Module-phase synchronization in hyperchaotic complex Lorenz system after modified complex projection

Applied Mathematics and Computation 232 (2014) 91–96

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Module-phase synchronization in hyperchaotic complex Lorenz system after modified complex projection Wang Xing-yuan ⇑, Zhang Hao, Lin Xiao-hui Faculty of Electronic Information & Electrical Engineering, Dalian University of Technology, Dalian 116024, China

a r t i c l e

i n f o

Keywords: Modified complex projective synchronization Module-phase synchronization Hyperchaotic complex Lorenz system Plural projective factor

a b s t r a c t This paper studies the modified complex projective synchronization which has plural projective factors and is meaningful to complex systems, and applies this kind of synchronization in a hyperchaotic complex Lorenz system. Based on the proposed synchronization, we also study the hybrid synchronization which contains modified complex projection and module-phase synchronization because hybrid synchronization not only deal with the synchronization real part and imaginary part, respectively, but also take module and phase of complex system into consideration. Owing to it is difficult to design the controller directly, our work involve an intermediary system. The asymptotic convergence of the errors between the states is proven and the computer simulation results present the effectiveness of our method. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction In the last decades chaotic synchronization has been a popular research area [1–2]. It is widely used in many fields of physics, engineering and secure communication [3–5]. Currently, more works have been done in the study of chaos synchronization and many different kinds of synchronizations have been discovered such as completed synchronization, generalized synchronization, phase synchronization, projective synchronization, lag synchronization and impulsive synchronization [6– 12] of most important, chaotic synchronization can be described as hybrid [13,14], high-dimension [15,16], fractional order [17], coupled complex system [18–21] and neural network [22–24]. Previous has been research focusing on plural state variable and ignore the fact that parameters in synchronization of complex systems also can be plural such as the projective factors in modified projective synchronization of two complex systems. Based on the proposed method, taking into module-phase synchronization which is proposed in our previous work [21], we also study the hybrid synchronization and get module-phase synchronization after modified complex projection. The rest of this paper is organized as follows. A brief description of complex system and modified complex projective synchronization is presented in Section 2. Section 3 presents the modified complex projective synchronization of two hyperchaotic complex Lorenz systems. Section 4 provides the module-phase synchronization of hyperchaotic complex Lorenz systems after modified complex projection. Finally, some concluding remarks are given in Section 5. 2. Complex system and synchronization descriptions 2.1. Complex system In this paper, we choose a hyperchaotic complex Lorenz system [25] to study, it has the following form _ y; _ z_ ; wÞ _ T ¼ Fðx; y; z; wÞ and can be described as x_ ¼ ðx; ⇑ Corresponding author. Address: No 2, Linggong Road, Dalian 116023, China. E-mail addresses: [email protected] (X.-y. Wang), [email protected] (H. Zhang). 0096-3003/$ - see front matter Ó 2014 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.12.191

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8 x_ ¼ aðy  xÞ þ jw > > > < y_ ¼ cx  y  xz þ jw ; Þ  bz > z_ ¼ 12 ðxy þ xy > > : _ ¼ 12 ðxy þ xy Þ  rw w

ð1Þ

pffiffiffiffiffiffiffi  are conjugate complex numbers of x and y. z and w are the coupled real part and the coupled imagx and y where j ¼ 1;  inary part of system (1). When a = 14, b = 5, c = 40, r = 13 and system (1) has two positive lyapunov exponents [25]. The attractors are presented in Fig. 1. 2.2. Modified complex projective synchronization Consider two chaotic systems

x_ ¼ f ðx; tÞ;

ð2Þ

y_ ¼ gðy; tÞ þ uðx; y; tÞ;

ð3Þ

and

where x = (x1, x2, . . ., xn)T 2 Rn and y = (y1, y2, . . ., yn)T 2 Rn are n-dimension state variables, f and g are nonlinear functions. u(x, y, t) = (u1, u2, . . ., un) is the controller. Suppose that systems (2) and (3) are the driving system and the response system, respectively. Define error e(t) = x  ay, where a = diag(a1, a2,   , an) presents nonzero complex matrix. When

lim eðtÞ ¼ 0;

t!1

we say that systems (2) and (3) get modified complex projective synchronization about nonzero complex matrix

a = diag(a1, a2,  , an). 3. Module-phase synchronization after modified complex projection 3.1. Modified complex projective synchronization Design driving system as

