Symplectic synchronization of different chaotic systems
Zheng-Ming Ge
*, Cheng-Hsiung Yang
Department of Mechanical Engineering, National Chiao Tung University, Hsinchu 300, Taiwan, ROC Accepted 29 October 2007
Communicated by Prof. Ji-Huan He
Abstract
In this paper, a new symplectic synchronization of chaotic systems is studied. Traditional generalized synchroniza-tions are special cases of the symplectic synchronization. A sufficient condition is given for the asymptotical stability of the null solution of an error dynamics. The symplectic synchronization may be applied to the design of secure commu-nication. Finally, numerical results are studied for a Quantum-CNN oscillators synchronized with a Ro¨ssler system in three different cases.
2007 Elsevier Ltd. All rights reserved.
1. Introduction
Many approaches have been presented for the synchronization of chaotic systems[2–6]. There are a chaotic master system and either an identical or a different slave system. Our goal is the synchronization of the chaotic master and the chaotic slave by coupling or by other methods.
Among many kinds of synchronizations[7], generalized synchronization is investigated[8–12]. There exists a func-tional relationship between the states of the master and that of the slave. In this paper, a new synchronization
y¼ H ðx; y; tÞ þ F ðtÞ ð1Þ
is studied, where x, y are the state vectors of the ‘‘master’’ and of the ‘‘slave’’, respectively, F(t) is a given function of time in different form, such as a regular or a chaotic function. When H(x, y, t) = x, Eq.(1)reduces to the generalized synchronization given in[1]. Therefore this paper is an extension of[1].
In Eq.(1), the final desired state y of the ‘‘slave’’ system not only depends upon the ‘‘master’’ system state x but also depends upon the ‘‘slave’’ system state y itself. Therefore the ‘‘slave’’ system is not a traditional pure slave obeying the ‘‘master’’ system completely but plays a role to determine the final desired state of the ‘‘slave’’ system. In other words, it plays an ‘‘interwined’’ role, so we call this kind of synchronization ‘‘symplectic synchronization’’1, and call the ‘‘master’’ system partner A, the ‘‘slave’’ system partner B.
0960-0779/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.10.055
* Corresponding author. Tel.: +886 3 5712121; fax: +886 3 5720634.
E-mail address:[email protected](Z.-M. Ge).
1
The term ‘‘symplectic’’ comes from the Greek for ‘‘interwined’’. H. Weyl first introduced the term in 1939 in his book ‘‘The Classical Groups’’ (p. 165 in both the first edition, 1939, and second edition, 1946, Princeton University Press).
Available online at www.sciencedirect.com
Chaos, Solitons and Fractals 40 (2009) 2532–2543
When H(x, y, t) = H(x, t), Eq.(1)becomes
y¼ Hðx; tÞ þ F ðtÞ ð2Þ
which reduces to generalized synchronization. Therefore generalized synchronization is a special case of the symplectic synchronization. There exists great potential of the application of the symplectic synchronization. For instance, when the symplectically synchronized chaotic signal is used as a signal carrier, the secure communication is more difficult to be deciphered.
As numerical examples, recently developed Quantum Cellular Neural Network (Quantum-CNN) chaotic oscillator is used to synchronize with different systems, respectively. Quantum-CNN oscillator equations are derived from a Schro¨dinger equation taking into account quantum dots cellular automata structures to which in the last decade a wide interest has been devoted, with particular attention towards quantum computing[13].
This paper is organized as follows. In Section2, by the Lyapunov asymptotical stability theorem, a symplectic syn-chronization scheme is given. In Section3, various feedback controllers are designed for the symplectic synchronization of the Quantum-CNN oscillator and a Ro¨ssler system. Numerical simulations are also given in Section3. Finally, some concluding remarks are given in Section4.
2. Symplectic synchronization scheme
There are two different nonlinear chaotic systems. The partner A controls the partner B partially. The partner A is given by
_x¼ f ðxÞ ð3Þ
where x = [x1, x2, . . . , xn]T2 Rnis a state vector and f is a vector function.
The partner B is given by
_y¼ gðyÞ ð4aÞ
where y = [y1, y2, . . . , yn]T2 Rnis a state vector, and g is a vector function different from f.
