coupling and pragmatical adaptive tracking
Zheng-Ming Ge and Cheng-Hsiung YangCitation: Chaos: An Interdisciplinary Journal of Nonlinear Science 18, 043129 (2008); doi: 10.1063/1.3049320 View online: http://dx.doi.org/10.1063/1.3049320
View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/18/4?ver=pdfcov Published by the AIP Publishing
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Synchronization of chaotic systems with uncertain chaotic parameters
by linear coupling and pragmatical adaptive tracking
Zheng-Ming Ge and Cheng-Hsiung Yang
Department of Mechanical Engineering, National Chiao Tung University, Hsinchu, Taiwan, Republic of China
共Received 28 January 2008; accepted 24 November 2008; published online 31 December 2008兲 We study the synchronization of general chaotic systems which satisfy the Lipschitz condition only, with uncertain chaotic parameters by linear coupling and pragmatical adaptive tracking. The un-certain parameters of a system vary with time due to aging, environment, and disturbances. A sufficient condition is given for the asymptotical stability of common zero solution of error dynam-ics and parameter update dynamdynam-ics by the Ge–Yu–Chen pragmatical asymptotical stability theorem based on equal probability assumption. Numerical results are studied for a Lorenz system and a quantum cellular neural network oscillator to show the effectiveness of the proposed synchroniza-tion strategy. © 2008 American Institute of Physics.关DOI:10.1063/1.3049320兴
Theoretical and experimental investigations have shown that synchronization, in particular chaos synchroniza-tion, has great potential in a large amount of application areas ranging from secure communications to modeling brain activity. In this paper, we introduce a synchroniza-tion of chaotic systems with uncertain chaotic parameters by linear coupling and pragmatical adaptive tracking. Based on pragmatical stability theorem and Lipschitz condition, some less conservative conditions for determin-ing linear coupldetermin-ing synchronization of general chaotic systems are obtained. Two examples are simulated to il-lustrate the validity of the theoretical analysis.
I. INTRODUCTION
The idea of synchronizing two identical chaotic systems with different initial conditions was introduced by Pecora and Carroll.1Since then there has been particular interest in chaotic synchronization, due to many potential applications in secure communication,2 chemical and biological systems.3,4 There are many control methods to synchronize chaotic systems, such as, linear coupling, for which the implementation is rather easy, adaptive control, impulsive control, sliding mode control, and other methods.5 Most of them are based on the exact knowledge of the system struc-ture and parameters. But in practice, some or all of the sys-tem parameters are uncertain. Moreover these parameters may change from time to time and become chaotic because of chaotic disturbances. For uncertain parameters, a lot of works have proceeded to solve this problem by adaptive synchronization.6–12 In the current scheme of adaptive synchronization,13–15 the traditional Lyapunov stability theo-rem and Barbalat lemma are used to prove that the error vector approaches zero as time approaches infinity. But the question, why the estimated parameters also approach the uncertain parameters, has remained without answer. From the Ge–Yu–Chen 共GYC兲 pragmatical asymptotical stability theorem,16–18the question is strictly answered. In this paper,
the synchronization of general chaotic systems which satisfy the Lipschitz condition only, with unknown parameters which are altered under some chaotic disturbances, by linear coupling and pragmatical adaptive tracking, is studied first.
As numerical examples, the Lorenz system and recently developed quantum cellular neural network共Quantum-CNN兲 chaotic oscillator are used. Pragmatical adaptive tracking is used to track chaotic parameters in unidirectional coupled systems. Two Lorenz systems and two Quantum-CNN sys-tems by pragmatical adaptive tracking are given as simula-tion examples. Quantum-CNN oscillator equasimula-tions are de-rived from a Schrödinger equation taking into account quantum dots cellular automata structures to which in the last decade a wide interest has been devoted with particular at-tention towards quantum computing.19–21
This paper is organized as follows: In Sec. II, by prag-matical asymptotical stability theorem and by using Lips-chitz conditions, theoretical analysis of synchronization is given. In Sec. III linear feedback controllers are used. By pragmatical adaptive tracking, chaos synchronization of two Lorenz systems and of two Quantum-CNN oscillator systems are achieved by numerical simulations. Conclusions are given in Sec. IV. GYC pragmatical asymptotical stability theorem is presented in the Appendix. Intuitively this theo-rem is different from traditional Lyapunov stability theotheo-rem at that when the points in the neighborhood of zero solution initiating trajectories not approaching zero with time are “not too many,” i.e., in a subset of Lebesque measure 0 in math-ematical language,22 we can neglect their existence, i.e., the zero solution is actually asymptotically stable.
