：林森生 指導教授：戈正銘

國立交通大學機械工程研究所

摘要

本論文研究超渾沌Lorenz系統藉由其本身渾沌激發而得到。應用以渾沌或規 則時間函數代替參數法可得Lorenz系統的渾沌控制與同步。以渾沌或規則時間函 數代替參數，也可得Lorenz系統之非耦合渾沌同步。最後應用本身渾沌及規則函 數之增項及取代法也可得Lorenz系統之渾沌與渾沌同步。

Hyperchaos and Chaos Control and Synchronization of

Lorenz System Obtained by Addition and Replacing of Its

Own Chaos and Regular Functions of Time

Student：Sen-Sheng Lin Advisor：Zheng-Ming Ge

Department of Mechanical Engineering National Chiao Tung University

ABSTRAST

Hyperchaotic Lorenz system is obtained by replacement of its parameters by its chaotic states.

The chaos control, synchronization and uncoupled synchronization of the systems are also obtained by replacement of its parameters by its chaotic states or by regular functions of time. Furthermore, by addition and replacement of its own chaotic states, chaos control and synchronization of Lorenz system can also be obtained.

ACKNOWLEDGMENT

此篇論文及碩士學業之完成，首先，必須感謝指導教授戈正銘老師的循循善誘與敦敦教 誨，使我在「渾沌」這個研究領域得以引導入門，讓我在陷入瓶頸時得以漸入佳境。老師對 學術研究與詩詞文學創作所投注的那份專注與散發之光彩更是令人覺得仰之彌高，鑽之彌深。 此外老師幽默樂觀的處世態度，對傳統戲劇文化的熱愛與隨遇而安的生活更是我效法的楷 模。 多年的在職研究生涯讓我得以完成，感謝我的一起跟著我學習與成研究的學長與學弟們， 感謝他們在學業與生活上的鼎力相助。感謝張晉民、鄭普建、楊振雄與陳炎生等諸位學長的 熱心指導，在研究陷入渾沌的困境時，幫助我走出渾沌的崎嶇道路。此外也感謝研究室其他 同學陪我渡過這段日子。 最重要的是要感謝父母與妻子的支持，讓我在工作之餘可以重拾書本利用的休息時間來 研究並完成學業。此外必須感謝一起工作過之同仁與推薦之長官，讓我有機會再進修並得以 繼續追尋人生的許多大夢，也讓我可以儲備新的能量邁向新的世紀。

CONTENTS：

ABSTRAST ... i  ACKNOWLEDGMENT ... ii  CONTENTS： ... iii  LIST OF FIGURES： ... iv Chapter 1 ... 1  Introduction ... 1 Chapter 2 ... 2 

Hyperchaos of a Lorenz System Obtained by Additions of Chaos ... 2 

2.1 Hyperchaos of a Lorenz System Obtained by Additions of Chaotic States of Its Own System ... 2 

2.2 Hyperchaos of a Lorenz System Excited by Double Additive Chaotic States of Chaotic Lorenz System... 4

Chapter 3 ... 14 

Chaos Control by Replacing Parameter by Chaotic States ... 14 

3.1 Chaos Control of a Lorenz System by Replacing Parameters with Chaos ... 14 

3.2 Chaos Control of a Lorenz System by Replacing a Parameter with Sum of Chaos and Regular Functions of Time ... 15

Chapter 4 ... 22

Chaos Synchronization of Two Lorenz Systems by Replacing Parameter and States by Chaotic and Regular Motion ... 22 

4.1 Chaos synchronization of Lorenz Systems by Replacing a Parameter with Regular Functions of Time ... 22 

4.2. Chaos synchronization of a Lorenz System by Replacing some Parameter and States with Chaotic state ... 23

Chapter 5 ... 36 

Uncoupled Chaos Synchronization of Two Lorenz Systems by Replacing of Parameter by Chaotic and Regular Function ... 36 

Chapter 6 ... 47 

Conclusions ... 47

LIST OF FIGURES：

Fig.2. 1 The Lyapunov exponents for system (2.3), where 1( ) 0, 2( ) 0, 3( ) 1,

u t = u t = u t =Az ... 6 Fig.2. 2 Phase portraits of excited system (2.2) where u t₁( )=0,u t₂( )=0,u t₃( )= −2z₁. . 6 Fig.2. 3 Lyapunov exponents for system (2.3), where

1( ) 0, 2( ) 0, 3( ) 1, 20 ~ 20

u t = u t = u t = Ay A= − . ... 7 Fig.2. 4 Local enlargement of Fig. 2.3. ... 7 Fig.2. 5 Lyapunov exponents for system (2.3), where

1( ) 1, 2( ) 0, 3( ) 0, 20 ~ 20

u t =Ay u t = u t = A= − . ... 8 Fig.2. 6 Local enlargement of Fig. 2.5. ... 8 Fig.2. 7 Lyapunov exponents for system (2.3), where

1( ) 0, 2( ) 1, 3( ) 0 20 ~ 20

u t = u t =Ay u t = A= − . ... 9 Fig.2. 8 Local enlargement of Fig. 2.7. ... 9 Fig.2. 9 Lyapunov exponents for system (2.3) , where

1( ) 1, 2( ) 0, 3( ) 1, 4 ~ 4

u t =x u t = u t = Az A= − . ... 10 Fig.2. 10 Local enlargement of Fig. 2.9. ... 10 Fig.2. 11 Lyapunov exponents for system (2.7), where

1( ) 0, 2( ) 1, 3( ) 0, 1( ) 0, 2( ) 2, 3( ) 0, 0 ~ 100

u t = u t = Ax u t = v t = v t = Ax v t = A= . ... 11

Fig.2. 12 Local enlargement of Fig. 2.12. ... 11 Fig.2. 13 Lyapunov exponents for system (2.7), where

1( ) 0, 2( ) 0, 3( ) 1, 1( ) 0, 2( ) 0, 3( ) 2, 0 ~ 100

u t = u t = u t =Ax v t = v t = v t =Ax A= . ... 12

Fig.2. 14 Local enlargement of Fig. 2.13. ... 12 Fig.2. 15 Lyapunov exponents for system (2.7), where

1( ) 0, 2( ) 0, 3( ) 1, 1( ) 0, 2( ) 0, 3( ) 2 2, 5 ~ 5.

u t = u t = u t = Az v t = v t = v t =x z A= − ... 13

Fig.3. 1 Phase portrait of x₂,y₂,z with ₂ σ′( )t =σ, ( )r t′ =r b t, ′( )= +x₁ 20y₁. ... 17

Fig.3. 2 Phase portrait of x₂,y₂,z with ₂ σ′( )t =σ, ( )r t′ =r b t, ′( )= 95x₁+0.1p ... 17

Fig.3. 3 Phase portrait of x₂,y₂,z₂σ′( )t =σ, ( )r t′ = 0.01x₁+0.1 ,b t′( )=b t( ) ... 18

Fig.3. 4 Phase portrait of x₂,y₂,z₂σ′( )t = 0.01x₁+0.1 , ( )r t′ =r t b t( ), ′( )=b t( ) ... 18

Fig.3. 5 Phase portrait of x₂,y₂,z₂σ′( )t =σ,r t′( )=r t b t( ), ′( )=b t( )+0.01sin(t+0.1) 19 Fig.3. 6 Phase portrait of x₂,y₂,z for ₂ 1 1 ( )t ,r t( ) 10 sin(0.5 ) 10t x y b t, ( ) b σ′ =σ ′ = + + ′ = . ... 19