After a controller u(t) is added, partner B becomes
_y¼ gðyÞ þ uðtÞ ð4bÞ
where u(t) = [u1(t), u2(t), . . . , un(t)]T2 Rnis the control vector.
Our goal is to design the controller u(t) so that the state vector y of the partner B asymptotically approaches H(x, y, t) + F(t), a given function H(x, y, t) plus a given vector function F(t) = [F1(t), F2(t), . . . , Fn(t)]Twhich is a regular
or a chaotic function of time. Define error vector e(t) = [e1, e2, . . . , en]T:
e¼ H ðx; y; tÞ y þ F ðtÞ ð5Þ
lim
t!1e¼ 0 ð6Þ
is demanded.
From Eq.(5), it is obtained that _e¼oH ox _xþ oH oy _yþ oH ot _y þ _F ðtÞ ð7Þ
By Eqs.(3), (4a) and (4b),(7)becomes _e¼oH oxfðxÞ þ oH oygðyÞ þ oH ot gðyÞ uðtÞ þ _F ðtÞ ð8Þ
A positive definite Lyapnuov function V(e) is chosen: VðeÞ ¼1
2e
T
e ð9Þ
Its derivative along any solution of Eq.(8)is _ VðeÞ ¼ eT oH oxfðxÞ þ oH oy gðyÞ þ oH ot gðyÞ þ _F ðtÞ uðtÞ : ð10Þ
In Eq.(10), u(t) is designed so that _V ¼ eTC
nnewhere Cn·nis a diagonal negative definite matrix. _V is a negative
lim
t!1e¼ 0
The symplectic synchronization is obtained[14–16].
3. Numerical results for the symplectic chaos synchronization of Quantum-CNN oscillator and Ro¨ssler System
Case I: A cubic symplectic synchronization
For a two-cell Quantum-CNN, following differential equations are obtained[13]
_x1¼ 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p sin x2 _x2¼ x1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 _x3¼ 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 p sin x4 _x4¼ x2ðx3 x1Þ þ 2a2 x3 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 p cos x4 8 > > > > > > > > < > > > > > > > > : ð11Þ
where x1, x3are polarizations, x2, x4are quantum phase displacements, a1and a2are proportional to the inter-dot
energy inside each cell and x1and x2are the parameters that weigh the effects on the cell of the difference of
polari-zation of the neighboring cells, like the cloning templates in traditional CNNs. When a1= 19.4, a2= 13.1, x1= 9.529
and x2= 7.94, the system is chaotic.
A chaotic Ro¨ssler system is described by _y1¼ y2 y3 _y2¼ y1 ay2þ y4 _y3¼ y1y3þ b _y4¼ cy3þ ry4 8 > > > < > > > : ð12Þ where a = 0.5, b = 0.52, c = 0.5, r = 0.05.
For symplectic synchronization of these two systems, u1, u2, u3and u4are added to the four equations of Eq.(12),
respectively: _y1¼ y2 y3þ u1 _y2¼ y1 ay2þ y4þ u2 _y3¼ y1y3þ b þ u3 _y4¼ cy3þ ry4þ u4 8 > > > < > > > : ð13Þ
The initial values of the states of the Quantum-CNN system and of the Ro¨ssler system are taken as x1(0) = 0.8,
x2(0) =0.77, x3(0) =0.72, x4(0) = 0.57, y1(0) = 0.3, y2(0) =0.4, y3(0) =0.7 and y4(0) = 0.15.
We take F1ðtÞ ¼ x34ðtÞ, F2ðtÞ ¼ x31ðtÞ, F3ðtÞ ¼ x32ðtÞ, and F4ðtÞ ¼ x33ðtÞ. They are chaotic functions of time.
