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Control and synchronization of Julia sets in coupled map lattice
Commun Nonlinear Sci Numer Simulat 16 (2011) 3344–3355
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Control and synchronization of Julia sets in coupled map lattice Ping Liu ⇑, Shutang Liu College of Control Science and Engineering, Shandong University, Jinan 250061, PR China
a r t i c l e
i n f o
Article history: Received 9 July 2010 Accepted 12 November 2010 Available online 18 November 2010 Keywords: Julia sets in coupled map lattice Gradient control Optimal control Synchronization
a b s t r a c t In this paper, we achieve the control and synchronization of Julia sets in coupled map lattice using gradient control and optimal control respectively. The control of the Julia sets is accomplished by controlling the stable space of the fixed plane. Moreover, the synchronization of two different Julia sets is also accomplished by their trajectories synchronization. To verify the feasibility of these control methods, we consider the Julia sets, whose lattice length is two, as examples to achieve their control and synchronization using different methods respectively. The numerical simulations are also shown to illustrate the effectiveness of these control methods. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction In the 1970s, French mathematician Gaston Julia [1] studied the simple iterative map znþ1 z2n þ c in complex plane and obtained the beautiful fractal sets, namely, Julia sets. In recent years, there have been a lot of developments in Julia sets, including qualitative characters, applications and controls. Lakhtakia et al. [2] examined the self-replicating properties of the Julia sets Jc(p) for the iterative processes z zp + c for integers p > 1 and showed the symmetries of the Julia sets Jc(p). The physical meaning for Julia sets was provided by studying the motion of a particle in double-well potential [3]. Wang and Meng [4] discussed the physical meaning of Julia sets based on the particle dynamics characteristics and found that the change law of the particle’s velocity can be reflected visually by the fractal construction characteristics of generalized Julia sets. The controls of Julia sets in complex plane were investigated by utilizing feedback, synchronization, coupling and so on [5,6]. In 1982, Alan Norton [7] displayed the spatial Julia sets in 4-D quaternions by the boundary tracking method firstly. Later the spatial Julia sets has been received extensive investigations. Hart et al. [8] obtained a deterministic 3-D Julia set by the ray tracing. The 3-D Julia sets are produced based on customized complexified quaternion [9]. Sui [10] constructed the Julia sets in coupled map lattice and gave its some properties. Moreover, many people are also interested in the structures and interior properties of the spatial Julia sets. Bogush et al. [11] investigated some algebraic and geometrical properties of 3-D Julia sets by means of the group analysis theory. The properties of the generalized quadratic Julia sets was also analyzed [12]. It is noted that the spatial Julia sets has been developed in the drawing of the graphics, the qualitative characters and the theory. However, the control of the spatial Julia sets is more less. Therefore, we discuss the control and synchronization of Julia sets in coupled map lattice by applying gradient control and optimal control in this paper. In next Section, we introduce the definitions and properties of the Julia sets in coupled map lattice and give two corollaries about the stable space and the fixed plane, respectively. In Section 3, we discuss the control of the Julia sets using the gradient control and optimal control and also give the simulations about the control of the Julia sets, whose lattice length ⇑ Corresponding author. E-mail address: [email protected] (P. Liu). 1007-5704/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2010.11.008
P. Liu, S. Liu / Commun Nonlinear Sci Numer Simulat 16 (2011) 3344–3355
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is two. In Section 4, the definition of the synchronization between two different Julia sets in coupled map lattice is given and the synchronization are also discussed using different methods. The simulations of the synchronization between two different Julia sets are also given. Finally, the conclusions are given in Section 5. 2. The basic theories of the Julia sets in coupled map lattice The following coupled map lattice system is introduced firstly,
e
xnþ1 ðiÞ ¼ ð1 eÞf ðxn ðiÞÞ þ ½f ðxn ði 1ÞÞ þ f ðxn ði þ 1ÞÞ; 2
ð1Þ
where n is the discrete time index, i is the lattice site, i = 1, 2, . . . , N, with N being the system size. e is a coupling strength, and e 2 [0, 1] is a real constant. In general, the periodic boundary condition, xn(0) = xn(N), is assumed. It is well known that the Julia sets is created by the iteration of a complex variable function. In order to obtain the Julia sets in coupled map lattice, the system (1) is generalized to the complex plane as follows
e
znþ1 ðiÞ ¼ ð1 eÞf ðzn ðiÞÞ þ ½f ðzn ði 1ÞÞ þ f ðzn ði þ 1ÞÞ; 2
ð2Þ
where f() is a nonlinear function in complex plane and the periodic boundary condition, zn(0) = zn(N), is taken. In this paper, we take f(z) = z2 + c, which is the classic complex quadratic polynomial. Hence, the system (2) reduces to
znþ1 ðiÞ ¼ ð1 eÞz2n ðiÞ þ
e 2
z2n ði 1Þ þ z2n ði þ 1Þ þ c:
ð3Þ
It is clear that the Julia sets in coupled map lattice is formed by the iteration of the system (3). For simplicity, the Julia sets of the system (3) is called CML Julia sets. The simulation figure of the classic Julia sets, which is obtained by the iteration of one function in z plane, is 2-D. Moreover, the CML Julia sets is created by the iteration of N initialization states, so it posses N state variables. The high dimensional Julia sets is created by investigating the state variable which is placed in two coordinates. Therefore, the CML Julia sets can be simulated by a 2N-dimensional figure. We recall the following basic concepts and conclusions, which will be helpful for the main results in next sections [10,13]. Definition 1. A state vector [zn(1), zn(2), . . . , zn(i), . . . , zn(N)] is said to be a repelling periodic state vector of the system (3), if the state vector [zn(1), zn(2), . . . , zn(i), . . . , zn(N)] of the system (3) shows the periodic and repelling properties in the iterative process. Definition 2. The closure of the repelling periodic state vector is called a Julia set in coupled map lattice. Definition 3. If w is a attractive fixed point of g(), the attractive domain is
AðwÞ ¼ fz 2 C : g s ðzÞ ! w; ðs ! 1Þg; where g() is a function with one variable and s is the iterative times. Lemma 1. The periodic state vector [zn(1), zn(2), . . . , zn(i), . . . , zn(N)] of the system (3) is a repelling periodic state vector if and only if jf0 (zn(i))j > 1 for every i = 1, 2, . . . , N. Lemma 2. If w is a attractive fixed point, then
@AðwÞ ¼ JðgÞ; where @A denotes the boundary of the attractive domain A and J denotes a Julia set. We obtain the following corollaries from above basic theories. Corollary 1. If . is a stable fixed plane, the stable space is
Sð.Þ ¼ fzðiÞ 2 C : F ðtÞ ðzðiÞÞ ! .; ðt ! 1Þg; where F() is a complex function about z(i) and t is the iterative times. Corollary 2. If . is a stable fixed plane, then
@Sð.Þ ¼ J s ðFÞ; where @S denotes the boundary of the stable space S and Js denotes a spatial Julia set.
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From the definition of the CML Julia sets and the above lemmas and corollaries, the control of the CML Julia sets can be obtained by controlling the iterative trajectory of state vectors. Moreover, the control of the iterative trajectory can be achieved by accomplishing the stability of the fixed plane. Hence, we achieve the control of the CML Julia sets via controlling the stability of the fixed plane.
3. Control of the Julia sets in coupled map lattice pffiffiffiffiffiffiffiffi As we mentioned in above section, we consider the stable problem of the fixed plane z ðiÞ ¼ 1 214c of the system (3) to achieve the control of the CML Julia sets. In other words, we are able to find a controlling item un(i) to ensure that the fixed plane z* of the following controlled system (4) is stable,
znþ1 ðiÞ ¼ ð1 eÞz2n ðiÞ þ
e 2
z2n ði 1Þ þ z2n ði þ 1Þ þ c þ un ðiÞ:
ð4Þ
Furthermore, z(i)n+1 ? z*(i), as n ? 1. Hence, we can achieve the control of the CML Julia sets by controlling the trajectory of the CML Julia sets. Next, we discuss the control of the CML Julia sets by using different methods. 3.1. The gradient control In this subsection, we accomplish the control of the CML Julia sets by using the gradient control [6], so the controlling k item un ¼ 1þk ðznþ1 ðiÞ z ðiÞÞ is taken. Then, the controlled system (4) reduces to
znþ1 ðiÞ ¼
i 1 h e k ð1 eÞz2n ðiÞ þ ðz2n ði 1Þ þ z2n ði þ 1ÞÞ þ c þ z ðiÞ; 1þk 1þk 2
i ¼ 1; 2 ; N:
ð5Þ
It is clear that z*(i) is also the fixed plane of the system (5). According to Corollary 2, we can achieve the control of the trajectory of the CML Julia sets via controlling the stability of the fixed plane of the controlled system (5). Therefore, we consider the problem of stability of the fixed plane as follows. Denote
FðzðiÞÞ ¼
i 1 h e k ð1 eÞz2 ðiÞ þ ðz2 ði 1Þ þ z2 ði þ 1ÞÞ þ c þ z ðiÞ; 1þk 1þk 2
i ¼ 1; 2 ; N:
ð6Þ
The Jacobin matrix JF of F(z(i)) is given by
0 2ð1eÞ B B B B B B B JF ¼ B B B B B B @
1þk
e 1þk
zð1Þ
zð1Þ 0 .. .
e
zð2Þ
1þk 2ð1eÞ zð2Þ 1þk
e
zð2Þ .. .