Fig.3. 7 Phase portrait of x₂,y₂,z for ₂ 1 1 ( )t , ( )r t r b t, ( ) 1.1sin(0.5 ) 0.1t x 19.5y σ′ =σ ′ = ′ = + + . ... 20

Fig.3. 8 Phase portrait of x₂,y₂,z for ₂ σ′( )t = 200sin(t+0.1) ,r t′( )= r b t, ′( )=b. ... 20

Fig.3. 9 Phase portrait of x₂,y₂,z for ₂ σ′( )t = 10.5sin(1.5t+0.1) ,r t′( )=r b t, ′( )=b. ... 21

Fig.4. 1 Time histories of 1, 2, 3 e e e , σ′( )t =σ, ( )r t′ =r t b t( ), ′( )=10sin(t+0.5) ... 25

Fig.4. 2 Phase portrait of x₂,y₂,z for ₂ σ′( )t =σ, ( )r t′ =r t b t( ), ′( )=10sin(t+0.5) ... 25

Fig.4. 3 Time histories of 1, 2, 3 e e e ,σ′( )t =σ, ( )r t′ =r t b t( ), ′( )= 5sin(5t+0.1) ... 26

Fig.4. 4 Phase portrait of x₂,y₂,z for ₂ σ′( )t =σ, ( )r t′ =r t b t( ), ′( )= 5sin(5t+0.1) ... 26

Fig.4. 5 Time histories of 1, 2, 3 e e e ,f = , x₂ σ′( )t =σ,r t′( )=r b t, ′( )=b. ... 27

Fig.4. 6 Phase portrait of x₂,y₂,z for₂ f = , x₂ σ′( )t =σ,r t′( )=r b t, ′( )=b. ... 27

Fig.4. 7 Time histories of 1, 2, 3 e e e ,f =10x₂, σ′( )t =σ,r t′( )=r b t, ′( )=b. ... 28

Fig.4. 8 Phase portrait of x₂,y₂,z for ₂ f =10x₂, σ′( )t =σ,r t′( )=r b t, ′( )=b. ... 28 Fig.4. 9Time histories of

1, 2, 3

e e e , f x( )= y₂, σ′( )t =σ,r t′( )=r b t, ′( )=b. ... 29 Fig.4. 10 Phase portrait of x₂,y₂,z for ₂ f x( )=y₂, σ′( )t =σ,r t′( )=r b t, ′( )=b. ... 29 Fig.4. 11Time histories of

1, 2, 3

e e e ,f =10y₂, σ′( )t =σ,r t′( )=r b t, ′( )=b. ... 30 Fig.4. 12 Phase portrait of x₂,y₂,z for ₂ f =10y₂, σ′( )t =σ,r t′( )=r b t, ′( )=b. ... 30 Fig.4. 13 Time histories for

1, 2, 3

e e e ，f₁ = , x₂ σ′( )t =σ,r t′( )=28,b t′( )=b, f₂ =0.85x₂. ... 31 Fig.4. 14 Phase portrait of x₂,y₂,z for₂ f₁= , x₂ σ′( )t =σ,r t′( )=28,b t′( )=b,

2 0.85 2

f = x . ... 31 Fig.4. 15 Time histories of

1, 2, 3

e e e ， f1= , x2 σ′( )t =σ,r t′( )=28,b t′( )=b, f2 =1.6x2. ... 32 Fig.4. 16 Phase portrait of x₂,y₂,z for₂ f₁= , x₂ σ′( )t =σ,r t′( )=28,b t′( )=b,

2 1.6 2

f = x . ... 32 Fig.4. 17 Time histories of

1, 2, 3

e e e ， f₁ = y₂, σ′( )t =σ,r t′( )=28,b t′( )=b, f₂ = y₂. 33 Fig.4. 18 Phase portrait of x₂,y₂,z for₂ f₁= y₂, σ′( )t =σ,r t′( )=28,b t′( )=b,f₂ = y₂.

... 33 Fig.4. 19 Time histories of

1, 2, 3

e e e ， f₁= , z₂ σ′( )t =σ,r t′( )=28,b t′( )=b, f₂ = . . 34 z₂

Fig.4. 20 Phase portrait of x₂,y₂,z for₂ f₁= ,x₂ σ′( )t =σ,r t′( )=28,b t′( )=b, f₂ = .x₂

... 34 Fig.4. 21 Time histories of

1, 2, 3

e e e ,r t′ =( ) r t( )=0.1x₂+20y₂,σ′( )t =σ,b t′( )=b. ... 35 Fig.4. 22 Phase portrait of x₂,y₂,z ,₂ r t′ =( ) r t( )=0.1x₂+20y₂,σ′( )t =σ,b t′( )=b. .... 35

Fig.5. 2 Phase portrait of x2,y2,z for 2

r t

( )

=

r t

′

( )

=

25sin(20

t

+

0.5)

... 38

Fig.5. 3 Time histories of 1, 2, 3 e e e for

b t

'( )

=

b t

( )

=

0.5sin(10 )

t

... 39

Fig.5. 4 Phase portrait of x2,y2,z for 2

b t

'( )

=

b t

( )

=

0.5sin(10 )

t

... 39

Fig.5. 5 Time histories of 1, 2, 3 e e e for ₃ 3 '( ) ( ) 1/ 2 sin(1/ 2 ) . b t =b t = x t + _y ... 40

Fig.5. 6 Phase portrait of x₂,y₂,z for _{2 3} 3 '( ) ( ) 1/ 2 sin(10 ) . b t =b t = x t + _y ... 40

Fig.5. 7 Time histories of 1, 2, 3 e e e for

b t

'( ) 10sin(0.5sin(20 )).

=

t

... 41

Fig.5. 8 Phase portrait of x₂,y₂,z for ₂

b t

'( ) 10sin(0.5sin(20 )).

=

t

... 41

Fig.5. 9 Time histories of x₂,y₂,z for ₂

b t

'( )

=

2 sin(9.99 sin(10

+

t

+

0.6)).

... 42

Fig.5. 10Phase portrait of x₂,y₂,z for ₂

b t

'( )

=

2 sin(9.99 sin(10

+

t

+

0.6)).

... 42

Fig.5. 11Time histories of x₂, y₂,z when₂ σ'( )t =17 sin(9.99+2 sin(10t+1.2)). ... 43

Fig.5. 12 Phase portrait of x₂,y₂,z for ₂ σ'( )t =17 sin(9.99+2 sin(10t+1.2)). ... 43

Fig.5. 13 Time histories of x₂,y₂,z when₂ r t( )=r t'( ) 32sin(9.99 2sin(10= + t+1.2)). 44 Fig.5. 14 Phase portrait of x₂,y₂,z for ₂ r t( )=r t'( ) 32sin(9.99 2sin(10= + t+1.2)) .. 44

Fig.5. 15Time histories of x2,y2,z when 2 b=b t'( )= 7 sin(5t+0.1) /

x

3. ... 45

Fig.5. 16Phase portrait of x2,y2,z for 2 b=b t'( )= 7 sin(5t+0.1) /

x

3. ... 45

Fig.5. 17 Time histories of x₂,y₂,z for ₂

b t

'( )

=

10

y

2

sin(0.5(sin(sin( ))))

t

... 46

Chapter 1

Introduction

Chaos has been observed in a lot of nonlinear dynamical systems [1], and it is quite useful in many applications such as heart beat regulation [2], fluid mixing [3], human brain [4], etc. When it is undesirable, chaos control is used [5], while when it is desirable, chaotification is used. During the past decades, chaos synchronization has been applied in many fields such as secure communication [6], chemical and biological systems [7], etc.