Hiðx; y; tÞ ¼ x2iyiði ¼ 1; 2; 3; 4Þ are given. By Eq.(6)we have
lim t!1ei¼ limt!1ðx 2 iyi yiþ x 3 jÞ ¼ 0; i¼ 1; 2; 3; 4 j ¼ 4; i¼ 1 i 1; i–1 ð14Þ From Eq.(7)we have
_ei¼ 2_xixiyi x 2 i_yi _yiþ 3_xjx2j; i¼ 1; 2; 3; 4 j ¼ 4; i¼ 1 i 1; i–1 ð15Þ Eq.(8)can be expressed as
_e1¼ 2y1x1 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 q sin x2 þ ðy2þ y3Þx 2 1þ y2þ y3 u1 þ 3x2 4 x2ðx3 x1Þ þ 2a2 x3 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 p cos x4 !
_e2¼ 2y2x2 x1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 ! ðy1 ay2þ y4Þx22 y1þ ay2 y4 u2þ 3x21 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 q sin x2 _e3¼ 2y3x3 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 q sin x4 ðy1y3þ bÞx 3 2 y1y3 b u3 þ 3x2 2 x1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 ! _e4¼ 2y4x4 x2ðx3 x1Þ þ 2a2 x3 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 p cos x4 ! ðcy3þ ry4Þx 2 4 cy3 ry4 u4þ 3x23 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 q sin x4 where e1¼ x21y1 y1þ x34, e2¼ x22y2 y2þ x31, e3¼ x23y3 y3þ x32and e4¼ x24y4 y4þ x33.
Choose a positive definite Lyapunov function: Vðe1; e2; e3; e4Þ ¼ 1 2ðe 2 1þ e 2 2þ e 2 3þ e 2 4Þ ð17Þ
Its time derivative along any solution of Eq.(16)is _ V ¼ e1 ( 2y1x1 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 q sin x2 þ ðy2þ y3Þx21þ y2þ y3 þ3x2 4 x2ðx3 x1Þ þ 2a2 x3 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 p cos x4 ! u1 ) þ e2 2y2x2 x1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 ! ðy1 ay2þ y4Þx22 ( y1þ ay2 y4þ 3x 2 1 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 q sin x2 u2 ) þ e3 ( 2y3x3 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 q sin x4 ðy1y3þ bÞx 3 2 y1y3 b þ3x2 2 x1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 ! u3 ) þ e4 2y4x4 x2ðx3 x1Þ þ 2a2 x3 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 p cos x4 ! ðcy3þ ry4Þx 2 4 cy3 ( ry4þ 3x23 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 q sin x4 u4 ) ð18Þ Choose u1¼ 2y1x1 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 q sin x2 þ ðy2þ y3Þx 2 1þ y2þ y3 þ 3x2 4 x2ðx3 x1Þ þ 2a2 x3 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 p cos x4 ! y1x 2 1 y1þ x 3 4 u2¼ 2y2x2 x1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 ! ðy1 ay2þ y4Þx 2 2 y1 y4þ 3x 2 1 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 q sin x2 aðy2x 2 2 x 3 1Þ
u3¼ 2y3x3 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 q sin x4 ðy1y3þ bÞx32 y1y3 b þ 3x2 2 x1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 ! y3x 2 3 y3þ x 3 2 u4¼ 2y4x4 x2ðx3 x1Þ þ 2a2 x3 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 p cos x4 ! ðcy3þ ry4Þx24 cy3 þ 3x2 3 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 q sin x4 rðy4x 2 4þ 2y4 x 3 3Þ Eq.(18)becomes _ V ¼ ðe2 1þ ae 2 2þ e 2 3þ re 2 4Þ < 0 ð19Þ
which is negative definite. The Lyapunov asymptotical stability theorem is satisfied. Cubic symplectic synchronization of the Quantum-CNN system and the Ro¨ssler system is achieved. The numerical results are shown inFig. 1. After 5 s, the motion trajectories enter a chaotic attractor.
Case II: A time delay symplectic synchronization
We take F1(t) = x1(t T), F2(t) = x2(t T), F3(t) = x3(t T) and F4(t) = x4(t T). They are chaotic functions of
time, where time delay T = 1 s is a positive constant. Hiðx; y; tÞ ¼ ðx2i þ yiÞðetþ 2Þ ði ¼ 1; 2; 3; 4Þ are given. By Eq.