0
0
0
0
zð1Þ 1þk
0
e
1þk
e 1þk
zðNÞ
1
C C C C 0 C C C .. C; . C C C 0 C ð1 kÞezðNÞ C A 2ð1eÞ zðNÞ 1þk 0
ð7Þ
where JF is an N N matrix. From the periodic boundary condition zn(0) = zn(N) and the Jacobin matrix JF, the homogeneous solution of the matrix JF is a circulant matrix [14]. Thus, the matrix JF can be diagonalized by a fourier matrix. We take the fourier matrix GNN, then the diagonal matrix bJ F of JF is
bJ F ¼ G1 J G: F The elements of bJ F is
bJ F ðl; lÞ ¼
2 2 ezðlÞ cosðhl Þ; ð1 eÞzðlÞ þ 1þk 1þk
ð8Þ
where hl ¼ 2Npl ; l ¼ 1; 2; . . . ; N. It is noted that the stability condition of the fixed plane of the controlled system (5) is the eigenvalues of JF at the fixed plane within the unit circle in complex plane. According to (8), the module of eigenvalues of 2 2 ð1 2eÞz and 1þk z . Hence, the fixed plane z* of the controlled system JF at the fixed plane are bounded between 1þk 2 2 z < 1. (5) should satisfy 1þk ð1 2eÞz < 1 and 1þk pffiffiffiffiffiffiffiffi Take z ¼ z ðiÞ ¼ 1 214c for simplicity, we get
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 4c 1 ð1 2eÞ ð1 2eÞð1 1 4cÞ < 1 ¼ 1 þ k 1þk 2
ð9Þ
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and
2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4c 1 < 1: ¼ ð1 1 4c Þ 1 þ k 2 1þk
ð10Þ
From (9) and (10), we obtain
j1 þ kj > jð1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4cÞð1 2eÞj
and
j1 þ kj > j1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4cj:
Thus, the range of the control parameter k is
n pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffio j1 þ kj > max ð1 1 4cÞð1 2eÞ; 1 1 4c :
ð11Þ
Therefore, we can choose appropriate k to control the trajectory of CML Julia sets, so as to achieve the control of the CML Julia sets. As we mentioned in the previous section, the CML Julia sets can be simulated by a 2N-dimensional figure. It increases the complexity of simulating the CML Julia sets and runs short of the intuition. In order to void those troubles, we take the lattice length of the system (3) as two, as an example, to illustrate the control of the CML Julia sets. Hence, the controlled systems (5) reduce to
(
1 k znþ1 ð1Þ ¼ 1þk ð1 eÞz2n ð1Þ þ ez2n ð2Þ þ c þ 1þk z ð1Þ; 1 k znþ1 ð2Þ ¼ 1þk ð1 eÞz2n ð2Þ þ ez2n ð1Þ þ c þ 1þk z ð2Þ:
ð12Þ
Also, the Eq. (6) becomes
FðzðiÞÞ ¼
1 k ð1 eÞz2 ðiÞ þ eðz2 ði þ 1Þ þ c þ z ðiÞ; 1þk 1þk
i ¼ 1; 2:
ð13Þ
Based on the periodic boundary condition z(1) = z(3), we have
JF ¼
2ð1eÞ zð1Þ 1þk 2e zð1Þ 1þk
2e zð2Þ 1þk 2ð1eÞ zð2Þ 1þk
! ð14Þ
:
Hence, the diagonal matrix bJ F of JF is
bJ F ¼
2ð12eÞ zð1Þ 1þk
0
0
2 zð2Þ 1þk
!
ð15Þ
:
eÞ eÞ 2 From (15), the eigenvalues of JF are k1 ¼ 2ð12 zð1Þ and k2 ¼ 1þk zð2Þ. Hence, we have jk1 j ¼ 2ð12 z ð1Þ < 1 and 1þk 1þk 2 jk2 j ¼ 1þk z ð2Þ < 1 according to the stability condition of the fixed plane. Therefore, we obtain the range of control parampffiffiffiffiffiffiffiffi eter k to satisfy the stability of the fixed plane z ðiÞ ¼ 1 214c ; i ¼ 1; 2 for the controlled system (12). When we take e = 0.4 and c = 0.2 + 0.1i, then the k satisfies k 2 {k < 1.38} [ {k > 0.62}. Fig. 1(a–f) illustrates the feasibility of this method. It is clear that the control parameter k can control the bifurcation of the CML Julia sets from the simulation figures. The bifurcation is changing from unconspicuously to obviously and the numbers of bifurcation becomes less with the increasing of k. Therefore, we can choose appropriate k to control the CML Julia sets, so as to satisfy the real requirements of the engineering technology and demands of fractal graphs for the industry production. 3.2. The optimal control In above subsection, the control of the CML Julia sets has been studied by applying the gradient control. But the fixed point of the system needs to be known. In the following, the control of the CML Julia sets is accomplished by using the optimal control. For this control method, we do not need to know the fixed point of the controlled system (3). Hence, we take un = k[zn+1(i) zn(i)] [15], then the controlled system (4) reduces to
znþ1 ðiÞ ¼ ð1 kÞ½ð1 eÞz2n ðiÞ þ
e 2
z2n ði 1Þ þ z2n ði þ 1Þ þ c þ kzn ðiÞ;
i ¼ 1; 2; . . . ; N:
ð16Þ
Denote
e
FðzðiÞÞ ¼ ð1 kÞ½ð1 eÞz2 ðiÞ þ ðz2 ði 1Þ þ z2 ði þ 1ÞÞ þ c þ kzðiÞ; 2 then the Jacobin matrix JF of F(z(i)) is given by
i ¼ 1; 2; . . . ; N;
ð17Þ
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2
2
1.5
1.5
1
1
0.5
0.5 0
0 2 0 −2
−3
−2
−1
1
0
2
2
3
0 −2
2
2
1.5
1.5
1
1
0.5
0.5
0
−3
−2
−1
0
1
2
0 2 0 −2
−3
−2
−1
1
0
2
2
3
0 −2
2
2
1.5
1.5
1
1
0.5
0.5
0
−3
−2
−1
0
1
2
0 −2
−3
−2
−1
1
0
2
2
3
0 −2
−3
−2
−1
0
1
2
Fig. 1. when e = 0.4 and c = 0.2 + 0.1i, the figures of the controlled system (12) with the changes of k.
B B B B B B JF ¼ B B B B B B @
2ð1 kÞð1 eÞzð1Þ þ k ð1 kÞezð1Þ
ð1 kÞezð2Þ
2ð1 kÞð1 eÞzð2Þ þ k
ð1 kÞezðNÞ 0
0 .. .
ð1 kÞezð2Þ .. .
0
0
0
0
0
ð1 kÞezðNÞ
ð1 kÞezð1Þ
0
2ð1 kÞð1 eÞzðNÞ þ k
0 .. .
where JF is an N N matrix. Similarly, the diagonal matrix bJ F of JF is obtained by
bJ F ¼ G1 J G F
3
0 2
0
3
1 C C C C C C C; C C C C C A
3
P. Liu, S. Liu / Commun Nonlinear Sci Numer Simulat 16 (2011) 3344–3355
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and the elements of bJ F is
bJ F ðl; lÞ ¼ 2ð1 kÞð1 eÞzðlÞ þ k þ 2ð1 kÞezðlÞ cosðhl Þ;
ð18Þ
2pl ; N
where hl ¼ l ¼ 1; 2; . . . ; N. Hence, the stable condition of the fixed plane for the controlled system (16) is the eigenvalues of JF at the fixed plane in the unit circle of the complex plane. According to (18), the module of eigenvalues of JF at the fixed plane is bounded between j2(1 k)(1 2e)z* + kj and j2(1 k)z* + kj. Hence, we get j2(1 k)(1 2e)z* + kj < 1 and j2(1 k)z* + kj < 1. pffiffiffiffiffiffiffiffi In the same way, we take z ¼ z ðiÞ ¼ 1 214c for simplicity, then the range of the control parameter k is
jkj < min
( pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) 1 jð1 2eÞð1 1 4cÞj 1 j1 1 4cj pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; : j1 ð1 2eÞð1 1 4cÞj j 1 4cj
Therefore, we can choose appropriate k to control the trajectory of CML Julia sets, so as to achieve the control of the CML Julia sets. Similarly, we take the lattice length of the system (3) as two, as an example, to illustrate the control of the CML Julia sets using the optimal control method. Hence, the controlled system (16) reduces to
(
znþ1 ð1Þ ¼ ð1 kÞ ð1 eÞz2n ð1Þ þ ez2n ð2Þ þ c þ kzn ð1Þ; znþ1 ð2Þ ¼ ð1 kÞ ð1 eÞz2n ð2Þ þ ez2n ð1Þ þ c þ kzn ð2Þ:
ð19Þ
At the same time, the formula (17) becomes
FðzðiÞÞ ¼ ð1 kÞ½ð1 eÞz2 ðiÞ þ eðz2 ði þ 1Þ þ c þ kzðiÞ;
i ¼ 1; 2:
ð20Þ
According to the periodic boundary condition z(1) = z(3), we have
JF ¼
2ð1 kÞð1 eÞzð1Þ þ k
2ð1 kÞezð2Þ
2ð1 kÞezð1Þ
2ð1 kÞð1 eÞzð2Þ þ k
:
ð21Þ
Hence, the diagonal matrix bJ F of JF is
bJ F ¼
2ð1 kÞð1 2eÞzð1Þ þ k
0
0
2ð1 kÞzð2Þ þ k
:
ð22Þ
The eigenvalues of k1 = 2(1 k)(1 2e)z(1) + k and k2 = 2(1 k)z(2) + k are obtained from (22). Hence, we get jk1j = j2(1 k) (1 2e)z*(1) + kj < 1 and jk2j = j2(1 k)z*(2) + kj < 1 according to the stability condition of the fixed plane. Similarly, we obtain the range of the control parameter k. For example, we take e = 0.4 and c = 0.2 + 0.1i, then the k satisfies k 2 { 0.4567 < k < 0.4567}. The control of the system (19) is shown by Fig. 2. Fig. 2(a–f) illustrate the effectiveness of the optimal control. It is clearly that the parameter k can control the bifurcation of the CML Julia sets. The number of bifurcation becomes less with the increasing of k. Therefore, the control of the CML Julia sets is achieved by choosing the appropriate k. 4. Synchronization of the Julia sets in coupled map lattice Firstly, the definition of the synchronization of CML Julia sets is given as follows. Definition 4. Consider two CML Julia sets
znþ1 ðiÞ ¼ ð1 eÞz2n ðiÞ þ
e 2
wnþ1 ðiÞ ¼ ð1 eÞw2n ðiÞ þ
z2n ði 1Þ þ z2n ði þ 1Þ þ c1 ;
e 2
w2n ði 1Þ þ w2n ði þ 1Þ þ c2 ;
ð23Þ ð24Þ
where c1 and c2 are complex parameters, and c1 – c2; e is the coupling strength and e 2 [0, 1]. A coupling term p[zn(i), wn(i), c2, e; k] is added to the system (23), then
znþ1 ðiÞ ¼ ð1 eÞz2n ðiÞ þ
e 2
z2n ði 1Þ þ z2n ði þ 1Þ þ c1 þ pðzn ðiÞ; wn ðiÞ; c2 ; e; kÞ;
ð25Þ
where p() is a coupling item about zn(i), wn(i), c2 and e. It is clear that there exists a Julia set corresponding to every k. The Julia sets of (24) and (25) are denoted by J 1s and J 2s respectively. If the CML Julia sets J 2s becomes the same with J 1s when k tends to k0, namely,
lim J 2s [ J 1s J 2s \ J 1s ¼ ; k¼k0
for some k0, the CML Julia sets of the systems (24) and (25) achieve synchronization.