II. STRATEGY OF THE CHAOTIC SYNCHRONIZATION Consider a nonautonomous system in the form as follows:
x˙ = F关t,x,B共t兲兴. 共1兲
The slave system is given by
1054-1500/2008/18共4兲/043129/11/$23.00 18, 043129-1 © 2008 American Institute of Physics
y˙ = F关t,y,Bˆ共t兲兴 + Kˆ共x − y兲, 共2兲
where x =关x1, x2, . . . , xn兴T苸Rn, y =关y1, y2, . . . , yn兴T苸Rn, and
B =关B1, B2, . . . , BM兴T苸RM is a vector of uncertain chaotic coefficients in F, Bˆ =关Bˆ1, Bˆ2, . . . , BˆM兴T苸RM is a vector of estimated coefficients in F, F :⍀傺R+⫻Rn⫻RM→Rn satis-fies Lipschitz conditions储F共t,xI, B兲−F共t,xII, B兲储艋G储xI− xII储, where xI and xII are two neighbor state vectors, and 储F共t,x,B兲−F共t,x,Bˆ兲储艋G储B−Bˆ储 in ⍀ with Lipschitz con-stant G. Kˆ =diag关Kˆ1, . . . , Kˆi, . . . , Kˆn兴 is a constant matrix.
Kˆ 共x−y兲 is the estimated linear coupling term. ⍀ is the
do-main containing the origin. For given共t0, x0, y0, B0兲苸⍀, the solutions关xT共t,t
0, x0, B0兲,yT共t,t0, x0, y0, B0兲兴Tof Eqs.共1兲and
共2兲 exist for t艌t0.
If the synchronization can be accomplished when t→⬁, the limit of the error vector e共t兲=关e1, e2, . . . , en兴T must ap-proach zero, lim t→⬁ e = 0, 共3兲 where e = x − y . 共4兲
From Eqs.共1兲,共2兲, and共4兲, we have
e˙ = x˙ − y˙ , 共5兲
e˙ = F共t,x,B兲 − F共t,x − e,Bˆ兲 − Kˆ共x − y兲. 共6兲
A Lyapunov function V共e,B˜,G˜ 兲 is chosen as a positive
definite function
V共e,B˜,G˜ 兲 =12eTe +1 2B˜
TB˜ +1
2G˜2, 共7兲
where G˜ =G−Gˆ; Gˆ is the estimated Lipschitz constant, B˜ = B − Bˆ .
When M = n, the time derivative of V along any solution of the differential equation system consisting of Eq.共6兲 and update differential equations for B˜ and G˜ is
V˙ 共e,B˜,G˜兲 = eT关F共t,x,B兲 − F共t,x − e,B兲 + F共t,x − e,B兲
− F共t,x − e,Bˆ兲 − Kˆe兴 + B˜T
B ˜˙ + G˜ G˜˙
= eT关F共t,x,B兲 − F共t,x − e,B兲 − Kˆe兴 + G˜ G˜˙ + eT关F共t,x − e,B兲 − F共t,x − e,Bˆ兲兴 + B˜TB˜˙ . 共8兲
From the Lipschitz condition,
V˙ 共e,B˜,G˜兲 艋 G储e储2− eTKˆ e + G˜ G˜˙ + eT关F共t,x − e,B兲
− F共t,x − e,Bˆ兲兴 + B˜TB˜˙ . 共9兲 Since
eT关F共t,x − e,B兲 − F共t,x − e,Bˆ兲兴
艋 兩e1兩 · 兩F1共t,x − e,B兲 − F1共t,x − e,Bˆ兲兩
+ ¯ + 兩en兩 · 兩Fn共t,x − e,B兲 − Fn共t,x − e,Bˆ兲兩 共10兲
by Schwarz inequality and Lipschitz condition, it is obtained that
兩e1兩 · 兩F1共t,x − e,B兲 − F1共t,x − e,Bˆ兲兩
+ ¯ + 兩en兩 · 兩Fn共t,x − e,B兲 − Fn共t,x − e,Bˆ兲兩 艋 储e储 · 储F共t,x − e,B兲 − F共t,x − e,Bˆ兲储 艋 G储e储 · 储B˜储.
共11兲 Therefore,
V˙ 共e,B˜,G˜兲 艋 G储e储2− eTKˆ e + G˜ G˜˙ + G储e储 · 储B˜储 + B˜1B˜˙1
+ ¯ + B˜nB˜˙n. 共12兲 Choosing G˜˙ = − eTe, Kˆ = diag关Gˆ + G兴 共13兲 and choosing B ˜˙ 1= − GB˜1储e储/储B˜储, ¯ ,B˜˙n= − GB˜n储e储/储B˜储, 共14兲 we have B ˜T B ˜˙ = − G共B˜12+ ¯ B˜n2兲储e储/储B˜储 = − G储B˜储2·储e储/储B˜储 = − G储e储 · 储B˜储. 共15兲
Introducing Eqs.共15兲and共13兲in Eq.共12兲, we get
V˙ 共e,B˜,G˜兲 艋 G储e储2− diag关Gˆ + G兴储e储2− G˜ 储e储2
+ G储e储 · 储B˜储 − G储e储 · 储B˜储 = − G储e储2
= − G共e12+ ¯ + en2兲, 共16兲
V˙ is a negative semidefinite function of e,B˜,G˜ . By GYC
pragmatical asymptotical stability theorem 共see Appendix兲, the solution e = 0, B˜ =0, G˜ =0 is asymptotically stable, which means that the two coupled systems are synchronized even if different initial conditions are used and the estimation of the parameters is not exact.