Hyperchaotic phenomena and chaos control of Lorenz systems are studied in this thesis. The first hyperchaotic system is hyperchaotic Rössler system [8], after that, many hyperchaotic systems have been found such as the hyperchaotic MCK circuit [9], the hyperchaotic Chen system [10], etc. The above hyperchaotic systems are developed by introducing a new state to the original chaotic systems. Different from that, the hyperchaotic Lorenz system studied here is obtained from excitation by its own chaos. In this thesis, hyperchaos is obtained by replacement of its parameters by its chaotic states. The chaos control, synchronization and uncoupled synchronization of the systems are also obtained by replacement of its parameters by its chaotic states or by regular function of time. Furthermore, by addition and replacement of its own chaotic states, chaos control and synchronization of Lorenz system can also be obtained.

This paper is organized as follows. Chapter 2 contains the hyperchaos of Lorenz system excited by its own chaos. First, the second Lorenz system is excited by the first system. Second, the third Lorenz system is excited by the second system. In Chapter 3, chaos control of Lorenz system is achieved by chaos excitation and excitation of sum of chaos and regular functions of time. In Chapter 4, chaos synchronization will be obtained by replacing parameter by chaotic and regular function of time. In Chapter 5 uncoupled chaos synchronization of two Lorenz systems is obtained by addition and replacement of its own chaotic states.

Chapter 2

Hyperchaos of a Lorenz System Obtained by Additions of

Chaos

In order to generate hyperchaotic phenomena of a Lorenz system, two types of excitation are investigated. First, a second Lorenz system is excited by additive chaotic states of the first system. Second, the third Lorenz system is excited by the second system.

2.1 Hyperchaos of a Lorenz System Obtained by Additions of Chaotic

States of Its Own System

Chaotic behavior is excited by adding chaotic signals from chaos supply system. The chaos supply system is a Lorenz system:

1 1 1 1 1 1 1 1 1 1 1 1 ( ) x x y y rx y x z z x y bz σ = − − ⎧ ⎪ = − − ⎨ ⎪ = − ⎩    (2.1)

and the chaos excited system is:

2 2 2 1 2 2 2 2 2 2 2 2 2 2 3 ( ) ( ) ( ) ( ) x x y u t y rx y x z u t z x y bz u t σ = − − + ⎧ ⎪ = − − + ⎨ ⎪ = − + ⎩    (2.2)

1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 1 2 2 2 2 2 2 2 2 2 2 3 ( ) ( ) ( ) ( ) ( ) x x y y rx y x z z x y bz x x y u t x rx y x z u t z x y bz u t σ σ = − − ⎧ ⎪ = − − ⎪ ⎪ = − ⎪ ⎨ = − − + ⎪ ⎪ = − − − + ⎪ = − + ⎪⎩       (2.3)

where σ =36, r=3, b =20, and the initial condition are x (0)=0.001, ₁ y (0)=0.001, ₁ z (0)=0, ₁ x₂

(0)=0.001, y (0)=0.001, ₂ z (0)=0. 2 (1) u₁(t)=0,u₂(t)=0,u₃(t)=Az₁

Six Lyapunov exponents for system (2.3) are shown in Fig. 2.1 in which three black horizontal lines are the Lyapunov exponents of system (2.1), three remained colored curves are Lyapunov exponents of system (2.2). The phase portraits system (2.2) where u t₁( )=0,u t₂( )=0,u t₃( )= −2z₁ are shown in Fig. 2.2. When -100<A<0, there are two positive Lyapunov exponents as shown in Fig 2.1. Hyperchaos is obtained.

(2) u t₁( )=0,u t₂( )=0,u t₃( )=Ay₁

The Lyapunov exponents for system (2.3) with u t₁( )=0,u t₂( )=0,u t₃( )= Ay₁ are shown in Fig. 2.3. The local enlargement of Fig. 2.3 is shown in Fig. 2.4. When -1.5<A<1.5 hyperchaos occasionally appears.

(3) u t₁( )=Ay u t₁, ₂( )=0,u t₃( )= 0

The Lyapunov exponents for system (2.3) are shown in Fig. 2.5. The local enlargement of Fig. 2.5 is shown in Fig. 2.6. When -0.3<A<0.3 hyperchaos occasionally appears.

(4) u t₁( )=0,u t₂( )=Ay u t₁, ₃( )=0 ,A= −20 ~ 20

The Lyapunov exponents for system (2.3) are shown in Fig. 2.7. The local enlargement of Fig. 2.7 is shown in Fig. 2.8. When -0.7<A<0.7, hyperchaos appears frequently.

(5) u₁(t)=x₁,u₂(t)=0,u₃(t)=Az₁

2.9 is shown in Fig. 2.10. When 1.7<A < 2.4, hyperchaos appears always.

From the above five cases, it is tentatively concluded that the more additive chaotic terms, the more hyperchaos.

2.2 Hyperchaos of a Lorenz System Excited by Double Additive

Chaotic States of Chaotic Lorenz System

Hyperchaotic is excited by additive chaotic states of excited Lorenz system. The chaos supply system is a Lorenz system:

1 1 1 1 1 1 1 1 1 1 1 1 ( ) x x y y rx y x z z x y bz σ = − − ⎧ ⎪ = − − ⎨ ⎪ = − ⎩    (2.4)

and the first chaos excited system is:

2 2 2 1 2 2 2 2 2 2 2 2 2 2 3 ( ) ( ) ( ) ( ) x x y u t y rx y x z u t z x y bz u t σ = − − + ⎧ ⎪ = − − + ⎨ ⎪ = − + ⎩    (2.5)

and the second chaos excited system is:

3 2 2 1 3 2 2 2 2 2 3 2 2 2 3 ( ) ( ) ( ) ( ) x x y v t y rx y x z v t z x y bz v t σ = − − + ⎧ ⎪ = − − + ⎨ ⎪ = − + ⎩    (2.6)

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ + − = + − − = + − − = + − = + − − = + − − = − = − − = − − = ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 3 3 3 3 3 2 3 3 3 3 3 1 3 3 3 3 2 2 2 2 2 2 2 2 2 2 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 t v bz y x z t v z x y rx y t v y x x t u bz y x z t u z x y rx y t u y x x bz y x z z x y rx y y x x          σ σ σ (2.7)

where σ =36, r=3, b =20, and the initial conditions are x (0)=0.002, ₁ y (0)=0.001, ₁ z (0)=0.003, ₁

2

x (0)=0.004, y (0)=0.002, 2 z (0)=0.001, 2 x (0)=0.001, 3 y (0)=0.002, 3 z (0)=0.001 3 (1) 0u₁(t)=0,u₂(t)= Ax₁,u₃(t)=0,v₁(t)=0,v₂(t)=Ax₂,v₃(t)=

The Lyapunov exponents for systems (2.7) are shown in Fig. 2.11. The local enlargement of Fig. 2.11 is shown in Fig. 2.12. When 40<A<100, hyperchaos occurs.

(2) u₁(t)=0,u₂(t)=0,u₃(t)=Ax₁,v₁(t)=0,v₂(t)=0,v₃(t)=Ax₂

The Lyapunov exponents for system (2.7) are shown in Fig. 2.13. The local enlargement of Fig. 2.13 is shown in Fig. 2.14. When 0<A<1.2, hyperchaos occurs.