ð26Þ
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P. Liu, S. Liu / Commun Nonlinear Sci Numer Simulat 16 (2011) 3344–3355
2
2
1.5
1.5
1
1
0.5
0.5
0
0 2 0 −2
−3
−2
−1
2
1
0
2
3
0 −2
2
2
1.5
1.5
1
1
0.5
0.5
0
−3
−2
−1
0
1
2
3
0 2 0 −2
−3
−2
−1
2
1
0
2
3
0 −2
2
2
1.5
1.5
1
1
0.5
0.5
0
−3
−2
−1
0
1
2
3
0 2 0 −2
−3
−2
−1
2
1
0
3
2 0 −2
−3
−2
−1
0
1
2
3
Fig. 2. when e = 0.4 and c = 0.2 + 0.1i, the figures of the controlled system (19).
The synchronization of the CML Julia sets is achieved by taking the limitation of k in (26) based on Definition 4. Therefore, we can accomplish the synchronization of CML Julia sets by its trajectory synchronization. The change of k ? k0 illustrates the case of the synchronization of CML Julia sets. In other words, the synchronization of CML Julia sets is also the synchronization case of the CML Julia sets J 2s . Moreover, we take the same iterative initialization vector values of the systems (23) and (24). 4.1. Synchronization of the Julia sets in coupled map lattice using gradient control In the following, we discuss the synchronization between CML Julia sets of the systems (23) and (24) by applying the gradient control. For simplicity, we denote
Fðzn ðiÞ; zn ði 1Þ; zn ði þ 1ÞÞ ¼ ð1 eÞz2n ðiÞ þ
e 2
z2n ði 1Þ þ z2n ði þ 1Þ þ c1 ;
Gðwn ðiÞ; wn ði 1Þ; wn ði þ 1ÞÞ ¼ ð1 eÞw2n ðiÞ þ
e 2
w2n ði 1Þz þ w2n ði þ 1Þ þ c2 ;
ð27Þ ð28Þ
P. Liu, S. Liu / Commun Nonlinear Sci Numer Simulat 16 (2011) 3344–3355
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where c1 and c2 are complex parameters and c1 – c2. k We take the coupling term pðÞ ¼ 1þk ½Fðzn ðiÞ; zn ði 1Þ; zn ði þ 1ÞÞ Gðwn ðiÞ; wn ði 1Þ; wn ði þ 1ÞÞ: Then the coupled system (25) reduces to
znþ1 ðiÞ ¼ Fðzn ðiÞ; zn ði 1Þ; zn ði þ 1ÞÞ
k ½Fðzn ðiÞ; zn ði 1Þ; zn ði þ 1ÞÞ Gðwn ðiÞ; wn ði 1Þ; wn ði þ 1ÞÞ: 1þk
ð29Þ
From (24), (25) and (29), we have
1 ½Fðzn ðiÞ; zn ði 1Þ; zn ði þ 1ÞÞ Gðwn ðiÞ; wn ði 1Þ; wn ði þ 1ÞÞ 1þk e e 1 ½ð1 eÞ z2n ðiÞ w2n ðiÞ þ z2n ði 1Þ w2n ði 1Þ þ ðz2n ði þ 1Þ w2n ði þ 1ÞÞ þ c1 c2 ¼ 1þk 2 2 ( 1 e ð1 eÞðzn ðiÞ wn ðiÞÞðzn ðiÞ þ wn ðiÞÞ þ ½ðzn ði 1Þ wn ði 1ÞÞðzn ði 1Þ þ wn ði 1ÞÞ ¼ 1þk 2 )
znþ1 ðiÞ wnþ1 ðiÞ ¼
þðzn ði þ 1Þ wn ði þ 1ÞÞðzn ði þ 1Þ wn ði þ 1ÞÞ þ c1 c2 :
ð30Þ
It is noted that CML Julia sets is closure, which can be obtained by calculating the iteration of the vector points in a bounded space S. We only consider the vector points whose trajectories are in S since z(i) R Js(F) if there exists an n0 such that F n0 ðzðiÞÞ R S. Since S is a bounded space, there exists M > 0 such that jz(i)j < M for all z(i) 2 S. Hence, we obtain
1 ½2Mð1 eÞjzn ðiÞ wn ðiÞj þ M eðjzn ði 1Þ wn ði 1Þj þ jzn ði þ 1Þ wn ði þ 1ÞjÞ þ jc1 c2 j j1 þ kj 1 1 ð2Mð1 eÞjzn ðiÞ wn ðiÞj þ 4M2 e þ jc1 c2 jÞ ¼ 2Mð1 eÞjzn ðiÞ wn ðiÞj < j1 þ kj j1 þ kj 1 1 1 þ ð4M 2 e þ jc1 c2 jÞ < 2Mð1 eÞ 2Mð1 eÞjzn1 ðiÞ wn1 ðiÞj j1 þ kj j1 þ kj j1 þ kj
2 1 1 1 2Mð1 eÞ jzn1 ðiÞ wn1 ðiÞj ð4M 2 e þ jc1 c2 jÞ þ ð4M2 e þ jc1 c2 jÞ ¼ þ j1 þ kj j1 þ kj 1 þ k n 1 1 ð4M2 e þ jc1 c2 jÞ 1 2Mð1 eÞ þ 1 < < þ e Þ jz1 ðiÞ w1 ðiÞj 2Mð1 1 þ k 1 þ k j1 þ kj " # n1 1 1 1 1 2 þ ð4M e þ jc1 c2 jÞ 2Mð1 eÞ 2Mð1 eÞ þ 1 ¼ jc1 þ þ j1 þ kj j1 þ kj j1 þ kj j1 þ kj " # n n1 1 1 1 2Mð1 eÞ þ 2Mð1 eÞ 2Mð1 eÞ þ 1 c2 j þ þ j1 þ kj j1 þ kj j1 þ kj " # n1 1 1 1 2 þ þ 4M e 2Mð1 eÞ 2Mð1 eÞ þ 1 þ j1 þ kj j1 þ kj j1 þ kj h inþ1 h in 1 1 1 j1þkj 2Mð1 eÞ 1 j1þkj 2Mð1 eÞ 1 1 jc1 c2 j þ 4M2 e: ð31Þ ¼ 1 1 1 j1þkj 2Mð1 eÞ j1 þ kj 1 j1þkj 2Mð1 eÞ j1 þ kj
jznþ1 ðiÞ wnþ1 ðiÞj 6
when
1 2Mð1 j1þkj
lim
eÞ < 1, then the limit in the right formula of (31) is
8
nþ1 > 1 <1 j1þkj 2Mð1 eÞ
n!1 >
:
1 1 j1þkj 2Mð1 eÞ
9
n > 1 1 = 1 j1þkj 2Mð1 eÞ ðjc1 c2 j þ 4M 2 eÞ 1 1 j1þkj jc1 c2 j þ 4M2 e ¼ : 1 1 > j1 þ kj 1 j1þkj 2Mð1 eÞ j1 þ kj 2Mð1 eÞ 1 j1þkj ;
ð32Þ
It is noted that if
1 ðjc1 c2 j þ 4M 2 eÞ ! 0; j1 þ kj namely, j1 + kj ? 1, that is, k ? 1, we obtain jzn+1(i) wn+1(i)j ? 0 from (31) and (32). Therefore, the trajectories of the CML Julia sets of the systems (23) and (24) achieve synchronization. Similarly, we take the lattice length of the system (3) as two, as an example, to illustrate the case of synchronization by using the gradient control. Hence, the systems (23) and (24) reduce to
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(
znþ1 ð1Þ ¼ ð1 eÞz2n ð1Þ þ ez2n ð2Þ þ c1 ;
ð33Þ
znþ1 ð2Þ ¼ ð1 eÞz2n ð2Þ þ ez2n ð1Þ þ c1 : (
wnþ1 ð1Þ ¼ ð1 eÞw2n ð1Þ þ ew2n ð2Þ þ c2 ;
ð34Þ
wnþ1 ð2Þ ¼ ð1 eÞw2n ð2Þ þ ew2n ð1Þ þ c2 :
k k We add the coupling term p1 ðÞ ¼ 1þk ½ð1 eÞz2n ð1Þ þ ez2n ð2Þ þ c1 ðð1 eÞw2n ð1Þ þ ew2n ð2Þ þ c2 Þ and p2 ðÞ ¼ 1þk ½znþ1 ð2Þ 2 2 2 2 ¼ ð1 eÞzn ð2Þ þ ezn ð1Þ þ c1 ðð1 eÞwn ð2Þ þ ewn ð1Þ þ c2 Þ to (33) respectively, then the system (33) becomes
2
2
1.5
1.5
1
1
0.5
0.5
0
0 2 0 −2
−2
−1
2
1
0
2
3
0 −2
−2
−1
0
1
2
3
Fig. 3. The CML Julia sets of the systems (33) and (34).