When M⫽n, all the other terms in Eq.共9兲 are kept un-changed, and only the last two terms will be reduced as follows. When M⬎n, we put
eT=关e1, . . . ,en,en+1, . . . ,eM兴T, 共17兲 where en+1= en+2=¯ =eM= 0. Then we have
eT关F共t,x − e,B兲 − F共t,x − e,Bˆ兲兴 艋 兩e1兩 · 兩F1共t,x − e,B兲 − F1共t,x − e,Bˆ兲兩 + ¯ + 兩en兩 · 兩Fn共t,x − e,B兲 − Fn共t,x − e,Bˆ兲兩
+兩en+1兩 · 兩Fn+1共t,x − e,B兲 − Fn+1共t,x − e,Bˆ兲兩 + ¯ + 兩eM兩 · 兩FM共t,x − e,B兲 − FM共t,x − e,Bˆ兲兩
艋 G储e储 · 储B˜储. 共18兲
In Eq.共18兲, the last term is obtained by the Schwarz inequal-ity. Similar to Eqs.共14兲and共15兲in which n is substituted by
M, we choose B˜1¯B˜M, then B ˜T B ˜˙ = − G储e储 · 储B˜储 共19兲 is obtained.
Introducing Eqs. 共18兲 and共19兲 in Eq. 共9兲, we can also get, lastly,
V˙ 共e,B˜,G˜兲 艋 − G共e12+ . . . + en2兲. 共20兲
By the same reasoning as when M = n, the solution e = 0, B˜ = 0, G˜ =0 is asymptotically stable.
When M⬍n, we put
Fi共t,x − e,B兲 − Fi共t,x − e,Bˆ兲 = 0, i = M + 1, ... ,n 共21兲
since BM+1,¯Bndoes not exist,
B ˜ M+1= ¯ = B˜n= 0, 共22兲 储B˜储2= B˜ 1 2 + ¯ + B˜M 2 + B˜M+12 + ¯ + B˜n 2 . 共23兲
Then by the Schwarz inequality, we can obtain the same result as Eq.共18兲except that n and M are exchanged. Simi-larly, choose B ˜˙ 1= − GB˜1储e储/储B˜储, ... ,B˜˙M= − GB˜M储e储/储B˜储, 共24兲 B ˜˙ M+1= − GB˜M+1储e储/储B˜储, ... ,B˜˙n= − GB˜n储e储/储B˜储, B ˜TB˜˙ = − G共B˜ 1 2+ ¯ B˜ n 2兲储e储/储B˜储 = − G储B˜储2储e储/储B˜储 = − G储e储 · 储B˜储. 共25兲
Introducing Eq. 共18兲 in which n and M are exchanged and Eq. 共25兲in Eq.共9兲, we can also get lastly
V˙ 共e,B˜,G˜兲 艋 − G共e12+ ¯ + en2兲 = − GeT
e. 共26兲
By the same reasoning as the case M = n, the solution e = 0,
B
˜ =0, G˜ =0 is asymptotically stable.
Remark. In the current scheme of adaptive synchronization,13–15 traditional Lyapunov stability theorem and Barbalat lemma are used to prove the error vector ap-proaches zero, as time apap-proaches infinity. But the question, why the estimated parameters also approach uncertain pa-rameters, remains no answer. By GYC pragmatical asymp-totical stability theorem, the question can be answered
strictly. Moreover, the asymptotical stability is global, see the Appendix.
III. NUMERICAL SIMULATIONS
Case I: Chaotic parameters for the Lorenz system, M ⬍n共2⬍3兲.
The master Lorenz system with uncertain chaotic param-eters is
x˙1= − A1共t兲共x1− x2兲,
x˙2= A2共t兲x1− x2− x1x3,
x˙3= x1x2− A3共t兲x3,
共27兲 where A1共t兲 and A2共t兲 are uncertain parameters, A3共t兲 is the given parameter. In simulation, we take
A1共t兲 =共1 + d1z1兲,
A2共t兲 =␥共1 + d2z2兲, 共28兲
A3共t兲 = b共1 + d3z3兲,
where d1, d2, and d3 are positive constants. The chaotic signals z1, z2, z3, are the states of
z˙1= −1共z1− z2兲,
z˙2=␥1z1− z2− z1z3,
z˙3= z1z2− b1z3,
共29兲 where1= 8,␥1= 27, b1= 3.2, and关z0T兴T =关222兴T.