(6)we have lim t!1ei¼ limt!1ððx 2 i þ yiÞðe tþ 2Þ y iþ xiðt T ÞÞ ¼ 0; i¼ 1; 2; 3; 4 ð20Þ
From Eq.(7)we have
_ei¼ ð2xi_xiþ _yiÞðetþ 2Þ etðx2i þ yiÞ _yiþ _xiðt T Þ; i¼ 1; 2; 3; 4 ð21Þ Eq.(8)is expressed as _e1¼ 2x1 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 q sin x2 ðetþ 2Þ þ ðy 2 y3Þðetþ 2Þ ðx 2 1þ y1Þet þ y2þ y3 u1 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1ðt T Þ q sin x2ðt T Þ _e2¼ 2x2 x1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 ! ðetþ 2Þ þ ðy 1 ay2þ y4Þðetþ 2Þ ðx2 2þ y2Þet y1þ ay2 y4 u2 x1ðx1ðt T Þ x3ðt T ÞÞ þ 2a1 x1ðt T Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1ðt T Þ p cos x2ðt T Þ _e3¼ 2x3 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 q sin x4 ðetþ 2Þ þ ðy 1y3þ bÞðetþ 2Þ ðx23þ y3Þet y1y3 b u3 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3ðt T Þ q sin x4ðt T Þ _e4¼ 2x4 x2ðx3 x1Þ þ 2a2 x3 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 p cos x4 ! ðetþ 2Þ þ ðcy 3þ ry4Þðetþ 2Þ ðx2 4þ y4Þet cy3 ry4 u4 x2ðx3ðt T Þ x1ðt T ÞÞ þ 2a2 x3ðt T Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3ðt T Þ p cos x4ðt T Þ ð22Þ
where e1¼ ðx21þ y1Þðetþ 2Þ y1þ x1ðt T Þ, e2¼ ðx22þ y2Þðetþ 2Þ y2þ x2ðt T Þ, e3¼ ðx23þ y3Þðetþ 2Þ y3þ
x3ðt T Þ, e4¼ ðx24þ y4Þðetþ 2Þ y4þ x4ðt T Þ.
Choose a positive definite Lyapunov function: Vðe1; e2; e3; e4Þ ¼ 1 2ðe 2 1þ e 2 2þ e 2 3þ e 2 4Þ ð23Þ
0 2 4 6 8 10 12 14 16 18 20 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 x4 y4 F4 H4 x4 ,y4 ,F4 ,H 4 0 2 4 6 8 10 12 14 16 18 20 -1 -0.5 0 0.5 1 1.5 x3 y3 F3 H3 x3 ,y3 ,F 3 ,H 3 0 2 4 6 8 10 12 14 16 18 20 -5 -4 -3 -2 -1 0 1 2 3 4 e1 e2 e3 e4 e1 ,e2 ,e3 ,e4 Time (sec)
Time (sec) Time (sec)
Time (sec) Time (sec)
0 2 4 6 8 10 12 14 16 18 20 -5 -4 -3 -2 -1 0 1 2 x1 y1 F1 H1 x1 ,y1 ,F 1 ,H 1 0 2 4 6 8 10 12 14 16 18 20 -1.5 -1 -0.5 0 0.5 1 1.5 2 x2 y2 F2 H2 x2 ,y2 ,F2 ,H 2
a
b
c
d
e
Its time derivative along any solution of Eq.(22)is _ V ¼ e1 2x1 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 q sin x2 ðetþ 2Þ þ ðy 2 y3Þðetþ 2Þ ðx 2 1þ y1Þet þy2þ y3 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1ðt T Þ q sin x2ðt T Þ u1 þ e2 2x2 x1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 ! ðetþ 2Þ þ ðy 1 ay2þ y4Þðetþ 2Þ ðx 2 2þ y2Þet ( y1þ ay2 y4 x1ðx1ðt T Þ x3ðt T ÞÞ þ 2a1 x1ðt T Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1ðt T Þ p cos x2ðt T Þ u2 ) þ e3 2x3 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 q sin x4 ðetþ 2Þ þ ðy 1y3þ bÞðetþ 2Þ ðx 2 3þ y3Þet y1y3 b 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3ðt T Þ q sin x4ðt T Þ u3 þ e4 2x4 x2ðx3 x1Þ þ 2a2 x3 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 p cos x4 ! ðetþ 2Þ þ ðcy 3þ ry4Þðetþ 2Þ ðx 2 4þ y4Þet ( cy3 ry4 x2ðx3ðt T Þ x1ðt T ÞÞ þ 2a2 x3ðt T Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3ðt