2
2
1.5
1.5
1
1
0.5
0.5
0
0 2 0 −2
−3
−2
−1
0
1
2
2
3
0 −2
2
2
1.5
1.5
1
1
0.5
0.5
0
−3
−2
−1
0
1
2
3
0 2 0 −2
−3
−2
−1
0
1
2
3
2 0 −2
−3
−2
−1
0
1
Fig. 4. The synchronization of the CML Julia set with changes of k by using the gradient control.
2
3
P. Liu, S. Liu / Commun Nonlinear Sci Numer Simulat 16 (2011) 3344–3355
(
1 k znþ1 ð1Þ ¼ 1þk ½ð1 eÞz2n ð1Þ þ ez2n ð2Þ þ c1 þ 1þk ½ð1 eÞw2n ð1Þ þ ew2n ð2Þ þ c2 ; 1 k znþ1 ð2Þ ¼ 1þk ½ð1 eÞz2n ð2Þ þ ez2n ð1Þ þ c1 þ 1þk ½ð1 eÞw2n ð2Þ þ ew2n ð1Þ þ c2 :
3353
ð35Þ
1 Hence, j1þkj j2Mð1 eÞj < 1 and k ? 1, the CML Julia sets of the systems (33) and (34) achieve synchronization. The case of synchronization can be seen from the simulations. Fig. 3(a) and Fig. 3(b) are the original CML Julia sets of the systems (33) and (34), respectively. From the simulations of Fig. 4, it is clear that the CML Julia set of the system (35) changes gradually toward to the Julia set of (34) with the increasing of k and become the Julia set of (34) as k ? 1. Therefore, the CML Julia set of the synchronization is obtained. We take the limitation of (26), the synchronization of the CML Julia sets is achieved. Therefore, we can accomplish the synchronization of CML Julia sets by its trajectory synchronization, and the change of k ? k0 illustrates the case of the synchronization of CML Julia sets. It is also the synchronization case of the CML Julia sets J 2s . Here we take the same iterative original vector values of the systems (33) and (34).
4.2. Synchronization of the Julia sets in coupled map lattice using optimal control In this subsection, we discuss the synchronization between the system (23) and the system (24) via the optimal control method. We take the coupling term p() = k[F(zn(i), zn(i 1), zn(i + 1)) G(wn(i), wn(i 1), wn(i + 1))] [15]. Then the coupled system (25) reduces to
znþ1 ðiÞ ¼ ð1 eÞz2n ðiÞ þ
e 2
z2n ði 1Þ þ z2n ði þ 1Þ þ c1 k½Fðzn ðiÞ; zn ði 1Þ; zn ði þ 1ÞÞ Gðwn ðiÞ; wn ði 1Þ; wn ði þ 1ÞÞ: ð36Þ
In the same way, we obtain
jznþ1 ðiÞ wnþ1 ðiÞj 6 j1 kj½2Mð1 eÞjzn ðiÞ wn ðiÞj þ M eðjzn ði 1Þ wn ði 1Þj þ jzn ði þ 1Þ wn ði þ 1ÞjÞ þ jc1 c2 j < j1 kjð2Mð1 eÞjzn ðiÞ wn ðiÞj þ 4M 2 e þ jc1 c2 jÞ ¼ j1 kj2Mð1 eÞjzn ðiÞ wn ðiÞj þ j1 kjð4M2 e þ jc1 c2 jÞ < j1 kj2Mð1 eÞ½j1 kj2Mð1 eÞjzn1 ðiÞ wn1 ðiÞj þ j1 kjð4M2 e þ jc1 c2 jÞ þ j1 kjð4M2 e þ jc1 c2 jÞ ¼ ðj1 kj2Mð1 eÞÞ2 jzn1 ðiÞ wn1 ðiÞj þ j1 kjð4M2 e þ jc1 c2 jÞðj1 kj2Mð1 eÞ þ 1Þ < < ðj1 kj2Mð1 eÞÞn jz1 ðiÞ w1 ðiÞj þ j1 kjð4M2 e þ jc1 c2 jÞðj1 kj2Mð1 eÞn1 þ þ j1 kj2Mð1 eÞ þ 1Þ ¼ j1 kjjc1 c2 j½ðj1 kj2Mð1 eÞÞn þ ðj1 kj2Mð1 eÞÞn1 þ þ j1 kj2Mð1 eÞ þ 1 þ j1 kj4M 2 e½ðj1 kj2Mð1 eÞÞn1 þ þ j1 kj2Mð1 eÞ þ 1 ¼
1 ðj1 kj2Mð1 eÞÞnþ1 1 ðj1 kj2Mð1 eÞÞn j1 kjjc1 c2 j þ j1 kj4M 2 e: 1 j1 kj2Mð1 eÞ 1 j1 kj2Mð1 eÞ
ð37Þ
when j1 kj2M(1 e) < 1, then the limit in the right of (37) is
( ) 1 ðj1 kj2Mð1 eÞÞnþ1 1 ðj1 kj2Mð1 eÞÞn j1 kjðjc1 c2 j þ 4M 2 eÞ 2 lim j1 kjjc1 c2 j þ j1 kj4M e ¼ : n!1 1 j1 kj2Mð1 eÞ 1 j1 kj2Mð1 eÞ 1 j1 kj2Mð1 eÞ ð38Þ It is noted that if
j1 kjðjc1 c2 j þ 4M 2 eÞ ! 0; namely, k ? 1, we obtain jzn+1(i) wn+1(i)j ? 0 from (37) and (38). Therefore, the trajectories of the CML Julia sets of the systems (23) and (24) achieve synchronization. Similarly, we take the lattice length of the system (3) as two, as an example, to illustrate the case of synchronization using the optimal control method. The coupling term p1 ðÞ ¼ k½ð1 eÞz2n ð1Þ þ ez2n ð2Þ þ c1 ðð1 eÞw2n ð1Þ þ ew2n ð2Þ þ c2 Þ and p2 ðÞ ¼ k½znþ1 ð2Þ ¼ ð1 eÞz2n ð2Þ þ ez2n ð1Þ þ c1 ðð1 eÞw2n ð2Þ þ ew2n ð1Þ þ c2 Þ are added to the system (33), then the system (33) becomes
(
znþ1 ð1Þ ¼ ð1 kÞ ð1 eÞz2n ð1Þ þ ez2n ð2Þ þ c1 þ k ð1 eÞw2n ð1Þ þ ew2n ð2Þ þ c2 ; znþ1 ð2Þ ¼ ð1 kÞ ð1 eÞz2n ð2Þ þ ez2n ð1Þ þ c1 þ k ð1 eÞw2n ð2Þ þ ew2n ð1Þ þ c2 :
ð39Þ
From the previous discussion, when j1 kj2M(1 e) < 1 and k ? 1, the CML Julia sets of the systems (33) and (34) achieve synchronization. We take c1 = 1 and c2 = 0.2 + 0.1i, then the case of synchronization is demonstrated as Fig. 5. From the
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2
2
1.5
1.5
1
1
0.5
0.5
0
0 2 0 −2
−3
−2
−1
0
1
2
2
3
0 −2
2
2
1.5
1.5
1
1
0.5
0.5
0
−3
−2
−1
0
1
2
3
0 2 0 −2
−3
−2
−1
0
1
2
3
2 0 −2
−3
−2
−1
0
1
2
3
Fig. 5. The synchronization of the CML Julia sets with changes of parameters k by using the optimal control.
Fig. 5, it is clear that the CML Julia set of the system (35) changes toward to the Julia set of the system (34) with the increasing of k, namely, k ? 1. That is to say, the CML Julia set of the synchronization is obtained. A plane z(2) and the tangent plane of x(2) = y(2) are taken for above all simulations. Since the direction of main bifurcation is related with the tangent plane chosen, then the direction of bifurcation in Figs. 1–5 is the same. The discussion of other cases are similar and will be omitted. 5. Conclusions In this paper, we achieve the control and synchronization of CML Julia sets by applying the gradient control and optimal control. The definition of synchronization of two different Julia sets in coupled map lattice is given. Specially, we take the lattice length of the system (3) as two, as examples, to illustrate the effectiveness and feasibility to achieve control and synchronization. These control methods and their theories are successfully applied to the spatial fractal, which have important practical significance for further study of the spatial fractal and explain the corresponding complicated phenomena. Acknowledgements The work was partially supported by the NNSF of China (Nos. 60874009 and 10971120) and a foundation for author of National Excellent Doctoral Dissertation of PR China (FANEDD) (No. 200444). References [1] [2] [3] [4]
Mandelbrot BB. The fractal geometry of nature. San Francisco: Freeman; 1982. Lakhtakia A, Varadan VV, Messier R, Varadan VK. On the symmetries of the Julia sets for the process z zp + c. J Phys A: Math Gen 1987;20:3533–5. Beck C. Physical meaning for Mandelbrot and Julia sets. Physica D 1999;125:171–82. Wang XY, Meng QY. Research on physical meaning for the generalized Mandelbrot-Julia sets based on Langevin problem. Acta Physica Sinica 2004;53:388–95. [5] Zhang YP, Liu ST, Sun WH. Feedback control in fractal mechanics. Chaos Soliton Fract 2009;40:337–43. [6] Zhang YP, Liu ST. Gradient control and synchronization of Julia sets. Chin Phys B 2008;17:543–9. [7] Norton A. Generation and display of geometric fractals in 3-D. Comput Graph 1982;16:61–7.