From Eq.共2兲, the slave Lorenz system is
y˙1= − Aˆ1共t兲共y1− y2兲 + 共Gˆ + G兲共x1− y1兲,
y˙2= Aˆ2共t兲y1− y2− y1y3+共Gˆ + G兲共x2− y2兲,
y˙3= y1y2− A3共t兲y3+共Gˆ + G兲共x3− y3兲,
共30兲
where Aˆ1共t兲 and Aˆ2共t兲 are estimated parameters. The initial condition be关x0Ty0TAˆT0Gˆ0兴T = 关111 000 00 0兴T.
Subtracting Eq. 共30兲 from Eq. 共27兲, we obtain an error dynamics,
e˙1= − A1共t兲共x1− x2兲 + Aˆ1共t兲共y1− y2兲 − 共Gˆ + G兲共x1− y1兲,
e˙2= A2共t兲x1− x2− x1x3− Aˆ2共t兲y1+ y2+ y1y3
−共Gˆ + G兲共x2− y2兲, 共31兲
e˙3= x1x2− A3共t兲x3− y1y2+ A3共t兲y3−共Gˆ + G兲共x3− y3兲, where e1= x1− y1, e2= x2− y2, e3= x3− y3.
Our aim is
lim
t→⬁ei= limt→⬁共xi− yi兲 = 0, i = 1,2,3. 共32兲 Let the adaptive law be
G˜˙ = G˙ − Gˆ˙ = − Gˆ˙ = − eTe. 共33兲
Since G is constant, G˙ =0. Define
A
˜ 共t兲 = 关A˜1共t兲A˜2共t兲兴T, 共34兲
A ˜
1共t兲 = A1共t兲 − Aˆ1共t兲, A˜2共t兲 = A2共t兲 − Aˆ2共t兲, 共35兲 then
A ˜˙
1共t兲 =d1z˙1− Aˆ˙1共t兲, A˜˙2共t兲 =␥d2z˙2− Aˆ˙2共t兲. 共36兲 Choose A˜˙1共t兲 and A˜˙2共t兲 as
A ˜˙
1= − GA˜1储e储/储A˜储, A˜˙2= − GA˜2储e储/储A˜储. 共37兲 A Lyapunov function is given in the form of the positive definite function, V共e1,e2,e3,A˜1,A˜2,G˜ 兲 = 1 2共e1 2 + e22+ e32+ A˜21+ A˜22+ G˜2兲. 共38兲 Its time derivative along any solution of Eqs. 共31兲,共32兲, and
共37兲is
V˙ = e1关− A1共t兲共x1− x2兲 + Aˆ1共t兲共y1− y2兲 − 共Gˆ + G兲共x1− y1兲兴 + e2关A2共t兲x1− x2− x1x3− Aˆ2共t兲y1+ y2+ y1y3−共Gˆ + G兲共x2− y2兲兴
+ e3关x1x2− A3共t兲x3− y1y2+ A3共t兲y3−共Gˆ + G兲共x3− y3兲兴 + A˜1A˜˙1+ A˜2A˜˙2− G˜ Gˆ˙,
V˙ = e1关− A1共t兲共x1− x2兲 + A1共t兲共y1− y2兲 − 共Gˆ + G兲共x1− y1兲兴 + e2关A2共t兲x1− x2− x1x3− A2共t兲y1+ y2+ y1y3−共Gˆ + G兲共x2− y2兲兴
+ e3关x1x2− A3共t兲x3− y1y2+ A3共t兲y3−共Gˆ + G兲共x3− y3兲兴 + A˜1共y1− y2兲e1− A˜2y1e2− G储e储共A˜12+ A˜22兲/储A˜储 − G˜ Gˆ˙,
V˙ 艋 G储e储2−共Gˆ + G兲储e储2+ G储e储储A˜储 − G储e储储A˜储2/储A˜储 − G˜ Gˆ˙.
V˙ can be rewritten as
V˙ 共ei兲 艋 − G储e储2. 共39兲
V˙ is a negative semidefinite function of e,A˜,G˜ . The
Lyapunov asymptotical stability theorem is not satisfied. We cannot obtain that the common origin of error dynamics共31兲, adaptive laws共33兲, and parameter dynamics 共37兲is asymp-totically stable. Now, D is a 3-manifold, n = 6 and the number of error state variables p = 3. When ei= 0,共i=1,2,3兲 and A˜j,
G˜ , 共j=1,2兲 take arbitrary values, V˙=0, so X is a 3-manifold,
m = n − p = 6 − 3 = 3. m + 1⬍n is satisfied. By GYC
pragmati-cal asymptotipragmati-cal stability theorem, error vector e approaches zero and the estimated parameters also approach the uncer-tain parameters. The pragmatical generalized synchroniza-tion is obtained. The equilibrium point ei= A˜j= G˜ =0 共i = 1 , 2 , 3; j = 1 , 2兲 is asymptotically stable. Moreover, the re-sult is global asymptotically stable 共see Appendix兲. The nu-merical results of the time series of states, state errors, pa-rameters, and estimated Lipschitz constant Gˆ are shown in Figs.1and2. The chaos synchronization is accomplished at 0.6 s. Gˆ approaches constant near 0.5 s. The coupling strength required is K = 2G = 39.34.