T Þ p cos x4ðt T Þ u4 ) ð24Þ Choose u1¼ 2x1 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 q sin x2 ðetþ 2Þ þ ðy 2 y3Þðetþ 2Þ ðx 2 1þ y1Þetþ y2þ y3 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1ðt T Þ q sin x2ðt T Þ þ ðx21þ y1Þðe tþ 2Þ y 1þ x1ðt T Þ u2¼ 2x2 x1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 ! ðetþ 2Þ þ ðy 1 ay2þ y4Þðetþ 2Þ ðx 2 2þ y2Þet y1 y4 x1ðx1ðt T Þ x3ðt T ÞÞ þ 2a1 x1ðt T Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1ðt T Þ p cos x2ðt T Þ þ aððx2 2þ y2Þðe tþ 2Þ þ x 2ðt T ÞÞ u3¼ 2x3 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 q sin x4
ðetþ 2Þ þ ðy1y3þ bÞðe
tþ 2Þ ðx2 3þ y3Þe t y1y3 b 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3ðt T Þ q sin x4ðt T Þ þ ðx23þ y3Þðe tþ 2Þ y 3þ x3ðt T Þ u4¼ 2x4 x2ðx3 x1Þ þ 2a2 x3 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 p cos x4 ! ðetþ 2Þ þ ðcy 3þ ry4Þðetþ 2Þ ðx 2 4þ y4Þet cy3 x2ðx3ðt T Þ x1ðt T ÞÞ þ 2a2 x3ðt T Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3ðt T Þ p cos x4ðt T Þ þ rððx2 4þ y4Þðe tþ 2Þ 2y 4þ x4ðt T ÞÞ Eq.(24)becomes _ V ¼ ðe2 1þ ae 2 2þ e 2 3þ re 2 4Þ < 0 ð25Þ
which is negative definite. The Lyapunov asymptotical stability theorem is satisfied. Time delay symplectic synchroni-zation of the Quantum-CNN system and the Ro¨ssler system is achieved. The numerical results are shown inFig. 2. After 5 s, the motion trajectories enter a chaotic attractor.
Case III: A cubic time delay symplectic synchronization
We take F1(t) = x4(t)x1(t T), F2(t) = x1(t)x2(t T), F3(t) = x2(t)x3(t T) and F4(t) = x3(t)x4(t T), where
T = 1 sec is a positive constant time delay. They are chaotic functions of time. Hiðx; y; tÞ ¼ x3i
ðy3
lim t!1ei¼ limt!1ðx 3 i ðy 3 isin -it 1Þ sin -it yiþ xjxiðt T ÞÞ ¼ 0; i¼ 1; 2; 3; 4; j¼ 4; i¼ 1 i 1; i–1 ð26Þ 0 2 4 6 8 10 12 14 16 18 20 -8 -6 -4 -2 0 2 4 6 x1 y1 F1 H1 x1 ,y1 ,F 1 ,H 1 0 2 4 6 8 10 12 14 16 18 20 -3 -2 -1 0 1 2 3 4 5 6 x2 y2 F2 H2 0 2 4 6 8 10 12 14 16 18 20 -6 -4 -2 0 2 4 6 x4 y4 F4 H4 0 2 4 6 8 10 12 14 16 18 20 -4 -3 -2 -1 0 1 2 3 4 5 6 x3 y3 F3 H3 x3 ,y3 ,F 3 ,H 3 x4 ,y4 ,F 4 ,H 4 x2 ,y2 ,F 2 ,H 2 0 2 4 6 8 10 12 14 16 18 20 -4 -3 -2 -1 0 1 2 3 4 5 6 e1 e2 e3 e4 e1 ,e2 ,e3 ,e4 Time (sec)
Time (sec) Time (sec)
Time (sec) Time (sec)
a
b
c
d
e
From Eq.(7)we have
_ei¼ ð3_xix2i ð3 _yiyi2sin -itþ y3i-icos -itÞ sin -it ðy3i sin -it 1Þ-icos -it _yiþ _xjxiðt T Þ þ xj_xiðt T Þ;