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Hart JC, Sandin DJ, Kauffman LH. Ray tracing deterministic 3-D fractals. Comput Graph 1989;23:289–96. Gintz TW. Artists statement CQUATSA non-distributive quad algebra for 3-D rendering of Mandelbrot and Julia sets. Comput Graph 2002;26:367–70. Sui SG. Spatial Fractal of complex dynamic system. Master’s thesis of shandong universty, 2005 (In Chinese). Bogush A, Gazizov AZ, Kurochkin YA, Stosui VT. On symmetry properties of quaternionic analogs of Julia sets, Proceedings of the 9th Annual Seminar NPCS-2000. Belarus, Minsk, 2000, p. 304–309. Wang XY, Sun YY. The general quaternionic M–J sets on the mapping z za + c(a 2 N). Comput Math Appl 2007;54:1718–32. Falconer K. Fractals geometry-mathematical foundations and applications. New York: Wiley; 1990. Amritkar RE, Gade PM, Gangal AD, Nandkumaran VM. Stability of periodic orbits of coupled-map lattices. Phys Rev A 1991;44(6):3407–10. de Sousa VM, Lichtenberg AJ. Controlling chaos using nonlinear feedback with delay. Phys Rev E 1996;54(2):1200–8.
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Control and synchronization of Julia sets in coupled map lattice Ping Liu ⇑, Shutang Liu College of Control Science and Engineering, Shandong University, Jinan 250061, PR China
a r t i c l e
i n f o
Article history: Received 9 July 2010 Accepted 12 November 2010 Available online 18 November 2010 Keywords: Julia sets in coupled map lattice Gradient control Optimal control Synchronization
a b s t r a c t In this paper, we achieve the control and synchronization of Julia sets in coupled map lattice using gradient control and optimal control respectively. The control of the Julia sets is accomplished by controlling the stable space of the fixed plane. Moreover, the synchronization of two different Julia sets is also accomplished by their trajectories synchronization. To verify the feasibility of these control methods, we consider the Julia sets, whose lattice length is two, as examples to achieve their control and synchronization using different methods respectively. The numerical simulations are also shown to illustrate the effectiveness of these control methods. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction In the 1970s, French mathematician Gaston Julia [1] studied the simple iterative map znþ1 z2n þ c in complex plane and obtained the beautiful fractal sets, namely, Julia sets. In recent years, there have been a lot of developments in Julia sets, including qualitative characters, applications and controls. Lakhtakia et al. [2] examined the self-replicating properties of the Julia sets Jc(p) for the iterative processes z zp + c for integers p > 1 and showed the symmetries of the Julia sets Jc(p). The physical meaning for Julia sets was provided by studying the motion of a particle in double-well potential [3]. Wang and Meng [4] discussed the physical meaning of Julia sets based on the particle dynamics characteristics and found that the change law of the particle’s velocity can be reflected visually by the fractal construction characteristics of generalized Julia sets. The controls of Julia sets in complex plane were investigated by utilizing feedback, synchronization, coupling and so on [5,6]. In 1982, Alan Norton [7] displayed the spatial Julia sets in 4-D quaternions by the boundary tracking method firstly. Later the spatial Julia sets has been received extensive investigations. Hart et al. [8] obtained a deterministic 3-D Julia set by the ray tracing. The 3-D Julia sets are produced based on customized complexified quaternion [9]. Sui [10] constructed the Julia sets in coupled map lattice and gave its some properties. Moreover, many people are also interested in the structures and interior properties of the spatial Julia sets. Bogush et al. [11] investigated some algebraic and geometrical properties of 3-D Julia sets by means of the group analysis theory. The properties of the generalized quadratic Julia sets was also analyzed [12]. It is noted that the spatial Julia sets has been developed in the drawing of the graphics, the qualitative characters and the theory. However, the control of the spatial Julia sets is more less. Therefore, we discuss the control and synchronization of Julia sets in coupled map lattice by applying gradient control and optimal control in this paper. In next Section, we introduce the definitions and properties of the Julia sets in coupled map lattice and give two corollaries about the stable space and the fixed plane, respectively. In Section 3, we discuss the control of the Julia sets using the gradient control and optimal control and also give the simulations about the control of the Julia sets, whose lattice length ⇑ Corresponding author. E-mail address: [email protected] (P. Liu). 1007-5704/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2010.11.008
P. Liu, S. Liu / Commun Nonlinear Sci Numer Simulat 16 (2011) 3344–3355
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is two. In Section 4, the definition of the synchronization between two different Julia sets in coupled map lattice is given and the synchronization are also discussed using different methods. The simulations of the synchronization between two different Julia sets are also given. Finally, the conclusions are given in Section 5. 2. The basic theories of the Julia sets in coupled map lattice The following coupled map lattice system is introduced firstly,
e
xnþ1 ðiÞ ¼ ð1 eÞf ðxn ðiÞÞ þ ½f ðxn ði 1ÞÞ þ f ðxn ði þ 1ÞÞ; 2
ð1Þ
where n is the discrete time index, i is the lattice site, i = 1, 2, . . . , N, with N being the system size. e is a coupling strength, and e 2 [0, 1] is a real constant. In general, the periodic boundary condition, xn(0) = xn(N), is assumed. It is well known that the Julia sets is created by the iteration of a complex variable function. In order to obtain the Julia sets in coupled map lattice, the system (1) is generalized to the complex plane as follows
e
znþ1 ðiÞ ¼ ð1 eÞf ðzn ðiÞÞ þ ½f ðzn ði 1ÞÞ þ f ðzn ði þ 1ÞÞ; 2
ð2Þ
where f() is a nonlinear function in complex plane and the periodic boundary condition, zn(0) = zn(N), is taken. In this paper, we take f(z) = z2 + c, which is the classic complex quadratic polynomial. Hence, the system (2) reduces to
znþ1 ðiÞ ¼ ð1 eÞz2n ðiÞ þ
e 2
z2n ði 1Þ þ z2n ði þ 1Þ þ c:
ð3Þ
It is clear that the Julia sets in coupled map lattice is formed by the iteration of the system (3). For simplicity, the Julia sets of the system (3) is called CML Julia sets. The simulation figure of the classic Julia sets, which is obtained by the iteration of one function in z plane, is 2-D. Moreover, the CML Julia sets is created by the iteration of N initialization states, so it posses N state variables. The high dimensional Julia sets is created by investigating the state variable which is placed in two coordinates. Therefore, the CML Julia sets can be simulated by a 2N-dimensional figure. We recall the following basic concepts and conclusions, which will be helpful for the main results in next sections [10,13]. Definition 1. A state vector [zn(1), zn(2), . . . , zn(i), . . . , zn(N)] is said to be a repelling periodic state vector of the system (3), if the state vector [zn(1), zn(2), . . . , zn(i), . . . , zn(N)] of the system (3) shows the periodic and repelling properties in the iterative process. Definition 2. The closure of the repelling periodic state vector is called a Julia set in coupled map lattice. Definition 3. If w is a attractive fixed point of g(), the attractive domain is
AðwÞ ¼ fz 2 C : g s ðzÞ ! w; ðs ! 1Þg; where g() is a function with one variable and s is the iterative times. Lemma 1. The periodic state vector [zn(1), zn(2), . . . , zn(i), . . . , zn(N)] of the system (3) is a repelling periodic state vector if and only if jf0 (zn(i))j > 1 for every i = 1, 2, . . . , N. Lemma 2. If w is a attractive fixed point, then
@AðwÞ ¼ JðgÞ; where @A denotes the boundary of the attractive domain A and J denotes a Julia set. We obtain the following corollaries from above basic theories. Corollary 1. If . is a stable fixed plane, the stable space is
Sð.Þ ¼ fzðiÞ 2 C : F ðtÞ ðzðiÞÞ ! .; ðt ! 1Þg; where F() is a complex function about z(i) and t is the iterative times. Corollary 2. If . is a stable fixed plane, then
@Sð.Þ ¼ J s ðFÞ; where @S denotes the boundary of the stable space S and Js denotes a spatial Julia set.
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From the definition of the CML Julia sets and the above lemmas and corollaries, the control of the CML Julia sets can be obtained by controlling the iterative trajectory of state vectors. Moreover, the control of the iterative trajectory can be achieved by accomplishing the stability of the fixed plane. Hence, we achieve the control of the CML Julia sets via controlling the stability of the fixed plane.
3. Control of the Julia sets in coupled map lattice pffiffiffiffiffiffiffiffi As we mentioned in above section, we consider the stable problem of the fixed plane z ðiÞ ¼ 1 214c of the system (3) to achieve the control of the CML Julia sets. In other words, we are able to find a controlling item un(i) to ensure that the fixed plane z* of the following controlled system (4) is stable,
znþ1 ðiÞ ¼ ð1 eÞz2n ðiÞ þ
e 2
z2n ði 1Þ þ z2n ði þ 1Þ þ c þ un ðiÞ:
ð4Þ
Furthermore, z(i)n+1 ? z*(i), as n ? 1. Hence, we can achieve the control of the CML Julia sets by controlling the trajectory of the CML Julia sets. Next, we discuss the control of the CML Julia sets by using different methods. 3.1. The gradient control In this subsection, we accomplish the control of the CML Julia sets by using the gradient control [6], so the controlling k item un ¼ 1þk ðznþ1 ðiÞ z ðiÞÞ is taken. Then, the controlled system (4) reduces to
znþ1 ðiÞ ¼
i 1 h e k ð1 eÞz2n ðiÞ þ ðz2n ði 1Þ þ z2n ði þ 1ÞÞ þ c þ z ðiÞ; 1þk 1þk 2
i ¼ 1; 2 ; N:
ð5Þ
It is clear that z*(i) is also the fixed plane of the system (5). According to Corollary 2, we can achieve the control of the trajectory of the CML Julia sets via controlling the stability of the fixed plane of the controlled system (5). Therefore, we consider the problem of stability of the fixed plane as follows. Denote
FðzðiÞÞ ¼
i 1 h e k ð1 eÞz2 ðiÞ þ ðz2 ði 1Þ þ z2 ði þ 1ÞÞ þ c þ z ðiÞ; 1þk 1þk 2
i ¼ 1; 2 ; N:
ð6Þ
The Jacobin matrix JF of F(z(i)) is given by
0 2ð1eÞ B B B B B B B JF ¼ B B B B B B @
1þk
e 1þk
zð1Þ
zð1Þ 0 .. .
e
zð2Þ
1þk 2ð1eÞ zð2Þ 1þk
e
zð2Þ .. .