Case II: Chaotic parameters for the Quantum-CNN sys-tem, M = n共4=4兲.
For a two-cell Quantum-CNN, the following differential equations are obtained:8–20
x˙1= − 2a1
冑1 − x
1 2 sin x2, x˙2= −1共x1− x3兲 + 2a1冑1 − x
x1 1 2cos x2, x˙3= − 2a2冑1 − x
3 2 sin x4, x˙4= −2共x3− x1兲 + 2a2 x3冑1 − x
32cos x4, 共40兲where x1, x3are polarizations, x2, x4are quantum phase
dis--20 -15 -10 -5 0 5 10 15 20 -30 -20 -10 0 10 20 305 10 15 20 25 30 35 40 45 50 x2 x3
FIG. 1. 共Color online兲 Phase portrait for the Lorenz system with= 10,␥ = 28, b = 8/3.
placements, a1and a2are proportional to the interdot energy inside each cell and 1 and 2 are parameters that weigh effects on the cell of the difference of the polarization of neighboring cells, like the cloning templates in traditional
CNNs. When a1= 6.8, a2= 4.3,1= 4.7,2= 3.9, the system is chaotic as shown in Fig.3.
The master Quantum-CNN system with uncertain cha-otic parameters is (a) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -15 -10 -5 0 5 10 15 20 25 30 x2 y2 (b) Time (sec) x2 ,y2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -15 -10 -5 0 5 10 15 20 25 x1 y1 x1 ,y1 Time (sec) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 5 10 15 20 25 30 35 40 45 50 x3 y3 (c) Time (sec) x3 ,y3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -1 0 1 2 3 4 5 6 7 8 9 e1 e2 e3 (d) Time (sec) e1 ,e2 ,e 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 2 4 6 8 10 12 14 16 18 20 (f) Time (sec) 30 (e) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 5 10 15 20 25 A1 A2 a r A A1 ,A 2 ,A 3 , 1 , 2 , 3 1 A2 Â1 Â2 Time (sec)
FIG. 2.共Color online兲 Time series of states, state errors, A1, A2, A3, Aˆ1, Aˆ2, Aˆ3and estimated Lipschitz constant Gˆ for Case I.
x˙1= − 2A1共t兲冑1 − x12sin x2, x˙2= − A3共t兲共x1− x3兲 + 2A1共t兲 x1
冑1 − x
12 cos x2, x˙3= − 2A2共t兲冑1 − x3 2 sin x4, x˙4= − A4共t兲共x3− x1兲 + 2A2共t兲冑1 − x
x3 3 2cos x4, 共41兲where A1共t兲, A2共t兲, A3共t兲, and A4共t兲 are uncertain parameters 共see Fig.4兲. In simulation, we take
A1共t兲 = a1共1 + d1z1兲, A2共t兲 = a2共1 + d2z2兲,
共42兲
A3共t兲 =1共1 + d3z3兲, A4共t兲 =2共1 + d4z4兲,
where d1, d2, and d3 are positive constants. Take d1= 0.039,
d2= 0.043, d3= 0.045, and d4= 0.038. This system is chaotic as shown in Fig.5.
The chaotic signals z1, z2, z3, z4are the states of
z˙1= − 2a21
冑1 − z
12sin z2, z˙2= −21共z1− z3兲 + 2a21 z1冑1 − z
12 cos z2, z˙3= − 2a22冑1 − z
3 2 sin z4, z˙4= −22共z3− z1兲 + 2a22冑1 − z
z3 3 2cos z4, 共43兲 where a21= 5.2, a22= 4.2,21= 4.7, and22= 3.5. From Eq.共2兲, the slave Quantum-CNN system isy˙1= − 2aˆ1
冑1 − y
12sin y2+共Gˆ + G兲共x1− y1兲,y˙2= −ˆ1共y1− y3兲 + 2aˆ1
y1
冑1 − y
12 cos y2+共Gˆ + G兲共x2− y2兲, y˙3= − 2aˆ2冑1 − y
3 2 sin y4+共Gˆ + G兲共x3− y3兲,y˙4= −ˆ2共y3− y1兲 + 2aˆ2
y3
冑1 − y
32cos y4+共Gˆ + G兲共x4− y4兲. 共44兲Subtracting Eq. 共44兲 from Eq. 共41兲, we obtain an error dy-namics. The initial values are taken as x1共0兲=0.8, x2共0兲= −0.77, x3共0兲=−0.72, x4共0兲=0.57, y1共0兲=−0.2, y2共0兲=0.41, -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 (a) x1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1.5 -1 -0.5 0 0.5 1 1.5 (b) x1 x2 x4 3 x x2 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1.5 -1 -0.5 0 0.5 1 1.5 (d) x2 x3 x4 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1.5 -1 -0.5 0 0.5 1 1.5 (c) x1 x3 x4
FIG. 3. 共Color online兲 Projections of phase portraits for chaotic system共40兲.