(3) u₁(t)=0,u₂(t)=0,u₃(t)= Az₁,v₁(t)=0,v₂(t)=0,v₃(t)=x₂z₂

The Lyapunov exponents for system (2.7) are shown in Fig. 2.15. The local enlargement of Fig. 2.15 is shown in Fig. 2.16. When -0.3<A<0.3, 2.1<A, 2.3, hyperchaos occurs.

Fig.2. 1 The Lyapunov exponents for system (2.3), where u t₁( )=0,u t₂( )=0,u t₃( )= Az₁, 100 ~ 100

A= − .

Fig.2. 2 Phase portraits of excited system (2.2) where u t₁( )=0,u t₂( )=0,u t₃( )= −2z₁.

-100 -80 -60 -40 -20 0 20 40 60 80 100 -30 -20 -10 0 10 20 30 40 50 A l y a punov ex po nent s X1(t) y1(t) z1(t) x2(t) y2(t) z2(t) -30 -20 -10 0 10 20 30 -50 0 50 x y -50 0 50 -20 0 20 40 60 80 y z -30 -20 -10 0 10 20 30 -20 0 20 40 60 80 x z -40 -20 0 20 40 -50 0 50 -50 0 50 100 x Lorenz system excitation phase: "z ." +az

y

Fig.2. 3 Lyapunov exponents for system (2.3), where u t₁( )=0,u t₂( )=0,u t₃( )= Ay A₁, = −20 ~ 20.

Fig.2. 4 Local enlargement of Fig. 2.3.

-20 -15 -10 -5 0 5 10 15 20 -14 -12 -10 -8 -6 -4 -2 0 2 4 A l y apu nov ex pon ent s x1 y1 z1 x2 y2 z2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -0.2 0 0.2 0.4 0.6 0.8 A l y a pun ov e x po nen ts x1 y1 z1 x2 y2 z2

Fig.2. 5 Lyapunov exponents for system (2.3), where u t₁( )=Ay u t₁, ₂( )=0,u t₃( )=0, A= −20 ~ 20.

Fig.2. 6 Local enlargement of Fig. 2.5.

-20 -15 -10 -5 0 5 10 15 20 -14 -12 -10 -8 -6 -4 -2 0 2 4 A l y a punov e x ponen ts X1 Y1 Z1 X2 Y2 Z2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Parm : A l y apunov ex pon ent s x1 y1 z1 x2 y2 z2

Fig.2. 7 Lyapunov exponents for system (2.3), where u t₁( )=0,u t₂( )= Ay u t₁, ₃( )=0 A= −20 ~ 20.

Fig.2. 8 Local enlargement of Fig. 2.7.

-20 -15 -10 -5 0 5 10 15 20 -15 -10 -5 0 5 10 A l y ap un ov e x p one nt s x1(t) y1(t) z1(t) x2(t) y2(t) z2(t) -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 A l y a punov ex po nent s x2 y2

Fig.2. 9 Lyapunov exponents for system (2.3) , where u t₁( )=x u t₁, ₂( )=0,u t₃( )= Az A₁, = −4 ~ 4.

Fig.2. 10 Local enlargement of Fig. 2.9.

-4 -3 -2 -1 0 1 2 3 4 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 A L y ap un ov ex p on ent X2 Y2 Z2 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 Ly apunov ex p onent s x2(t) y2(t) z2(t) A

Fig.2. 11 Lyapunov exponents for system (2.7), where

1( ) 0, 2( ) 1, 3( ) 0, 1( ) 0, 2( ) 2, 3( ) 0, 0 ~ 100

u t = u t = Ax u t = v t = v t = Ax v t = A= .

Fig.2. 12 Local enlargement of Fig. 2.12.

0 10 20 30 40 50 60 70 80 90 100 -20 -15 -10 -5 0 5 10 15 20 25 A l y apu nov ex pon ent s x1 y1 z1 x2 y2 z2 x3 y3 z3 34 36 38 40 42 44 -1 -0.5 0 0.5 1 1.5 2 2.5 3 A l y apu nov ex pon en ts x1 y1 z1 x3 y3 z3

Fig.2. 13 Lyapunov exponents for system (2.7), where

1( ) 0, 2( ) 0, 3( ) 1, 1( ) 0, 2( ) 0, 3( ) 2, 0 ~ 100

u t = u t = u t =Ax v t = v t = v t =Ax A= .

Fig.2. 14 Local enlargement of Fig. 2.13.

0 10 20 30 40 50 60 70 80 90 100 -20 -15 -10 -5 0 5 10 15 20 A l y apunov ex ponent s x1 y1 z1 x2 y2 z2 x3 y3 z3 -3 -2 -1 0 1 2 3 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 A l y apu nov ex pon ent s x1 y1 z1 x3 y3 z3

Fig.2. 15 Lyapunov exponents for system (2.7), where

1( ) 0, 2( ) 0, 3( ) 1, 1( ) 0, 2( ) 0, 3( ) 2 2, 5 ~ 5.

u t = u t = u t =Az v t = v t = v t =x z A= −

Fig.2. 16 Local enlargement of Fig. 2.15.

-5 -4 -3 -2 -1 0 1 2 3 4 5 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 A l y a pu no v ex po ne n ts x1 y1 z1 x2 y2 z2 x3 y3 z3 -0.5 0 0.5 1 1.5 2 -1.5 -1 -0.5 0 0.5 1 A l y apu nov ex pon en ts x1 y1 z1 x3 y3 z3

Chapter 3

Chaos Control by Replacing Parameter by Chaotic States

In this Chapter, chaos control of a Lorenz system is achieved by chaos excitation and excitation of sum of chaos and regular functions of time.

3.1 Chaos Control of a Lorenz System by Replacing Parameters with

Chaos

In order to control the chaos of a Lorenz system, some parameters of Lorenz system are replaced by chaotic signals from a chaos supply system which is also a Lorenz system:

1 1 1 1 1 1 1 1 1 1 1 1 ( ) bz y x z z x y rx y y x x − = − − = − − =    σ (3.1)

For Eqs. (3.1) withσ =36, r=3, b =20, the initial condition are x (0) = 0.001, ₁ y (0) = 0.002, 1 z1 (0) = 0. The chaos excited Lorenz system is:

2 2 2 2 2 2 2 2 2 2 2 2 ( )( ) ( ) ( ) x t x y y r t x y x z z x y b t z σ′ = − − ′ = − − ′ = −    (3.2)

For Eqs. (3.2), the initial conditions are x (0) = 0.001, ₂ y (0) = 0.002, ₂ z (0) = 0.001, and ₂

( )t

σ′ , r t′( ), b t′( ) are the chaotic parameters formed by chaotic states of system (3.1).

Forσ′( )t =σ ,r t′( )=r, b t′( )= +x₁ 20y₁ the phase portrait of x y z is shown in Fig 3.1, ₂, ₂, ₂ which converges to a fix point (0,0,8.3351).

Forσ′( )t =σ, ( )r t′ =r b t, ′( )= kx₁+ , withp k=95,p=0.1, the phase portrait is shown in Fig. 3.2, which converges to a fix point (506.2837, 506.5360, 29.5925).

1

z (0) = 1. For Eqs. (3.2), the initial conditions are x (0) = 0.001, 2 y (0) = 0.002, 2 z (0) = 0.001, 2 and σ′( )t , r t′( ), b t′( ) are the chaotic parameters formed by chaotic states of system (3.1).