i¼ 1; 2; 3; 4; j¼ 4; i¼ 1 i 1; i–1 ð27Þ Eq.(8)is expressed as _e1¼ 3x21 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 q sin x2 3y2 1ðy2 y3Þ sin 2 -1t y31-1sin 2-1t þ -1cos -1tþ y2þ y3 u1þ x2ðx3 x1Þ þ 2a2 x3 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 p cos x4 ! x1ðt T Þ 2a1x4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1ðt T Þ q sin x2ðt T Þ _e2¼ 3x22 x1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 ! 3y2 2ðy1 ay2þ y4Þ sin 2 -2t y3 2-2sin 2-2tþ -2cos -2t y1þ ay2 y4 u2 2a1x2ðt T Þ ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 q sin x2 þ x1ðx1ðx1ðt T Þ x3ðt T ÞÞ þ 2a1 x1ðt T Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1ðt T Þ p cos x2ðt T Þ _e3¼ 3x23 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 q sin x4 3y2 3ðy1y3þ bÞ sin 2 -3tþ y33-3sin 2-3t þ -3cos -3t y1y3 b u3þ x1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 ! x3ðt T Þ 2a2x2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3ðt T Þ q sin x4ðt T Þ _e4¼ 3x24 x2ðx3 x1Þ þ 2a2 x3 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 p cos x4 ! 3y2 4ðcy3þ ry4Þ sin 2 -4t þ y3 4-4sin 2-4tþ -4cos -4t cy3 ry4 u4 2a2x4ðt T Þ ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 q sin x4 þ x3 x2ðx3ðt T Þ x1ðt T ÞÞ þ 2a2 x3ðt T Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3ðt T Þ p cos x4ðt T Þ ! ð28Þ where e1¼ x31 ðy 3 1sin -1t 1Þ sin -1t y1þ x4ðtÞx1ðt T Þ e2¼ x32 ðy 3 2sin -2t 1Þ sin -2t y2þ x1ðtÞx2ðt T Þ e3¼ x33 ðy 3 3sin -3t 1Þ sin -3t y3þ x2ðtÞx3ðt T Þ e4¼ x34 ðy 3 4sin -4t 1Þ sin -4t y4þ x3ðtÞx4ðt T Þ
Choose a positive definite Lyapunov function: Vðe1; e2; e3; e4Þ ¼ 1 2ðe 2 1þ e 2 2þ e 2 3þ e 2 4Þ ð29Þ
Its time derivative along any solution of Eq.(28)is _ V ¼ e1 3x21 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 q sin x2 3y2 1ðy2 y3Þ sin 2 -1t y31-1sin 2-1tþ -1cos -1tþ y2þ y3 ( þ x2ðx3 x1Þ þ 2a2 x3 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 p cos x4 ! x1ðt T Þ 2a1x4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1ðt T Þ q sin x2ðt T Þ u1 ) þ e2 3x22 x1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 ! 3y2 2ðy1 ay2þ y4Þ sin 2 -2t y32-2sin 2-2t (
þ-2cos -2t y1þ ay2 y4 2a1x2ðt T Þ ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 q sin x2þ x1ðx1ðx1ðt T Þ x3ðt T ÞÞ þ2a1 x1ðt T Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1ðt T Þ p cos x2ðt T Þ u2 ) þ e3 ( 3x2 3 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 q sin x4 3y2 3ðy1y3þ bÞ sin 2 -3tþ y33-3sin 2-3tþ -3cos -3t y1y3 b þ x1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 ! x3ðt T Þ 2a2x2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3ðt T Þ q sin x4ðt T Þ u3 ) þ e4 3x24 x2ðx3 x1Þ þ 2a2 x3 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 p cos x4 ! 3y2 4ðcy3þ ry4Þ sin 2 -4tþ y34-4sin 2-4t ( þ-4cos -4t cy3 ry4 2a2x4ðt T Þ ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 q sin x4þ x3ðx2ðx3ðt T Þ x1ðt T ÞÞ þ2a2 x3ðt T Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3ðt T Þ p cos x4ðt T ÞÞ u4 ) Choose u1¼ 3x21 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 q sin x2 3y2 1ðy2 y3Þ sin 2 -1t y31-1sin 2-1t þ -1cos -1tþ y2þ y3þ x2ðx3 x1Þ þ 2a2 x3 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 p cos x4 ! x1ðt T Þ 2a1x4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1ðt T Þ q sin x2ðt T Þ þ x31 ðy 3 1sin -1t 1Þ sin -1t y1þ x4ðtÞx1ðt T Þ u2¼ 3x22 x1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 ! 