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1.5 -1 -0.5 0 0.5 1 1.5 (b) xB1B xB2B x B4 B -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1.5 -1 -0.5 0 0.5 1 1.5 (d) xB2B xB3B x B4 B -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1.5 -1 -0.5 0 0.5 1 1.5 (c) xB1B xB3B x B4 B -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 (a) xB1B xB2B x B3 B
FIG. 4. 共Color online兲 Projections of phase portraits for chaotic system共41兲.
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -1 -0.5 0 0.5 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 (a) z1 z2 z3 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -1 -0.5 0 0.5 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 (b) z1 z2 z4 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -1 -0.5 0 0.5 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 (d) z2 z3 z4 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -1 -0.5 0 0.5 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 (c) z1 z3 z4
FIG. 5. 共Color online兲 Projections of phase portraits for chaotic system共42兲.
y3共0兲=0.25, y4共0兲=−0.81, z1共0兲=0.5, z2共0兲=−0.3, z3共0兲
= 0.1, z4共0兲=0.2, and 关aˆ10aˆ20ˆ10ˆ20Gˆ0兴T=关00 00 0兴T. The er-ror dynamics is
e˙1= − 2A1共t兲冑1 − x12sin x2+ 2aˆ1
冑1 − y
12sin y2 −共Gˆ + G兲e1, e˙2= − A3共t兲共x1− x3兲 + 2A1共t兲 x1冑1 − x
12 cos x2+ˆ1共y1− y3兲 − 2aˆ1 y1冑1 − y
12 cos y2−共Gˆ + G兲e2, 共45兲 e˙3= − 2A2共t兲冑1 − x3 2 sin x4+ 2aˆ2冑1 − y
3 2 sin y4 −共Gˆ + G兲e3, e˙4= − A4共t兲共x3− x1兲 + 2A2共t兲 x3冑1 − x
32cos x4+ˆ2共y3− y1兲 − 2aˆ2 y3冑1 − y
32cos y4−共Gˆ + G兲e4, where e1= x1− y1, e2= x2− y2, e3= x3− y3, e4− x4− y4. Our aim is lim t→⬁ ei= lim t→⬁共xi− yi兲 = 0, i = 1,2,3,4. 共46兲 Let the adaptive law beG˜˙ = G˙ − Gˆ˙ = − Gˆ˙ = − eTe. 共47兲
Since G is constant, G˙ =0. Define
A ˜ 共t兲 = 关a˜1共t兲 a˜2共t兲˜1共t兲˜2共t兲兴T, 共48兲 a ˜1= A1共t兲 − aˆ1, ˜a2= A2共t兲 − aˆ2, 共49兲 ˜1= A3共t兲 −ˆ1, ˜2= A4共t兲 −ˆ2, a ˜˙1= a1d1z˙1− aˆ˙1, ˜˙a2= a2d2z˙2− aˆ˙2, 共50兲 ˜˙1=1d3z˙3−ˆ˙1, ˜˙2=2d4z˙4−ˆ˙2. Choose a˜˙1, a˜˙2,˜˙1, and˜˙2as a ˜˙1= − Ga˜1储e储/储A˜共t兲储, ˜˙1= − G˜1储e储/储A˜共t兲储, 共51兲 a ˜˙2= − Ga˜2储e储/储A˜共t兲储, ˜˙2= − G˜2储e储/储A˜共t兲储.
A Lyapunov function is given in the form of a positive definite function, V共e1,e2,e3,e4,a˜1,a˜2,˜1,˜2,G˜ 兲 =12共e12+ e22+ e32+ e42+ a˜12+ a˜ 2 2+˜ 1 2+˜ 2 2+ G˜2兲. 共52兲 Its time derivative along any solution of Eqs. 共45兲,共47兲, and
共51兲is V˙ = e1关− 2A1共t兲冑1 − x1 2 sin x2+ 2aˆ1
冑1 − y
1 2 sin y2−共Gˆ + G兲e1兴 + e2冋
− A3共t兲共x1− x3兲 + 2A1共t兲 ⫻ x1冑1 − x
12cos x2+ˆ1共y1 − y3兲 − 2aˆ1 y1冑1 − y
12cos y2−共Gˆ + G兲e2册
+ e3关− 2A2共t兲冑1 − x3 2 sin x4+ 2aˆ2冑1 − y
3 2 sin y4−共Gˆ + G兲e3兴 + e4冋
− A4共t兲共x3− x1兲 + 2A2共t兲 x3冑1 − x
32cos x4+ˆ2共y3− y1兲 − 2aˆ2y3
冑1 − y
32cos y4−共Gˆ + G兲e4册
+ a˜1˜˙a1+ a˜2a˜˙2+˜1˜˙1 +˜2˜˙2− G˜ Gˆ˙,V˙ = e1关− 2A1共t兲冑1 − x12sin x2+ 2A1共t兲冑1 − y12sin y2−共Gˆ + G兲e1兴 + e2
冋
− A3共t兲共x1− x3兲 + 2A1共t兲 ⫻冑1 − x
x11 2cos x2+ A3共t兲共y1 − y3兲 − 2A1共t兲
冑1 − y
y1 1 2 cos y2−共Gˆ + G兲e2册
+ e3关− 2A2共t兲冑1 − x3 2sin x 4+ 2A2共t兲冑1 − y32sin y4−共Gˆ + G兲e3兴 + e4冋
− A4共t兲共x3− x1兲 + 2A2共t兲冑1 − x
x3 3 2cos x4+ A4共t兲共y3− y1兲 − 2A2共t兲 y3冑1 − y
32cos y4−共Gˆ + G兲e4册
+ a˜1冋
2冑1 − y12sin y2e1−冑1 − y
2y1 12cos y2e2
册
+˜1关共y1− y3兲e2兴 + a˜2冋
2冑1 − y3 2sin y4e3− 2y3
冑1 − y
32cos y4e4册
+˜2关共y3− y1兲e4兴 − G储e储共a˜12+ a˜22+˜1+˜2兲/储A˜储 − G˜ Gˆ˙,V˙ 艋 G储e储2−共Gˆ + G兲储e储2+ G储e储储A˜储 − G储e储储A˜储2/储A˜储 − G˜ Gˆ˙.