For σ′( )t =σ, ( )r t′ = kx₁+p b t, ′( )= withb k =0.01,p=0.1, the phase portrait is shown in Fig. 3.3, which converges to fixed point (0, 0, 0).

Forσ′( )t = kx₁+ p r t, ( )′ =r b t, ′( )= , withb k =0.01,p=0.1, the phase portrait is shown in Fig 3.4, which converges to fixed point (8.4853, 8.4853, 27).

For

σ

′

( )

t

=

σ

, ( )

r t

′

=

r b t

,

′

( )

= +

b

k

sin(

ω

t

+

p

)

, with k=0.01, ω =1,p=0.1 the

phase portrait is shown in Fig 3.5, which converges to a fixed point (0.00024, 0.00042, 0.6899).

3.2 Chaos Control of a Lorenz System by Replacing a Parameter with

Sum of Chaos and Regular Functions of Time

In order to control the chaos of a Lorenz system, a given parameter of Lorenz system are replaced by the sum of chaotic signals from chaos supply system, and regular function of time. The chaos supply system is also a Lorenz system:

1 1 1 1 1 1 1 1 1 1 1 1 ( ) x x y y rx y x z z x y bz σ = − − ⎧ ⎪ = − − ⎨ ⎪ = − ⎩    (3.3)

where σ =36, r=3, b =20, and the initial conditions are x (0) = 0.001, ₁ y (0) = 0.002, 1 z (0) = 0. 1 The Lorenz system with parameter replaced by sum of chaos and regular function of time is:

2 2 2 2 2 2 2 2 2 2 2 2 ( )( ) ( ) ( ) x t x y y r t x y x z z x y b t z σ′ = − − ⎧ ⎪ _{= ′ − −} ⎨ ⎪ _{= − ′} ⎩    (3.4)

unchanged :

1 1

( )t ,r t( ) 10 sin(0.5 ) 10t x y b t, ( ) b

σ′ =σ ′ = + + ′ =

The phase portrait, are shown in Fig. 3.6, which is a periodic function of time.

For σ′( )t =σ, ( )r t′ =r b t, ′( )=1.1sin(0.5 ) 0.1t + x₁+19.5y₁ the phase portrait is shown in Fig. 3.7, which converges to a fixed point (0, 0, 8.335).

For σ′( )t = A sin(ωt+p) ,r t′( ) = r b t, ′( ) = b and A=200, ω=1,p=0.1 , the phase portrait is shown in Fig. 3.8, which converges to a fixed point (0, 0, 8.335 ).

When A=10.5, ω=1.5,p=1 the phase portrait is shown in Fig. 3.9, which converges to a fixed point (0, 0, 8.335).

Fig.3. 1 Phase portrait of x₂, y₂,z with ₂ σ′( )t =σ, ( )r t′ =r b t, ′( )= +x₁ 20y₁.

Fig.3. 2 Phase portrait of x₂,y₂,z with ₂ σ′( )t =σ, ( )r t′ =r b t, ′( )= 95x₁+0.1p

-10 0 10 20 -20 0 20 40 0 10 20 30 40 50 60 x y z 0 200 400 600 0 200 400 600 0 5 10 15 20 25 30 2 x 2 y 2 z 2 x 2 y 2 z

Fig.3. 3 Phase portrait of x₂,y₂,z₂σ′( )t =σ, ( )r t′ = 0.01x₁+0.1 ,b t′( )=b t( )

Fig.3. 4 Phase portrait of x , y ,z σ′( )t = 0.01x +0.1 , ( )r t′ =r t b t( ), ′( )=b t( )

0 0.5 1 1.5 2 x 10-3 0 0.5 1 1.5 2 x 10-3 0 0.2 0.4 0.6 0.8 1 x 10-3 x2 y2 z2 0 2 4 6 8 10 0 10 20 30 0 10 20 30 40 2 x 2 y 2 z 2 x 2 y 2 z

Fig.3. 5 Phase portrait of x₂,y₂,z₂ σ′( )t =σ,r t′( )=r t b t( ), ′( )=b t( )+0.01sin(t+0.1)

Fig.3. 6 Phase portrait of x₂,y₂,z for ₂ σ′( )t =σ,r t′( )=10 sin(0.5 ) 10t + x₁+ y b t₁, ′( )= . b

-5 0 5 10 x 10-4 -5 0 5 10 x 10-4 -1 -0.5 0 0.5 1 -5 0 5 10 -5 0 5 10 0 5 10 15 x y z x y z 2 x 2 y 2 z 2 x 2 y 2 z

Fig.3. 7 Phase portrait of x₂,y₂,z for ₂ σ′( )t =σ, ( )r t′ =r b t, ′( )=1.1sin(0.5 ) 0.1t + x₁+19.5y₁.

Fig.3. 8 Phase portrait of x₂,y₂,z for ₂ σ′( )t = 200sin(t+0.1) ,r t′( )= r b t, ′( )= b.

0 5 10 15 20 25 0 10 20 30 40 0 10 20 30 40 50 x y z 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30 35 40 45 50 y x z 2 x 2 y 2 z 2 x 2 y 2 z

Fig.3. 9 Phase portrait of x₂,y₂,z for ₂ σ′( )t = 10.5sin(1.5t+0.1) ,r t′( )=r b t, ′( )= b. -10 0 10 20 30 -20 0 20 40 0 10 20 30 40 50 2 x 2 y 2 z

Chapter 4

Chaos Synchronization of Two Lorenz Systems by Replacing

Parameter and States by Chaotic and Regular Motion

4.1 Chaos synchronization of Lorenz Systems by Replacing a

Parameter with Regular Functions of Time

In order to synchronization the chaos of two Lorenz systems, a given parameter of Lorenz system are replaced by regular function of time. Two Lorenz systems are :

1 1 1 1 1 1 1 1 1 1 1 1 ( ) x x y y rx y x z z x y bz σ = − − ⎧ ⎪ = − − ⎨ ⎪ = − ⎩    (4.1) 2 2 2 2 2 2 2 2 2 2 2 2 ( ) ( ) x x y y r x y x z z x y b t z σ′ = − − ⎧ ⎪ _{= ′ − −} ⎨ ⎪ _{= − ′} ⎩    (4.2)

For Eqs.(4.1) with σ =36,r= ,3 b=20, where the initial conditions x (0) = 0.001, ₁ y (0) = 0.002, ₁

1

z (0) = 0, x2(0)=0.001 y (0) = 0.001,2 z (0)=0.001, 2 b t'( ) is the parameters replaced by regular functions of time, while σ′ , r′ remain unchanged . And we define

1 2 1

e =x −x

3

2 2 1 2 1

,e = y −y e, =z −z .

Forσ′=σ,r′= r b t, ′( ) = b t( ) = k sin (ωt+ p) ,ω=1,k =10,p=0.5, chaos synchronization is obtained as shown in Fig. 4.1 and Fig. 4.2.

Forσ′ =σ, r′ = r b t, ′( ) = b t( ) = k s in (ωt+ p) , withω=1.5,k =5,p=0.1, two systems are synchronized; the time histories and the phase portrait are shown in Fig. 4.3, Fig.4.4.

4.2. Chaos synchronization of a Lorenz System by Replacing some

Parameter and States with Chaotic state

In order to synchronization the chaos of a Lorenz system, some parameters and states of the Lorenz system are replaced by chaotic signals.