0
0
0
0
zð1Þ 1þk
0
e
1þk
e 1þk
zðNÞ
1
C C C C 0 C C C .. C; . C C C 0 C ð1 kÞezðNÞ C A 2ð1eÞ zðNÞ 1þk 0
ð7Þ
where JF is an N N matrix. From the periodic boundary condition zn(0) = zn(N) and the Jacobin matrix JF, the homogeneous solution of the matrix JF is a circulant matrix [14]. Thus, the matrix JF can be diagonalized by a fourier matrix. We take the fourier matrix GNN, then the diagonal matrix bJ F of JF is
bJ F ¼ G1 J G: F The elements of bJ F is
bJ F ðl; lÞ ¼
2 2 ezðlÞ cosðhl Þ; ð1 eÞzðlÞ þ 1þk 1þk
ð8Þ
where hl ¼ 2Npl ; l ¼ 1; 2; . . . ; N. It is noted that the stability condition of the fixed plane of the controlled system (5) is the eigenvalues of JF at the fixed plane within the unit circle in complex plane. According to (8), the module of eigenvalues of 2 2 ð1 2eÞz and 1þk z . Hence, the fixed plane z* of the controlled system JF at the fixed plane are bounded between 1þk 2 2 z < 1. (5) should satisfy 1þk ð1 2eÞz < 1 and 1þk pffiffiffiffiffiffiffiffi Take z ¼ z ðiÞ ¼ 1 214c for simplicity, we get
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 4c 1 ð1 2eÞ ð1 2eÞð1 1 4cÞ < 1 ¼ 1 þ k 1þk 2
ð9Þ
P. Liu, S. Liu / Commun Nonlinear Sci Numer Simulat 16 (2011) 3344–3355
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and
2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4c 1 < 1: ¼ ð1 1 4c Þ 1 þ k 2 1þk
ð10Þ
From (9) and (10), we obtain
j1 þ kj > jð1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4cÞð1 2eÞj
and
j1 þ kj > j1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4cj:
Thus, the range of the control parameter k is
n pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffio j1 þ kj > max ð1 1 4cÞð1 2eÞ; 1 1 4c :
ð11Þ
Therefore, we can choose appropriate k to control the trajectory of CML Julia sets, so as to achieve the control of the CML Julia sets. As we mentioned in the previous section, the CML Julia sets can be simulated by a 2N-dimensional figure. It increases the complexity of simulating the CML Julia sets and runs short of the intuition. In order to void those troubles, we take the lattice length of the system (3) as two, as an example, to illustrate the control of the CML Julia sets. Hence, the controlled systems (5) reduce to
(
1 k znþ1 ð1Þ ¼ 1þk ð1 eÞz2n ð1Þ þ ez2n ð2Þ þ c þ 1þk z ð1Þ; 1 k znþ1 ð2Þ ¼ 1þk ð1 eÞz2n ð2Þ þ ez2n ð1Þ þ c þ 1þk z ð2Þ:
ð12Þ
Also, the Eq. (6) becomes
FðzðiÞÞ ¼
1 k ð1 eÞz2 ðiÞ þ eðz2 ði þ 1Þ þ c þ z ðiÞ; 1þk 1þk
i ¼ 1; 2:
ð13Þ
Based on the periodic boundary condition z(1) = z(3), we have
JF ¼
2ð1eÞ zð1Þ 1þk 2e zð1Þ 1þk
2e zð2Þ 1þk 2ð1eÞ zð2Þ 1þk
! ð14Þ
:
Hence, the diagonal matrix bJ F of JF is
bJ F ¼
2ð12eÞ zð1Þ 1þk
0
0
2 zð2Þ 1þk
!
ð15Þ
:
eÞ eÞ 2 From (15), the eigenvalues of JF are k1 ¼ 2ð12 zð1Þ and k2 ¼ 1þk zð2Þ. Hence, we have jk1 j ¼ 2ð12 z ð1Þ < 1 and 1þk 1þk 2 jk2 j ¼ 1þk z ð2Þ < 1 according to the stability condition of the fixed plane. Therefore, we obtain the range of control parampffiffiffiffiffiffiffiffi eter k to satisfy the stability of the fixed plane z ðiÞ ¼ 1 214c ; i ¼ 1; 2 for the controlled system (12). When we take e = 0.4 and c = 0.2 + 0.1i, then the k satisfies k 2 {k < 1.38} [ {k > 0.62}. Fig. 1(a–f) illustrates the feasibility of this method. It is clear that the control parameter k can control the bifurcation of the CML Julia sets from the simulation figures. The bifurcation is changing from unconspicuously to obviously and the numbers of bifurcation becomes less with the increasing of k. Therefore, we can choose appropriate k to control the CML Julia sets, so as to satisfy the real requirements of the engineering technology and demands of fractal graphs for the industry production. 3.2. The optimal control In above subsection, the control of the CML Julia sets has been studied by applying the gradient control. But the fixed point of the system needs to be known. In the following, the control of the CML Julia sets is accomplished by using the optimal control. For this control method, we do not need to know the fixed point of the controlled system (3). Hence, we take un = k[zn+1(i) zn(i)] [15], then the controlled system (4) reduces to
znþ1 ðiÞ ¼ ð1 kÞ½ð1 eÞz2n ðiÞ þ
e 2
z2n ði 1Þ þ z2n ði þ 1Þ þ c þ kzn ðiÞ;
i ¼ 1; 2; . . . ; N:
ð16Þ
Denote
e
FðzðiÞÞ ¼ ð1 kÞ½ð1 eÞz2 ðiÞ þ ðz2 ði 1Þ þ z2 ði þ 1ÞÞ þ c þ kzðiÞ; 2 then the Jacobin matrix JF of F(z(i)) is given by
i ¼ 1; 2; . . . ; N;
ð17Þ
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2
2
1.5
1.5
1
1
0.5
0.5 0
0 2 0 −2
−3
−2
−1
1
0
2
2
3
0 −2
2
2
1.5
1.5
1
1
0.5
0.5
0
−3
−2
−1
0
1
2
0 2 0 −2
−3
−2
−1
1
0
2
2
3
0 −2
2
2
1.5
1.5
1
1
0.5
0.5
0
−3
−2
−1
0
1
2
0 −2
−3
−2
−1
1
0
2
2
3
0 −2
−3
−2
−1
0
1
2
Fig. 1. when e = 0.4 and c = 0.2 + 0.1i, the figures of the controlled system (12) with the changes of k.
B B B B B B JF ¼ B B B B B B @
2ð1 kÞð1 eÞzð1Þ þ k ð1 kÞezð1Þ
ð1 kÞezð2Þ
2ð1 kÞð1 eÞzð2Þ þ k
ð1 kÞezðNÞ 0
0 .. .
ð1 kÞezð2Þ .. .
0
0
0
0
0
ð1 kÞezðNÞ
ð1 kÞezð1Þ
0
2ð1 kÞð1 eÞzðNÞ þ k
0 .. .