V˙ can be rewritten as
V˙ 艋 − G共e12+ e22+ e32+ e42兲. 共53兲
V˙ is a negative semidefinite function of e,a˜,˜ , G˜ . The
Lyapunov asymptotical stability theorem is not satisfied. We
cannot obtain that the common origin of error dynamics共45兲, adaptive laws 共47兲, and parameter dynamics 共51兲is asymp-totically stable. Now, D is a 5-manifold, n = 9 and the number of error state variables p = 4. When ei= 0,共i=1,2,3,4兲, and
a
˜j,˜j, G˜ , 共i=1,2,3,4; j=1,2兲 take arbitrary values, V˙=0, so
0 1 2 3 4 5 6 7 8 9 10 -1.5 -1 -0.5 0 0.5 1 1.5 2 e1 e2 e3 e4 (e) Time (sec) e ,12 34 e ,e ,e 0 1 2 3 4 5 6 7 8 9 10 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x3 y3 (c) Time (sec) x3 ,y 3 0 1 2 3 4 5 6 7 8 9 10 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 (a) Time (sec) (b) 1 x1 y1 x11 ,y 0 1 2 3 4 5 6 7 8 9 10 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 x2 y2 2 ,y 2 x Time (sec) 1.5 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 3 (g) Time (sec) 0 1 2 3 4 5 6 7 8 9 10 -1.5 -1 -0.5 0 0.5 1 x4 y4 (d) Time (sec) 4 ,y 4 x Time (sec) (f) A1 ,A 2 ,A 3 ,A 4 â1 ,â 2 , 1 , 2 0 1 2 3 4 5 6 7 8 9 10 -1 0 1 2 3 4 5 6 7 8 A1 A2 A3 A4 a1 a2 w1 w2 â1 â2 1 2
FIG. 6. 共Color online兲 Time series of states, state errors, A1, A2, A3, A4, aˆ1, aˆ2, wˆ1, wˆ2and estimated Lipschitz constant Gˆ for Case II.
X is a 5-manifold, m = n − p = 9 − 4 = 5. m + 1⬍n is satisfied.
From the GYC pragmatical asymptotical stability theorem, error vector e approaches zero and the estimated parameters also approach the uncertain parameters. The equilibrium point ei= a˜j=˜j= G˜ =0 共i=1,2,3,4; j=1,2兲 is asymptoti-cally stable. Moreover, the result is global asymptotiasymptoti-cally stable共see Appendix兲. The numerical results of the time se-ries of states, state errors, parameters and estimated Lipschitz constant Gˆ are shown in Fig.6. The chaos synchronization is accomplished near 3 s. Gˆ approaches constant also near 3 s. The coupling strength required is K = 2G = 5.62.
IV. CONCLUSIONS
Using the Lipschitz condition, the synchronization of Lorenz chaotic systems and of Quantum-CNN chaotic oscil-lator systems with uncertain chaotic parameters by linear coupling and pragmatical adaptive tracking are accomplished by the GYC pragmatical asymptotical stability theorem. Tracking uncertain chaotic parameters is first studied in this paper. This is of practical interest, because system param-eters may be varied chaotically due to aging, environment, and disturbances. Two Lorenz systems are synchronized for chaotic parameters M⬍n. Two Quantum-CNN systems are synchronized for chaotic parameters M = n. The simulation results imply that this scheme is very effective. By GYC pragmatical asymptotical stability theorem, the question, why the estimated parameters approach the uncertain param-eters, has been strictly answered and verified by numerical simulations.