1 1 1 1 1 1 1 1 1 1 1 ( ) x f y y rx y x z z x y bz σ = − − ⎧ ⎪ = − − ⎨ ⎪ = − ⎩    (4.3)

where σ =10, r=28, b =8/3, and the initial conditions are x (0) = 0.001, ₁ y (0) = 0.002, ₁ z (0) = ₁

0. The chaos excited Lorenz system is:

2 2 2 2 2 2 2 2 2 2 2 ( )( ) ( ) ( ) x t f y y r t x y x z z x y b t z σ′ = − − ⎧ ⎪ _{= ′ − −} ⎨ ⎪ _{= − ′} ⎩    (4.4)

where the initial condition is x (0) = 0.002, ₂ y (0) = 0.003, ₂ z (0) = 0.001, and ₂ σ′ , r′, b′ are the parameters excited by chaotic signals. We define

1 2 1

e =x −x

3

2 2 1 2 1

,e = y −y e, =z −z . For f =kx₂,σ′( )t =σ,r t′( )=r b t, ′( )=b, with k= , two systems are synchronized , the time 1 histories and the phase portrait are shown in Fig. 4.5-4.6.

For Eqs. (4.3) Eqs.(4.4) where the initial conditions are x (0) = 0.002, ₂ y (0) = 0.003, 2 z (0) 2 = 0.001, and σ′ , r′, b′ are the parameters replaced , f is the state replaced.

For f =kx₂,σ′( )t =σ,r t′( )=r b t, ′( )=b, with k =10 two systems are synchronized, the time histories and the phase portrait is shown in Fig. 4.7-4.8

. For f x( )=ky₂,σ′( )t =σ,r t′( )=r b t, ′( )=b, with k = two systems are synchronized, the time 1 histories and the phase portrait are shown in Fig. 4.9-4.10.

For f x( )=ky₂,σ′( )t =σ,r t′( )=r b t, ′( )=b, with k=10 two systems are synchronized, the time histories and the phase portrait are shown in Fig. 4.11-4.12.

1 1 1 1 1 1 1 1 1 1 1 1 ( ) ( ) x x y y r f y f z z x y b t z σ′ = − − ⎧ ⎪ _{= ′ − −} ⎨ ⎪ _{= − ′} ⎩    (4.5) 2 2 2 2 1 2 2 2 2 2 2 2 ( )( ) ( ) ( ) x t x y y r t f y f z z x y b t z σ′ = − − ⎧ ⎪ _{= ′ − −} ⎨ ⎪ _{= − ′} ⎩    (4.6)

where the initial conditions are x (0) = 0.002, ₂ y (0) = 0.003, ₂ z (0) = 0 , and ₂ σ′ , r′, b′ are the parameters replaced, f f are the states replaced. ₁, ₂

For f₁ =x₂ ,σ′( )t =σ,r t′( )=k b t, ′( )=b , f₂ = px₂ with k=28,p=0.85 two systems are synchronized, the time histories and the phase portrait are shown in Fig. 4.13.- 4.14.

For f₁ =x₂ , σ′( )t =σ,r t′( )=k b t, ′( )=b , f₂ = px₂ , with k=28,p=1.6 two systems are synchronized, the time histories and the phase portrait are shown in Fig. 4.15.- 4.16.

For f₁ = y₂ , σ′( )t =σ,r t′( )=k b t, ′( )=b , f₂ = py₂ with k =28,p=1 two systems are synchronized, the time histories and the phase portrait are shown in Fig. 4.17 - 4.18.

For f₁=z₂ , σ′( )t =σ,r t′( )=k b t, ′( )=b , f₂ = pz₂ with k =28,p=1 two systems are synchronized, the time histories and the phase portrait are shown in Fig. 4.19 - 4.20.

If the chaos excited Lorenz systems becomes:

1 1 1 1 1 1 1 1 1 1 1 ( ) ( ) x x y y rx y x z z f x y bz σ = − − ⎧ ⎪ = − − ⎨ ⎪ = − ⎩    (4.7) 2 2 2 2 2 2 2 2 2 2 2 ( )( ) ( ) ( ) ( ) x t x y y r t x y x z z f x y b t z σ′ = − − ⎧ ⎪ _{= ′ − −} ⎨ ⎪ _{= − ′} ⎩    (4.8) Forσ'( )t =σ(t), r t′( )=r t b t( ), ′( )=b, f = px₂, with p=2

Forr t′ =( ) px₂+qy₂, with p=0.1, q=20two systems are synchronized the time histories and the phase portrait are shown in Fig. 4.21-4.22.

Fig.4. 1 Time histories of

1, 2, 3

e e e , σ′( )t =σ, ( )r t′ =r t b t( ), ′( )=10sin(t+0.5)

Fig.4. 2 Phase portrait of x₂,y₂,z for ₂ σ′( )t =σ, ( )r t′ =r t b t( ), ′( )=10sin(t+0.5)

0 50 100 150 200 250 300 -10 0 10 0 50 100 150 200 250 300 -20 0 20 0 50 100 150 200 250 300 -20 0 20 0 5 10 15 20 25 0 10 20 30 40 0 10 20 30 40 50 2 x 2 y 2 z 1

e

2

e

3

e

t

Fig.4. 3 Time histories of

1, 2, 3

e e e ,σ′( )t =σ, ( )r t′ =r t b t( ), ′( )= 5sin(5t+0.1)

Fig.4. 4 Phase portrait of x₂,y₂,z for ₂ σ′( )t =σ, ( )r t′ =r t b t( ), ′( )= 5sin(5t+0.1)

0 100 200 300 400 500 600 -40 -20 0 20 0 100 200 300 400 500 600 -40 -20 0 20 0 100 200 300 400 500 600 -20 0 20 40 -20 -10 0 10 20 30 -40 -20 0 20 40 0 10 20 30 40 50 1

e

2

e

3

e

2 x 2 y 2 z t

Fig.4. 5 Time histories of

1, 2, 3

e e e ,f = , x₂ σ′( )t =σ,r t′( )=r b t, ′( )=b.

Fig.4. 6 Phase portrait of x₂,y₂,z for₂ f = , x₂ σ′( )t =σ,r t′( )=r b t, ′( )=b.

0 100 200 300 400 500 600 -100 -50 0 0 100 200 300 400 500 600 -200 0 200 0 100 200 300 400 500 600 -200 -100 0 100 -20 -10 0 10 20 -40 -20 0 20 40 0 10 20 30 40 50 x2 y2 z2 1

e

2

e

3

e

t

Fig.4. 7 Time histories of

1, 2, 3

e e e ,f =10x₂, σ′( )t =σ,r t′( )=r b t, ′( )=b.

Fig.4. 8 Phase portrait of x₂, y₂,z for ₂ f =10x₂, σ′( )t =σ,r t′( )=r b t, ′( )=b.

0 100 200 300 400 500 600 -1 0 1 0 100 200 300 400 500 600 -2 0 2x 10 -3 0 100 200 300 400 500 600 -2 0 2x 10 -3 -200 -100 0 100 200 -20 -10 0 10 20 0 10 20 30 40 50 1

e

2

e

3

e

2 x 2 y 2 z t

Fig.4. 9Time histories of

1, 2, 3

e e e ,f x( )= y₂, σ′( )t =σ,r t′( )=r b t, ′( )=b.

Fig.4. 10 Phase portrait of x₂,y₂,z for ₂ f x( )=y₂, σ′( )t =σ,r t′( )=r b t, ′( )=b.