8 x_ m ¼ aðym  xm Þ þ jwm > > > < y_ m ¼ cxm  ym  xm zm þ jwm ; 1 m Þ  bzm > > z_ m ¼ 2 ðxm ym þ xm y > : _ m ¼ 12 ðxm ym þ xm y m Þ  rwm w

ð4Þ

the corresponding response system is

8 x_ s ¼ aðys  xs Þ þ jws þ L1 > > > < y_ ¼ cx  y  x z þ jw þ L s s s s 2 s s : s Þ  bzs þ L3 > z_ s ¼ 12 ðxs ys þ xs y > > : _ s ¼ 12 ðxs ys þ xs y s Þ  rws þ L4 w

Fig. 1. Attractors of hyperchaotic complex Lorenz system: (a) Attractor of the real part, (b) Attractor of the imaginary part.

ð5Þ

X.-y. Wang et al. / Applied Mathematics and Computation 232 (2014) 91–96

93

where L1, L2, L3, L4 are controllers. plurals x and y can be presented as xm = um1 + jum2, ym = um3 + jum4, zm = um5, wm = um7 and xs = us1 + jus2, ys = us3 + jus4, zs = us5, ws = us7, where umi(i = 1, 2, 3, 4, 5, 7) and usi(i = 1, 2, 3, 4, 5, 7) are coefficients of real parts and imaginary parts, respectively. Image that projective factors are P1 = c1 + jc2, P2 = c3 + jc4, P3 = c5, P4 = c7, where ci(i = 1, 2, 3, 4, 5, 7) is the corresponding coefficients of real parts and imaginary parts. Then systems (4) and (5) are translated into

8 x_ m ¼ ðaum3  aum1 Þ þ jðum7 þ aum4  aum2 Þ > > > < y_ ¼ cu  u  u u þ jðu þ cu  u  u u Þ m m1 m3 m1 m5 m7 m2 m4 m2 m5 ; > z_ m ¼ 12 ðum1 um3 þ um2 um4 Þ  bum5 > > : _ m ¼ 12 ðum1 um3 þ um2 um4 Þ  rum7 w

ð6Þ

8 x_ s ¼ ðaus3  aus1 Þ þ jðus7 þ aus4  aus2 Þ þ L1 > > > < y_ ¼ cu  u  u u þ jðu þ cu  u  u u Þ þ L s s1 s3 s1 s5 s7 s2 s4 s2 s5 2 : > z_ s ¼ 12 ðus1 us3 þ us2 us4 Þ  bus5 þ L3 > > : _ s ¼ 12 ðus1 us3 þ us2 us4 Þ  rus7 þ L4 w

ð7Þ

and

So the error system of modified complex projective synchronization can be shown as

8_ eu1 ¼ ðaus3  aus1 Þ  c1 ðaum3  aum1 Þ þ c2 ðum7 þ aum4  aum2 Þ > > > > e_ u2 ¼ jðus7 þ aus4  aus2  c1 ðum7 þ aum4  aum2 Þ  c2 ðaum3  aum1 ÞÞ > > > > < e_ ¼ cu  u  u u  c ðcu  u  u u Þ þ c ðu þ cu  u  u u Þ u3 s1 s3 s1 s5 3 m1 m3 m1 m5 4 m7 m2 m4 m2 m5 ; > e_ u4 ¼ jðus7 þ cus2  us4  us2 us5  c3 ðum7 þ cum2  um4  um2 um5 Þ  c4 ðcum1  um3  um1 um5 ÞÞ > > > > > e_ u5 ¼ 12 ðus1 us3 þ us2 us4 Þ  bus5  c25 ðum1 um3 þ um2 um4 Þ þ c5 bum5 > > : e_ u7 ¼ 12 ðus1 us3 þ us2 us4 Þ  rus7  c27 ðum1 um3 þ um2 um4 Þ þ c7 rum7