where JF is an N N matrix. Similarly, the diagonal matrix bJ F of JF is obtained by
bJ F ¼ G1 J G F
3
0 2
0
3
1 C C C C C C C; C C C C C A
3
P. Liu, S. Liu / Commun Nonlinear Sci Numer Simulat 16 (2011) 3344–3355
3349
and the elements of bJ F is
bJ F ðl; lÞ ¼ 2ð1 kÞð1 eÞzðlÞ þ k þ 2ð1 kÞezðlÞ cosðhl Þ;
ð18Þ
2pl ; N
where hl ¼ l ¼ 1; 2; . . . ; N. Hence, the stable condition of the fixed plane for the controlled system (16) is the eigenvalues of JF at the fixed plane in the unit circle of the complex plane. According to (18), the module of eigenvalues of JF at the fixed plane is bounded between j2(1 k)(1 2e)z* + kj and j2(1 k)z* + kj. Hence, we get j2(1 k)(1 2e)z* + kj < 1 and j2(1 k)z* + kj < 1. pffiffiffiffiffiffiffiffi In the same way, we take z ¼ z ðiÞ ¼ 1 214c for simplicity, then the range of the control parameter k is
jkj < min
( pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) 1 jð1 2eÞð1 1 4cÞj 1 j1 1 4cj pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; : j1 ð1 2eÞð1 1 4cÞj j 1 4cj
Therefore, we can choose appropriate k to control the trajectory of CML Julia sets, so as to achieve the control of the CML Julia sets. Similarly, we take the lattice length of the system (3) as two, as an example, to illustrate the control of the CML Julia sets using the optimal control method. Hence, the controlled system (16) reduces to
(
znþ1 ð1Þ ¼ ð1 kÞ ð1 eÞz2n ð1Þ þ ez2n ð2Þ þ c þ kzn ð1Þ; znþ1 ð2Þ ¼ ð1 kÞ ð1 eÞz2n ð2Þ þ ez2n ð1Þ þ c þ kzn ð2Þ:
ð19Þ
At the same time, the formula (17) becomes
FðzðiÞÞ ¼ ð1 kÞ½ð1 eÞz2 ðiÞ þ eðz2 ði þ 1Þ þ c þ kzðiÞ;
i ¼ 1; 2:
ð20Þ
According to the periodic boundary condition z(1) = z(3), we have
JF ¼
2ð1 kÞð1 eÞzð1Þ þ k
2ð1 kÞezð2Þ
2ð1 kÞezð1Þ
2ð1 kÞð1 eÞzð2Þ þ k
:
ð21Þ
Hence, the diagonal matrix bJ F of JF is
bJ F ¼
2ð1 kÞð1 2eÞzð1Þ þ k
0
0
2ð1 kÞzð2Þ þ k
:
ð22Þ
The eigenvalues of k1 = 2(1 k)(1 2e)z(1) + k and k2 = 2(1 k)z(2) + k are obtained from (22). Hence, we get jk1j = j2(1 k) (1 2e)z*(1) + kj < 1 and jk2j = j2(1 k)z*(2) + kj < 1 according to the stability condition of the fixed plane. Similarly, we obtain the range of the control parameter k. For example, we take e = 0.4 and c = 0.2 + 0.1i, then the k satisfies k 2 { 0.4567 < k < 0.4567}. The control of the system (19) is shown by Fig. 2. Fig. 2(a–f) illustrate the effectiveness of the optimal control. It is clearly that the parameter k can control the bifurcation of the CML Julia sets. The number of bifurcation becomes less with the increasing of k. Therefore, the control of the CML Julia sets is achieved by choosing the appropriate k. 4. Synchronization of the Julia sets in coupled map lattice Firstly, the definition of the synchronization of CML Julia sets is given as follows. Definition 4. Consider two CML Julia sets
znþ1 ðiÞ ¼ ð1 eÞz2n ðiÞ þ
e 2
wnþ1 ðiÞ ¼ ð1 eÞw2n ðiÞ þ
z2n ði 1Þ þ z2n ði þ 1Þ þ c1 ;
e 2
w2n ði 1Þ þ w2n ði þ 1Þ þ c2 ;
ð23Þ ð24Þ
where c1 and c2 are complex parameters, and c1 – c2; e is the coupling strength and e 2 [0, 1]. A coupling term p[zn(i), wn(i), c2, e; k] is added to the system (23), then
znþ1 ðiÞ ¼ ð1 eÞz2n ðiÞ þ
e 2
z2n ði 1Þ þ z2n ði þ 1Þ þ c1 þ pðzn ðiÞ; wn ðiÞ; c2 ; e; kÞ;
ð25Þ
where p() is a coupling item about zn(i), wn(i), c2 and e. It is clear that there exists a Julia set corresponding to every k. The Julia sets of (24) and (25) are denoted by J 1s and J 2s respectively. If the CML Julia sets J 2s becomes the same with J 1s when k tends to k0, namely,
lim J 2s [ J 1s J 2s \ J 1s ¼ ; k¼k0
for some k0, the CML Julia sets of the systems (24) and (25) achieve synchronization.
ð26Þ
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2
2
1.5
1.5
1
1
0.5
0.5
0
0 2 0 −2
−3
−2
−1
2
1
0
2
3
0 −2
2
2
1.5
1.5
1
1
0.5
0.5
0
−3
−2
−1
0
1
2
3
0 2 0 −2
−3
−2
−1
2
1
0
2
3
0 −2
2
2
1.5
1.5
1
1
0.5
0.5
0
−3
−2
−1
0
1
2
3
0 2 0 −2
−3
−2
−1
2
1
0
3
2 0 −2
−3
−2
−1
0
1
2
3
Fig. 2. when e = 0.4 and c = 0.2 + 0.1i, the figures of the controlled system (19).
The synchronization of the CML Julia sets is achieved by taking the limitation of k in (26) based on Definition 4. Therefore, we can accomplish the synchronization of CML Julia sets by its trajectory synchronization. The change of k ? k0 illustrates the case of the synchronization of CML Julia sets. In other words, the synchronization of CML Julia sets is also the synchronization case of the CML Julia sets J 2s . Moreover, we take the same iterative initialization vector values of the systems (23) and (24). 4.1. Synchronization of the Julia sets in coupled map lattice using gradient control In the following, we discuss the synchronization between CML Julia sets of the systems (23) and (24) by applying the gradient control. For simplicity, we denote
Fðzn ðiÞ; zn ði 1Þ; zn ði þ 1ÞÞ ¼ ð1 eÞz2n ðiÞ þ
e 2
z2n ði 1Þ þ z2n ði þ 1Þ þ c1 ;
Gðwn ðiÞ; wn ði 1Þ; wn ði þ 1ÞÞ ¼ ð1 eÞw2n ðiÞ þ
e 2
w2n ði 1Þz þ w2n ði þ 1Þ þ c2 ;
ð27Þ ð28Þ
P. Liu, S. Liu / Commun Nonlinear Sci Numer Simulat 16 (2011) 3344–3355
3351
where c1 and c2 are complex parameters and c1 – c2. k We take the coupling term pðÞ ¼ 1þk ½Fðzn ðiÞ; zn ði 1Þ; zn ði þ 1ÞÞ Gðwn ðiÞ; wn ði 1Þ; wn ði þ 1ÞÞ: Then the coupled system (25) reduces to
znþ1 ðiÞ ¼ Fðzn ðiÞ; zn ði 1Þ; zn ði þ 1ÞÞ
k ½Fðzn ðiÞ; zn ði 1Þ; zn ði þ 1ÞÞ Gðwn ðiÞ; wn ði 1Þ; wn ði þ 1ÞÞ: 1þk
ð29Þ
From (24), (25) and (29), we have
1 ½Fðzn ðiÞ; zn ði 1Þ; zn ði þ 1ÞÞ Gðwn ðiÞ; wn ði 1Þ; wn ði þ 1ÞÞ 1þk e e 1 ½ð1 eÞ z2n ðiÞ w2n ðiÞ þ z2n ði 1Þ w2n ði 1Þ þ ðz2n ði þ 1Þ w2n ði þ 1ÞÞ þ c1 c2 ¼ 1þk 2 2 ( 1 e ð1 eÞðzn ðiÞ wn ðiÞÞðzn ðiÞ þ wn ðiÞÞ þ ½ðzn ði 1Þ wn ði 1ÞÞðzn ði 1Þ þ wn ði 1ÞÞ ¼ 1þk 2 )
znþ1 ðiÞ wnþ1 ðiÞ ¼
þðzn ði þ 1Þ wn ði þ 1ÞÞðzn ði þ 1Þ wn ði þ 1ÞÞ þ c1 c2 :
ð30Þ
It is noted that CML Julia sets is closure, which can be obtained by calculating the iteration of the vector points in a bounded space S. We only consider the vector points whose trajectories are in S since z(i) R Js(F) if there exists an n0 such that F n0 ðzðiÞÞ R S. Since S is a bounded space, there exists M > 0 such that jz(i)j < M for all z(i) 2 S. Hence, we obtain
1 ½2Mð1 eÞjzn ðiÞ wn ðiÞj þ M eðjzn ði 1Þ wn ði 1Þj þ jzn ði þ 1Þ wn ði þ 1ÞjÞ þ jc1 c2 j j1 þ kj 1 1 ð2Mð1 eÞjzn ðiÞ wn ðiÞj þ 4M2 e þ jc1 c2 jÞ ¼ 2Mð1 eÞjzn ðiÞ wn ðiÞj < j1 þ kj j1 þ kj 1 1 1 þ ð4M 2 e þ jc1 c2 jÞ < 2Mð1 eÞ 2Mð1 eÞjzn1 ðiÞ wn1 ðiÞj j1 þ kj j1 þ kj j1 þ kj
2 1 1 1 2Mð1 eÞ jzn1 ðiÞ wn1 ðiÞj ð4M 2 e þ jc1 c2 jÞ þ ð4M2 e þ jc1 c2 jÞ ¼ þ j1 þ kj j1 þ kj 1 þ k n 1 1 ð4M2 e þ jc1 c2 jÞ 1 2Mð1 eÞ þ 1 < < þ e Þ jz1 ðiÞ w1 ðiÞj 2Mð1 1 þ k 1 þ k j1 þ kj " # n1 1 1 1 1 2 þ ð4M e þ jc1 c2 jÞ 2Mð1 eÞ 2Mð1 eÞ þ 1 ¼ jc1 þ þ j1 þ kj j1 þ kj j1 þ kj j1 þ kj " # n n1 1 1 1 2Mð1 eÞ þ 2Mð1 eÞ 2Mð1 eÞ þ 1 c2 j þ þ j1 þ kj j1 þ kj j1 þ kj " # n1 1 1 1 2 þ þ 4M e 2Mð1 eÞ 2Mð1 eÞ þ 1 þ j1 þ kj j1 þ kj j1 þ kj h inþ1 h in 1 1 1 j1þkj 2Mð1 eÞ 1 j1þkj 2Mð1 eÞ 1 1 jc1 c2 j þ 4M2 e: ð31Þ ¼ 1 1 1 j1þkj 2Mð1 eÞ j1 þ kj 1 j1þkj 2Mð1 eÞ j1 þ kj
jznþ1 ðiÞ wnþ1 ðiÞj 6
when
1 2Mð1 j1þkj
lim
eÞ < 1, then the limit in the right formula of (31) is
8
nþ1 > 1 <1 j1þkj 2Mð1 eÞ
n!1 >
:
1 1 j1þkj 2Mð1 eÞ
9
n > 1 1 = 1 j1þkj 2Mð1 eÞ ðjc1 c2 j þ 4M 2 eÞ 1 1 j1þkj jc1 c2 j þ 4M2 e ¼ : 1 1 > j1 þ kj 1 j1þkj 2Mð1 eÞ j1 þ kj 2Mð1 eÞ 1 j1þkj ;
ð32Þ
It is noted that if
1 ðjc1 c2 j þ 4M 2 eÞ ! 0; j1 þ kj namely, j1 + kj ? 1, that is, k ? 1, we obtain jzn+1(i) wn+1(i)j ? 0 from (31) and (32). Therefore, the trajectories of the CML Julia sets of the systems (23) and (24) achieve synchronization. Similarly, we take the lattice length of the system (3) as two, as an example, to illustrate the case of synchronization by using the gradient control. Hence, the systems (23) and (24) reduce to
3352
P. Liu, S. Liu / Commun Nonlinear Sci Numer Simulat 16 (2011) 3344–3355
(
znþ1 ð1Þ ¼ ð1 eÞz2n ð1Þ þ ez2n ð2Þ þ c1 ;
ð33Þ
znþ1 ð2Þ ¼ ð1 eÞz2n ð2Þ þ ez2n ð1Þ þ c1 : (
wnþ1 ð1Þ ¼ ð1 eÞw2n ð1Þ þ ew2n ð2Þ þ c2 ;
ð34Þ
wnþ1 ð2Þ ¼ ð1 eÞw2n ð2Þ þ ew2n ð1Þ þ c2 :
k k We add the coupling term p1 ðÞ ¼ 1þk ½ð1 eÞz2n ð1Þ þ ez2n ð2Þ þ c1 ðð1 eÞw2n ð1Þ þ ew2n ð2Þ þ c2 Þ and p2 ðÞ ¼ 1þk ½znþ1 ð2Þ 2 2 2 2 ¼ ð1 eÞzn ð2Þ þ ezn ð1Þ þ c1 ðð1 eÞwn ð2Þ þ ewn ð1Þ þ c2 Þ to (33) respectively, then the system (33) becomes
2
2
1.5
1.5
1
1
0.5
0.5
0
0 2 0 −2
−2
−1
2
1
0
2
3
0 −2
−2
−1
0
1
2
3
Fig. 3. The CML Julia sets of the systems (33) and (34).