0 100 200 300 400 500 600 -5 0 5x 10 -3 0 100 200 300 400 500 600 -0.01 0 0.01 0 100 200 300 400 500 600 -0.01 0 0.01 -20 -10 0 10 20 -40 -20 0 20 40 0 10 20 30 40 50 1

e

2

e

3

e

2 x 2 y 2 z t

Fig.4. 11Time histories of

1, 2, 3

e e e ,f =10y₂, σ′( )t =σ,r t′( )=r b t, ′( )=b.

Fig.4. 12 Phase portrait of x₂,y₂,z for ₂ f =10y₂, σ′( )t =σ,r t′( )=r b t, ′( )=b.

0 100 200 300 400 500 600 0 0.5 1x 10 -3 0 100 200 300 400 500 600 -5 0 5x 10 -3 0 100 200 300 400 500 600 -5 0 5x 10 -3 -200 -100 0 100 200 -20 -10 0 10 20 10 20 30 40 50 1

e

2

e

3

e

2 x 2 y 2 z t 2 x

Fig.4. 13 Time histories for

1, 2, 3

e e e ， f1= , x2 σ′( )t =σ,r t′( )=28,b t′( )=b,f2 =0.85x2.

Fig.4. 14 Phase portrait of x₂,y₂,z for₂ f₁= , x₂ σ′( )t =σ,r t′( )=28,b t′( )=b, f₂ =0.85x₂.

0 100 200 300 400 500 600 -0.2 0 0.2 0 100 200 300 400 500 600 -0.5 0 0.5 0 100 200 300 400 500 600 -0.5 0 0.5 -20 -15 -10 -5 0 5 10 15 20 -30 -20 -10 0 10 20 30 0 10 20 30 40 50 60 1

e

2

e

3

e

2 x 2 y 2 z t

Fig.4. 15 Time histories of

1, 2, 3

e e e ， f1 = , x2 σ′( )t =σ,r t′( )=28,b t′( )=b, f2 =1.6x2.

Fig.4. 16 Phase portrait of x₂, y₂,z for₂ f₁= , x₂ σ′( )t =σ,r t′( )=28,b t′( )=b, f₂ =1.6x₂.

0 100 200 300 400 500 600 -0.2 0 0.2 0 100 200 300 400 500 600 -0.5 0 0.5 0 100 200 300 400 500 600 -0.5 0 0.5 -20 -10 0 10 20 -20 0 20 40 0 5 10 15 20 25 30 1

e

2

e

3

e

t 2 x 2 y 2 z

Fig.4. 17 Time histories of

1, 2, 3

e e e ， f1= y2, σ′( )t =σ,r t′( )=28,b t′( )=b,f2 = y2.

Fig.4. 18 Phase portrait of x₂,y₂,z for₂ f₁ =y₂, σ′( )t =σ,r t′( )=28,b t′( )=b, f₂ =y₂.

0 100 200 300 400 500 600 -5 0 5x 10 -3 0 100 200 300 400 500 600 -0.01 0 0.01 0 100 200 300 400 500 600 -0.01 0 0.01 -20 -10 0 10 20 -40 -20 0 20 40 0 10 20 30 40 50 60 1

e

2

e

3

e

t 2 x 2 y 2 z

Fig.4. 19 Time histories of

1, 2, 3

e e e ， f1= , z2 σ′( )t =σ,r t′( )=28,b t′( )=b,f2 = . z2

Fig.4. 20 Phase portrait of x₂,y₂,z for₂ f₁ = ,x₂ σ′( )t =σ,r t′( )=28,b t′( )=b, f₂ = . x₂

0 100 200 300 400 500 600 0.01 0 0.01 0 100 200 300 400 500 600 0.01 0 0.01 0 100 200 300 400 500 600 -5 0 5x 10 -3 0.5 1 1.5 2 2.5 0 1 2 3 0.7 0.8 0.9 1 1.1 1.2 1.3 1

e

2

e

3

e

x y t 2 x 2 y 2 z

Fig.4. 21 Time histories of

1, 2, 3

e e e ,r t′ =( ) r t( )=0.1x₂ +20y₂,σ′( )t =σ,b t′( )=b.

Fig.4. 22 Phase portrait of x₂,y₂,z ,₂ r t′ =( ) r t( )=0.1x₂+20y₂,σ′( )t =σ,b t′( )=b.

0 100 200 300 400 500 600 -1 0 1x 10 -3 0 100 200 300 400 500 600 -5 0 5x 10 -3 0 100 200 300 400 500 600 -5 0 5x 10 -3 -50 0 50 -100 -50 0 50 50 100 150 200 250 300 350 1

e

2

e

3

e

x t 2 x 2 y 2 z

Chapter 5

Uncoupled Chaos Synchronization of Two Lorenz Systems

by Replacing of Parameter by Chaotic and Regular Function

In order to obtain uncoupled chaos synchronization of two Lorenz systems, the corresponding parameters are replaced by a function of regular function of time and state of a third Lorenz system.

There are two Lorenz systems of which two corresponding parameters are replaced by a regular function of time. Two identical Lorenz systems to be synchronized are:

1 1 1 1 1 1 1 1 1 1 1 1 ( ) x x y y rx y x z z x y bz σ = − − ⎧ ⎪ = − − ⎨ ⎪ = − ⎩    (5.1) 2 2 2 2 2 2 2 2 2 2 2 2 ( ) x x y y r x y x z z x y b z σ′ = − − ⎧ ⎪ _{= ′ − −} ⎨ ⎪ _{= − ′} ⎩    (5.2)

where σ =σ′ =10, r=r'=28, b = 'b =8/3, and the initial conditions are x (0)=0.001, ₁ y (0)=0.002, ₁

1

z (0)=0. x (0)=0.001, 2 y (0) = 0.002, 2 z (0) = 0.001, the third Lorenz system is: 2

3 3 3 3 3 3 3 3 3 3 3 3 ( ) x x y y rx y x z z x y bz σ = − − ⎧ ⎪ = − − ⎨ ⎪ = − ⎩    (5.3)

with initial condition x (0)=0.001, ₃ y (0)=0.002, ₃ z (0)=0.001. ₃

We define 1 2 1 e =x −x 3 2 2 1 2 1 ,e = y −y e, =z −z . (1) r t( ) = r t′( ) = Asin (ωt + p)

When A=25, ω =20, p=0.5 two system are synchronized. The time histories of e e e and the ₁, ₂, ₃ phase portrait are shown in Fig. 5.1-5.2.

When A=0.5,ω=10 two systems are synchronized. The time histories of e e e and the phase ₁, ₂, ₃ portrait are shown in Fig. 5.3-5.4.

(3) ₃

3 '( ) ( ) s i n ( )

b t = b t = k x ωt + _y

When A=0.5,ω=10 two systems are synchronized. The time histories of e e e and the phase ₁, ₂, ₃ portrait are shown in Fig. 5.5-5.6.

(4) b t( ) = b t'( ) = As in (ω s in (ω₁t))

When A=10,ω=0.5,

ω

₁=20 two systems are synchronized. The time histories of e e e and the ₁, ₂, ₃ phase portrait are shown in Fig. 5.7-5.8.

(5) b t( ) = b '( )t = A s i n (ω + k s i n (ω₁t + p) )

When A= 2,ω=9.99, ω₁ =10, k=1, p=0.6 two system are synchronized.