ð8Þ

Design controllers as

8 L1 ¼ aus3 þ c1 aum3  c2 um7  ac2 um4 þ jðus7  aus4 þ c1 um7 þ c1 aum4 þ c2 aum3 Þ > > > > > > < L2 ¼ cus1 þ us1 us5 þ c3 ðcum1  um1 um5 Þ  c4 ðum7 þ cum2  um2 um5 Þ  jðus7 þ cus2  us2 us5  c3 ðum7 þcum2  um2 um5 Þ  c4 ðcum1  um1 um5 ÞÞ : > > > L3 ¼  1 ðus1 us3 þ us2 us4 Þ þ c5 ðum1 um3 þ um2 um4 Þ > 2 2 > > : L4 ¼  12 ðus1 us3 þ us2 us4 Þ þ c27 ðum1 um3 þ um2 um4 Þ

ð9Þ

We will get the following theorem. Theorem 1. Under the controller of controllers (9), the corresponding response system (7) will get modified complex projective synchronization with system (6) about projective factors P1 = c1 + jc2, P2 = c3 + jc4, P3 = c5, P4 = c7. Proof. Substitute controllers (9) into system (7) and we will find that error system (8) is translated into

_ _  C xðtÞ _ eðtÞ ¼ yðtÞ ¼ aeu1  eu3  beu5  reu7 þ jðaeu2  eu4 Þ; from above, we know that a, b, r are all positive. Then the Jacobian matrix of error system is negative, according to the stable theory of linear system, the real parts and imaginary parts of error system will converge to zero, respectively. And synchronization of system (6) and (7) is achieved. h 3.2. Module-phase synchronization after modified complex projection Even though we can conclude that complex systems get synchronization after real parts and imaginary parts get synchronization, respectively. However, in some cases, we need to research generalized synchronization in complex systems or hybrid synchronization. Authors once proposed a kind of synchronization which focuses on the module and phase [21] of plural variables. In this paper, we will take both module-phase synchronization and modified projective synchronization into consideration. For convenience, bring in new system as



x_ t ¼ ðaut3  aut1 Þ þ jðus7 þ aut4  aut2 Þ þ L1 þ L5 ; y_ t ¼ cut1  ut3  ut1 ut5 þ jðus7 þ cut2  ut4  ut2 us5 Þ þ L2 þ L6

ð10Þ

we know form Theorem 1 that systems (6) and (7) will get modified complex projective synchronization, so the problem of module-phase synchronization after modified complex projection is changed into the problem of module-phase synchronization of system (10) and system (7) with L1, L2. Design modules and phases of systems (7) and (10) are

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Fig. 2. The diagrams of M-P synchronization after modified complex projection: (a) M(Xt) and M(Xm) after projection, (b) M(Xt) and M(Xm) after projection (c) P(Xt) and P(Xm) after projection (d) P(Xt) and P(Xm) after projection.

Mðxs Þ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2s1 þ u2s2 ;

Mðys Þ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2s3 þ u2s4 ;

Pðxs Þ ¼ tan1 ðus1 =us2 Þ;

Pðys Þ ¼ tan1 ðus3 =us4 Þ;

Mðxt Þ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2t1 þ u2t2 ;

Mðyt Þ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2t3 þ u2t4 ;

Pðxt Þ ¼ tan1 ðut1 =ut2 Þ;

Pðyt Þ ¼ tan1 ðut3 =ut4 Þ:

and

Then we can get module-phase matrixes as

HðYÞ ¼ ðMðxs Þ; Pðxs Þ; Mðys Þ; Pðys ÞÞT and

HðXÞ ¼ ðMðxt Þ; Pðxt Þ; Mðyt Þ; Pðyt ÞÞT : The error matrix is e = H(Y)  H(X), where Y_ and X_ present systems (10) and corresponding parts in system (7). For convenience, define the following equations

8 as1 > > > a s3 > > : as4 and

¼ aðus3  us1 Þ þ rðL1 Þ ¼ aðus3  us1 Þ þ us7 þ imðL1 Þ ¼ cus1  us3  us1 us5 þ rðL2 Þ ¼ cus2  us4  us2 us5 þ us7 þ imðL2 Þ

;

ð11Þ

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Fig. 3. The error diagrams of M-P synchronization after modified complex projection: (a) e(M(x)) after modified complex projection, (b) e(M(y)) after modified complex projection, (c) e(P(x)) after modified complex projection, (d) e(P(y)) after modified complex projection.