2
2
1.5
1.5
1
1
0.5
0.5
0
0 2 0 −2
−3
−2
−1
0
1
2
2
3
0 −2
2
2
1.5
1.5
1
1
0.5
0.5
0
−3
−2
−1
0
1
2
3
0 2 0 −2
−3
−2
−1
0
1
2
3
2 0 −2
−3
−2
−1
0
1
Fig. 4. The synchronization of the CML Julia set with changes of k by using the gradient control.
2
3
P. Liu, S. Liu / Commun Nonlinear Sci Numer Simulat 16 (2011) 3344–3355
(
1 k znþ1 ð1Þ ¼ 1þk ½ð1 eÞz2n ð1Þ þ ez2n ð2Þ þ c1 þ 1þk ½ð1 eÞw2n ð1Þ þ ew2n ð2Þ þ c2 ; 1 k znþ1 ð2Þ ¼ 1þk ½ð1 eÞz2n ð2Þ þ ez2n ð1Þ þ c1 þ 1þk ½ð1 eÞw2n ð2Þ þ ew2n ð1Þ þ c2 :
3353
ð35Þ
1 Hence, j1þkj j2Mð1 eÞj < 1 and k ? 1, the CML Julia sets of the systems (33) and (34) achieve synchronization. The case of synchronization can be seen from the simulations. Fig. 3(a) and Fig. 3(b) are the original CML Julia sets of the systems (33) and (34), respectively. From the simulations of Fig. 4, it is clear that the CML Julia set of the system (35) changes gradually toward to the Julia set of (34) with the increasing of k and become the Julia set of (34) as k ? 1. Therefore, the CML Julia set of the synchronization is obtained. We take the limitation of (26), the synchronization of the CML Julia sets is achieved. Therefore, we can accomplish the synchronization of CML Julia sets by its trajectory synchronization, and the change of k ? k0 illustrates the case of the synchronization of CML Julia sets. It is also the synchronization case of the CML Julia sets J 2s . Here we take the same iterative original vector values of the systems (33) and (34).
4.2. Synchronization of the Julia sets in coupled map lattice using optimal control In this subsection, we discuss the synchronization between the system (23) and the system (24) via the optimal control method. We take the coupling term p() = k[F(zn(i), zn(i 1), zn(i + 1)) G(wn(i), wn(i 1), wn(i + 1))] [15]. Then the coupled system (25) reduces to
znþ1 ðiÞ ¼ ð1 eÞz2n ðiÞ þ
e 2
z2n ði 1Þ þ z2n ði þ 1Þ þ c1 k½Fðzn ðiÞ; zn ði 1Þ; zn ði þ 1ÞÞ Gðwn ðiÞ; wn ði 1Þ; wn ði þ 1ÞÞ: ð36Þ
In the same way, we obtain
jznþ1 ðiÞ wnþ1 ðiÞj 6 j1 kj½2Mð1 eÞjzn ðiÞ wn ðiÞj þ M eðjzn ði 1Þ wn ði 1Þj þ jzn ði þ 1Þ wn ði þ 1ÞjÞ þ jc1 c2 j < j1 kjð2Mð1 eÞjzn ðiÞ wn ðiÞj þ 4M 2 e þ jc1 c2 jÞ ¼ j1 kj2Mð1 eÞjzn ðiÞ wn ðiÞj þ j1 kjð4M2 e þ jc1 c2 jÞ < j1 kj2Mð1 eÞ½j1 kj2Mð1 eÞjzn1 ðiÞ wn1 ðiÞj þ j1 kjð4M2 e þ jc1 c2 jÞ þ j1 kjð4M2 e þ jc1 c2 jÞ ¼ ðj1 kj2Mð1 eÞÞ2 jzn1 ðiÞ wn1 ðiÞj þ j1 kjð4M2 e þ jc1 c2 jÞðj1 kj2Mð1 eÞ þ 1Þ < < ðj1 kj2Mð1 eÞÞn jz1 ðiÞ w1 ðiÞj þ j1 kjð4M2 e þ jc1 c2 jÞðj1 kj2Mð1 eÞn1 þ þ j1 kj2Mð1 eÞ þ 1Þ ¼ j1 kjjc1 c2 j½ðj1 kj2Mð1 eÞÞn þ ðj1 kj2Mð1 eÞÞn1 þ þ j1 kj2Mð1 eÞ þ 1 þ j1 kj4M 2 e½ðj1 kj2Mð1 eÞÞn1 þ þ j1 kj2Mð1 eÞ þ 1 ¼
1 ðj1 kj2Mð1 eÞÞnþ1 1 ðj1 kj2Mð1 eÞÞn j1 kjjc1 c2 j þ j1 kj4M 2 e: 1 j1 kj2Mð1 eÞ 1 j1 kj2Mð1 eÞ
ð37Þ
when j1 kj2M(1 e) < 1, then the limit in the right of (37) is
( ) 1 ðj1 kj2Mð1 eÞÞnþ1 1 ðj1 kj2Mð1 eÞÞn j1 kjðjc1 c2 j þ 4M 2 eÞ 2 lim j1 kjjc1 c2 j þ j1 kj4M e ¼ : n!1 1 j1 kj2Mð1 eÞ 1 j1 kj2Mð1 eÞ 1 j1 kj2Mð1 eÞ ð38Þ It is noted that if
j1 kjðjc1 c2 j þ 4M 2 eÞ ! 0; namely, k ? 1, we obtain jzn+1(i) wn+1(i)j ? 0 from (37) and (38). Therefore, the trajectories of the CML Julia sets of the systems (23) and (24) achieve synchronization. Similarly, we take the lattice length of the system (3) as two, as an example, to illustrate the case of synchronization using the optimal control method. The coupling term p1 ðÞ ¼ k½ð1 eÞz2n ð1Þ þ ez2n ð2Þ þ c1 ðð1 eÞw2n ð1Þ þ ew2n ð2Þ þ c2 Þ and p2 ðÞ ¼ k½znþ1 ð2Þ ¼ ð1 eÞz2n ð2Þ þ ez2n ð1Þ þ c1 ðð1 eÞw2n ð2Þ þ ew2n ð1Þ þ c2 Þ are added to the system (33), then the system (33) becomes
(
znþ1 ð1Þ ¼ ð1 kÞ ð1 eÞz2n ð1Þ þ ez2n ð2Þ þ c1 þ k ð1 eÞw2n ð1Þ þ ew2n ð2Þ þ c2 ; znþ1 ð2Þ ¼ ð1 kÞ ð1 eÞz2n ð2Þ þ ez2n ð1Þ þ c1 þ k ð1 eÞw2n ð2Þ þ ew2n ð1Þ þ c2 :
ð39Þ
From the previous discussion, when j1 kj2M(1 e) < 1 and k ? 1, the CML Julia sets of the systems (33) and (34) achieve synchronization. We take c1 = 1 and c2 = 0.2 + 0.1i, then the case of synchronization is demonstrated as Fig. 5. From the
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Fig. 5. The synchronization of the CML Julia sets with changes of parameters k by using the optimal control.
Fig. 5, it is clear that the CML Julia set of the system (35) changes toward to the Julia set of the system (34) with the increasing of k, namely, k ? 1. That is to say, the CML Julia set of the synchronization is obtained. A plane z(2) and the tangent plane of x(2) = y(2) are taken for above all simulations. Since the direction of main bifurcation is related with the tangent plane chosen, then the direction of bifurcation in Figs. 1–5 is the same. The discussion of other cases are similar and will be omitted. 5. Conclusions In this paper, we achieve the control and synchronization of CML Julia sets by applying the gradient control and optimal control. The definition of synchronization of two different Julia sets in coupled map lattice is given. Specially, we take the lattice length of the system (3) as two, as examples, to illustrate the effectiveness and feasibility to achieve control and synchronization. These control methods and their theories are successfully applied to the spatial fractal, which have important practical significance for further study of the spatial fractal and explain the corresponding complicated phenomena. Acknowledgements The work was partially supported by the NNSF of China (Nos. 60874009 and 10971120) and a foundation for author of National Excellent Doctoral Dissertation of PR China (FANEDD) (No. 200444). References [1] [2] [3] [4]
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