ACKNOWLEDGMENTS
This research was supported by the National Science Council, Republic of China, under Grant No. NSC 96-2221-E-009-144-MY3.
APPENDIX: GYC PRAGMATICAL ASYMPTOTICAL STABILITY THEOREM
The stability for many problems in real dynamical sys-tems is actual asymptotical stability, although it may not be mathematical asymptotical stability. The mathematical as-ymptotical stability demands that trajectories from all initial states in the neighborhood of zero solution must approach the origin as t→⬁. If there is only a small part or even a few of the initial states from which the trajectories do not ap-proach the origin as t→⬁, the zero solution is not math-ematically asymptotically stable. If the probability of occur-rence of the event that the trajectories from the initial states are that they do not approach zero when t→⬁, i.e., these trajectories are not asymptotical stable for zero solution, is zero, the stability of zero solution is actual asymptotical sta-bility though it is not mathematical asymptotical stasta-bility. In order to analyze the asymptotical stability of the equilibrium point of such systems, the pragmatical asymptotical stability theorem is used. The conditions for pragmatical asymptotical
stability are more slack than that for traditional Lyapunov asymptotical stability.
Let X and Y be two manifolds of dimensions m and n 共m⬍n兲, respectively, and be a differentiable map from X to Y; then共X兲 is a subset of the Lebesque measure 0 of Y.22 For an autonomous system
x˙ = f共x1, . . . ,xn兲, 共A1兲
where x =关x1, . . . , xn兴T is a state vector, the function f =关f1, . . . , fn兴Tis defined on D傺Rn, an n-manifold.
Let x = 0 be an equilibrium point for the system 共A1兲. Then
f共0兲 = 0. 共A2兲
For a nonautonomous system,
x˙ = f共x1, . . . ,xn+1兲, 共A3兲
where x =关x1, . . . , xn+1兴T, the function f =关f
1, . . . , fn兴T is de-fined on D傺Rn⫻R
+, here t = xn+1傺R+. The equilibrium point is
f共0,xn+1兲 = 0. 共A4兲
Definition. The equilibrium point for the system is
pragmati-cally asymptotipragmati-cally stable provided that with initial points on C which is a subset of the Lebesque measure 0 of D, the behaviors of the corresponding trajectories cannot be deter-mined, while with initial points on D − C, the corresponding trajectories behave as those that agree with traditional as-ymptotical stability.
Theorem: Let V =关x1, x2, . . . , xn兴T: D→R
+ be positive definite and analytic on D, where x1, x2, . . . , xn are all space coordinates such that the derivative of V through Eqs.共A1兲
or 共A3兲, V˙ , is negative semidefinite of 关x1, x2, . . . , xn兴T.
For an autonomous system, let X be the m-manifold con-sisting of a point set for which ∀x⫽0, V˙共x兲=0 and D is an
m-manifold. If m + 1⬍n, then the equilibrium point of the
system is pragmatically asymptotically stable.
For a nonautonomous system, let X be the
m + 1-manifold consisting of the point set for which ∀x⫽0,
V˙ 共x1, x2, . . . , xn兲=0, and D is an n+1-manifold. If m+1+1 ⬍n+1, i.e., m+1⬍n, then the equilibrium point of the sys-tem is pragmatically asymptotically stable. Therefore, for both autonomous and nonautonomous systems, the formula
m + 1⬍n is universal. So the following proof is only for an
autonomous system. The proof for the nonautonomous sys-tem is similar.
Proof: Since every point of X can be passed by a
trajec-tory of Eq.共A1兲, which is one-dimensional, the collection of these trajectories, C, is an 共m+1兲-manifold.16,17
If m + 1⬍n, then the collection C is a subset of Lebesque measure 0 of D. By the above definition, the equilibrium point of the system is pragmatically asymptotically stable.
If an initial point is ergodicly chosen in D, the
probabil-ity of that the initial point falls on the collection C is zero. Here, equal probability is assumed for every point chosen as an initial point in the neighborhood of the equilibrium point. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
Hence, the event that the initial point is chosen from collec-tion C does not actually occur. Therefore, under the equal probability assumption, pragmatical asymptotical stability becomes actual asymptotical stability. When the initial point falls on D − C, V˙ 共x兲⬍0, the corresponding trajectories have as that agree with traditional asymptotical stability be-cause by the existence and uniqueness of the solution of the initial-value problem, these trajectories never meet C.
The Lyapunov function is a positive definite function of
n variables, i.e., p error state variables and n − p = m
differ-ences between unknown and estimated parameters, while V˙
= eTCe is a negative semidefinite function of n variables.
Since the number of error state variables is always more than one, p⬎1, m+1⬍n is always satisfied, by pragmatical as-ymptotical stability theorem we have
lim t→⬁
e = 0 共A5兲
and the estimated parameters approach the uncertain param-eters. Therefore, the equilibrium point of the system is
prag-matically asymptotically stable. Under the equal probability assumption, it is actually asymptotically stable for both error state variables and parameter variables.
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