8 s1 as1 s2 as2 A ¼ uMðx þ uMðx > sÞ sÞ > > > > < B ¼  us2 as12 þ us1 as22 Mðx Þ Mðx Þ s

s

s3 as3 s3 as3 > C ¼ uMðy þ uMðy > > sÞ sÞ > > u a : D ¼  s4 s3 þ us3 as4

Mðys Þ2

ð12Þ

;

Mðys Þ2

then we can get the following theorem. Theorem 2. Design controllers as

8 ut1 A s ÞMðxt ÞÞ > L ¼ ut1 ðMðx  ut2 ðPðxs Þ  Pðxt ÞÞ þ Mðx  ut2 B  ðaut3  aut1 Þ  rðL1 Þ > Mðxt Þ tÞ > 51 > > > < L52 ¼ ut2 ðMðxs ÞMðxt ÞÞ þ ut1 ðPðxs Þ  Pðxt ÞÞ þ ut2 A þ ut1 B  ðus7 þ aut4  aut2 Þ  imðL1 Þ Mðxt Þ Mðxt Þ ut3 C s ÞMðyt ÞÞ > L61 ¼ ut3 ðMðy  ut4 ðPðys Þ  Pðyt ÞÞ þ Mðx  ut4 D  ðcut1  ut3  ut1 ut5 Þ  rðL2 Þ > > Mðyt Þ tÞ > > > u ðMðy ÞMðy ÞÞ u C : L ¼ t4 s t t4 þ ut3 ðPðys Þ  Pðyt ÞÞ þ Mðx þ ut3 D  ðus7 þ cut2  ut4  ut2 us5 Þ  imðL2 Þ 62 Mðy Þ tÞ

;

t

where L5 = L51 + jL52, L6 = L61 + jL62 and the controlling matrix is L = (L5, L6)T. Then systems (10) and (6) will get module-phase synchronization after modified complex projection about P1, P2, P3 and P4. Proof. Substitute controllers into system (10), then we can get

_ ðYÞðFðxt ; zs ; ws Þ þ LÞ  H _ ðXÞðFðxs ; zs ; ws ÞÞ ¼ e; _ eðtÞ ¼H

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according to our previous research [21], system (10) and system (7) with L1, L2 will get module-phase synchronization. It is proven that systems (7) with controllers L1, L2 will get modified complex projective synchronization with system (6), so systems (10) will get module-phase synchronization after modified complex projection about P1, P2, P3 and P4 at last. h 4. Simulation results and analysis Choose the initial conditions of system (4) as (1 + 2i, 3 + i, 2, 2), plural projective factors are (1 + 2i, 2  i, 2, 3), initial conditions of intermediate system are (6  8i, 5 + 2i, 4, 1), and initial conditions of system (10) are (20 + 18i, 40  16i). The synchronization diagrams of systems (4) and (10) are shown in Fig. 2. The error diagrams of systems (4) and (10) are shown in Fig. 3. From Fig. 2 and Fig. 3, we can see that systems (10) and system (6) get module synchronization at 6 s; plural variables xm and xt get phase synchronization at 8 s and plural variables ym and yt get phase synchronization at 12 s. At last, hyperchaotic complex Lorenz systems get module-phase synchronization after modified complex projection. 5. Conclusions In this paper, we consider plural projective factors and bring forward the concept of modified complex projective synchronization which is particularly useful to study synchronization in complex systems. Based on this kind of synchronization and our previous work about module-phase synchronization, we apply the hybrid synchronization in a hyperchaotic complex Lorenz system and design the controllers step by step. Bringing intermediate system allows us to achieve the hybrid synchronization controllers indirectly and hybrid synchronization not only extend the concept of synchronization in complex systems but also avoid researching synchronization simply by dividing a complex system into a isolated real part system and a isolated imaginary part system. Simulation results show the effectiveness of our proposed method. Acknowledgement This research is supported by the National Natural Science Foundation of China (Nos: 61370145, 61173183, and 60973152), the Doctoral Program Foundation of Institution of Higher Education of China (No: 20070141014), Program for Liaoning Excellent Talents in University (No: LR2012003), the National Natural Science Foundation of Liaoning province (No: 20082165) and the Fundamental Research Funds for the Central Universities (No: DUT12JB06). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

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