3y2 2ðy1 ay2þ y4Þ sin 2 -2t y3 2-2sin 2-2tþ -2cos -2t y1 y4 2a1x2ðt T Þ ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 q sin x2 þ x1ðx1ðx1ðt T Þ x3ðt T ÞÞ þ 2a1 x1ðt T Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1ðt T Þ p cos x2ðt T Þ þ aðx3 2 ðy 3 2sin -2t 1Þ sin -2tþ x1ðtÞx2ðt T ÞÞ u3¼ 3x23 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 q sin x4 3y2 3ðy1y3þ bÞ sin 2 -3t þ y3 3-3sin 2-3tþ -3cos -3t y1y3 b þ x1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 ! x3ðt T Þ 2a2x2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3ðt T Þ q sin x4ðt T Þ þ x33 ðy 3 3sin -3t 1Þ sin -3t y3þ x2ðtÞx3ðt T Þ u4¼ 3x24 x2ðx3 x1Þ þ 2a2 x3 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 p cos x4 ! 3y2 4ðcy3 ry4Þ sin 2 -4t þ y3 4-4sin 2-4tþ -4cos -4t cy3 2a2x4ðt T Þ ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 q sin x4 þ x3 x2ðx3ðt T Þ x1ðt T ÞÞ þ 2a2 x3ðt T Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3ðt T Þ p cos x4ðt T Þ ! þ rðx3 4 ðy 3 4sin -4t 1Þ sin -4t 2y4þ x3ðtÞx4ðt T ÞÞ Eq.(30)becomes _ V ¼ ðe2 1þ ae 2 2þ e 2 3þ re 2 4Þ < 0 ð31Þ
which is negative definite. The Lyapunov asymptotical stability theorem is satisfied. Cubic time delay symplectic syn-chronization of the Quantum-CNN system and the Ro¨ssler system is achieved. The numerical results are shown in
0 2 4 6 8 10 12 14 16 18 20 -4 -2 0 2 4 6 8 10 12 14 x1 y1 F1 H1 x1 , y1 , F1 , H1 0 2 4 6 8 10 12 14 16 18 20 -6 -5 -4 -3 -2 -1 0 1 2 3 4 x4 y4 F4 H4 0 2 4 6 8 10 12 14 16 18 20 -10 -5 0 5 10 15 e1 e2 e3 e4 e1 , e2 , e3 , e4 Time (sec) 0 2 4 6 8 10 12 14 16 18 20 -4 -3 -2 -1 0 1 2 x2 y2 F2 H2 x2 , y2 , F2 , H2 0 2 4 6 8 10 12 14 16 18 20 -3 -2 -1 0 1 2 3 x3 y3 F3 H3 x3 , y3 , F3 , H3 x4 , y4 , F4 , H4
Time (sec) Time (sec)
Time (sec) Time (sec)
a
b
c
d
e
4. Conclusions
A new symplectic synchronization of a Quantum-CNN chaotic oscillator and a Ro¨ssler system is obtained by the Lyapunov asymptotical stability theorem. Two different chaotic dynamical systems, the Quantum-CNN system and the Ro¨ssler system, are in symplectic synchronization for three cases: the cubic symplectic synchronization, the time delay symplectic synchronization and the cubic time delay symplectic synchronization. Symplectic synchronization of chaotic systems can be used to increase the security of secret communication.
Acknowledgement
This research was supported by the National Science Council, Republic of China, under Grant Number 96-2221-E-009-144-MY3.
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