The time histories of e e e and the phase portrait are shown in Fig. 5.9-5.10. ₁, ₂, ₃ (6) σ ( )t = σ '( )t = As in (ω + k s i n (ω₁t + p) )

When A=17, ω=9.99, ω₁=10, k=2, p=1.2 two system are synchronized.

The time histories of e e e and the phase portrait are shown in Fig. 5.11-5.12. ₁, ₂, ₃ (7)b t'( ) = k y2 s in (

ω

+ (s in (

ω

1t + p)) ,

When k=32, ω=9.99,ω₁ = 1 0 ,p=1.2 two system are synchronized. The time histories of 1, 2, 3

e e e and the phase portrait are shown in Fig. 5.13-5.14.

(8) b = b '( )t = As i n (ωt + p) /

x

3

When A=7, ω=5,p=0.1 two system are synchronized.

The time histories of e e e and the phase portrait are shown in Fig. 5.15-5.16. ₁, ₂, ₃

When b t'( ) = k y2 s in (

ω

(s in (s in (

ω

1t)))) , when k=10,ω=0.5,ω1 = 1 two system are synchronized. The time histories of e e e and the phase portrait are shown in Fig. 5.17-5.18. ₁, ₂, ₃

Fig.5. 1 Time histories of

1, 2, 3

e e e for

r t

( )

=

r t

′

( )

=

25sin(20

t

+

0.5)

Fig.5. 2 Phase portrait of x2,y2,z for 2

r t

( )

=

r t

′

( )

=

25sin(20

t

+

0.5)

0 100 200 300 400 500 600 -10 -5 0 5 0 100 200 300 400 500 600 -20 -10 0 10 0 100 200 300 400 500 600 -20 0 20 -6.8 -6.6 -6.4 -6.2 -6 -5.8 -8 -7 -6 -5 -4 14.2 14.4 14.6 14.8 15 15.2 1

e

2

e

3

e

x t 2 x 2 y 2 z

Fig.5. 3 Time histories of

1, 2, 3

e e e for

b t

'( )

=

b t

( )

=

0.5sin(10 )

t

Fig.5. 4 Phase portrait of x2,y2,z for 2

b t

'( )

=

b t

( )

=

0.5sin(10 )

t

0 10 20 30 40 50 60 70 80 90 100 -10 0 10 20 0 10 20 30 40 50 60 70 80 90 100 -10 0 10 20 0 10 20 30 40 50 60 70 80 90 100 -10 0 10 20 -100 -50 0 50 100 -100 -50 0 50 100 0 10 20 30 40 50 60 1

e

2

e

3

e

X2 Y2 t 2 x 2 y 2 z

Fig.5. 5 Time histories of 1, 2, 3 e e e for ₃ 3 '( ) ( ) 1/ 2 sin(1/ 2 ) . b t =b t = x t + _y

Fig.5. 6 Phase portrait of x₂,y₂,z for _{2 3}

3 '( ) ( ) 1/ 2 sin(10 ) . b t =b t = x t + _y 0 10 20 30 40 50 60 70 80 90 100 -10 0 10 20 0 10 20 30 40 50 60 70 80 90 100 -10 0 10 20 0 10 20 30 40 50 60 70 80 90 100 -10 0 10 20 1

e

2

e

3

e

t -100 -50 0 50 100 -100 -50 0 50 100 0 20 40 60 80 x 2 x 2 y 2 z

Fig.5. 7 Time histories of

1, 2, 3

e e e for

b t

'( ) 10sin(0.5sin(20 )).

=

t

Fig.5. 8 Phase portrait of x₂,y₂,z for ₂

b t

'( ) 10sin(0.5sin(20 )).

=

t

0 50 100 150 200 250 300 -10 0 10 0 50 100 150 200 250 300 -20 0 20 0 50 100 150 200 250 300 -20 0 20 -40 -20 0 20 40 -50 0 50 100 0 20 40 60 80 1

e

2

e

3

e

t 2 x 2 y 2 z

Fig.5. 9 Time histories of x₂,y₂,z for ₂

b t

'( )

=

2 sin(9.99 sin(10

+

t

+

0.6)).

Fig.5. 10Phase portrait of x₂,y₂,z for ₂

b t

'( )

=

2 sin(9.99 sin(10

+

t

+

0.6)).

0 20 40 60 80 100 120 140 160 180 200 -20 0 20 0 20 40 60 80 100 120 140 160 180 200 -50 0 50 0 20 40 60 80 100 120 140 160 180 200 -50 0 50 -10 -8 -6 -4 -2 0 2 4 6 8 10 -15 -10 -5 0 5 10 10 15 20 25 30 35 40 45 1

e

2

e

3

e

t 2 x 2 y 2 z

Fig.5. 11Time histories of x₂,y₂,z when₂ σ'( )t =17 sin(9.99+2 sin(10t+1.2)).

Fig.5. 12 Phase portrait of x ,y ,z for σ'( )t =17 sin(9.99+2 sin(10t+1.2)).

0 20 40 60 80 100 120 140 160 180 200 -50 0 50 100 0 20 40 60 80 100 120 140 160 180 200 -50 0 50 0 20 40 60 80 100 120 140 160 180 200 -50 0 50 5 10 15 20 -5 0 5 10 15 15 20 25 30 35 1

e

2

e

3

e

t 2 x 2 y 2 z

Fig.5. 13 Time histories of x₂,y₂,z when₂ r t( )=r t'( ) 32sin(9.99 2sin(10= + t+1.2)).

Fig.5. 14 Phase portrait of x₂,y₂,z for ₂ r t( )=r t'( ) 32sin(9.99 2sin(10= + t+1.2))

0 20 40 60 80 100 120 140 160 180 200 -50 0 50 0 20 40 60 80 100 120 140 160 180 200 -50 0 50 0 20 40 60 80 100 120 140 160 180 200 -100 -50 0 50 -10 -5 0 5 10 15 -20 -10 0 10 20 0 5 10 15 20 25 1

e

2

e

3

e

x t 2 x 2 y 2 z

Fig.5. 15Time histories of x2, y2,z when 2 b=b t'( )= 7 sin(5t+0.1) /

x

3.

Fig.5. 16Phase portrait of x2,y2,z for 2 b=b t'( )= 7 sin(5t+0.1) /

x

3.

0 100 200 300 400 500 600 -20 -10 0 10 0 100 200 300 400 500 600 -40 -20 0 20 0 100 200 300 400 500 600 -20 0 20 40 -40 -20 0 20 40 -50 0 50 -20 0 20 40 60 1

e

2

e

3

e

t 2 x 2 y 2 z

Fig.5. 17 Time histories of x₂,y₂,z for ₂

b t

'( )

=

10

y

2

sin(0.5(sin(sin( ))))

t

Fig.5. 18 Phase portrait of x₂,y₂,z for ₂

b t

'( )

=

10

y

2

sin(0.5(sin(sin( ))))

t

0 100 200 300 400 500 600 -5 0 5 10 0 100 200 300 400 500 600 -20 0 20 0 100 200 300 400 500 600 -10 0 10 20 -50 0 50 -100 -50 0 50 100 0 20 40 60 80 2 x 2 y 2 z 1

e

2

e

3

e

Chapter 6

Conclusions

We can obtain by different ways hyperchaos of Lorenz system that excited by its own chaos. Chaos control of Lorenz system is achieved by chaos excitation and excitation of sum of chaos and regular functions of time. Chaos synchronization will be obtained by replacing parameter by chaotic and regular function of time. Uncoupled chaos synchronization of two Lorenz systems by replacing parameter by chaotic and regular function.

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