AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 193138 10.1155/2013/193138 193138 Research Article Enhanced Symplectic Synchronization between Two Different Complex Chaotic Systems with Uncertain Parameters Yang Cheng-Hsiung Akca Haydar Graduate Institute of Automation and Control National Taiwan University of Science and Technology 43 Section 4, Keelung Road, Taipei 106 Taiwan ntust.edu.tw/ 2013 12 5 2013 2013 12 10 2012 13 04 2013 2013 Copyright © 2013 Cheng-Hsiung Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

An enhanced symplectic synchronization of complex chaotic systems with uncertain parameters is studied. The traditional chaos synchronizations are special cases of the enhanced symplectic synchronization. A sufficient condition is given for the asymptotical stability of the null solution of error dynamics. The enhanced symplectic synchronization may be applied to the design of secure communication. Finally, numerical simulations results are performed to verify and illustrate the analytical results.

1. Introduction

A synchronized mechanism that enables a system to maintain a desired dynamical behavior (the goal or target) even when intrinsically chaotic has many applications ranging from biology to engineering . Thus, it is of considerable interest and potential utility to devise control techniques capable of achieving the desired type of behavior in nonlinear and chaotic systems. Many approaches have been presented for the synchronization of chaotic systems . 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.

The symplectic chaos synchronization concept  (1)y=H(t,x,y)+F(t) is studied, where x, y are the state vectors of the master system and of the slave system, respectively, and F(t) is a given function of time in different form. The F(t) may be a regular motion function or a chaotic motion function. When H(t,x,y)+F(t)=x and H(t,x,y)=x, (1) reduces to the generalized chaos synchronization and the traditional chaos synchronization given in , respectively. In this paper, a new enhance symplectic chaos synchronization: (2)y=H(t,x˙,y˙,x,y)+F(t).

As numerical examples, we select hyperchaotic Chen system  and hyperchaotic Lorenz system  as the master system and the slave system, respectively.

This paper is organized as follows. In Section 2, by the Lyapunov asymptotical stability theorem, a symplectic synchronization scheme is given. In Section 3, various feedbacks of nonlinear controllers are designed for the enhanced symplectic synchronization of a hyperchaotic Chen system with uncertain parameters and a hyperchaotic Lorenz system. Numerical simulations are also given in Section 3. Finally, some concluding remarks are given in Section 4.

2. Enhanced Symplectic Synchronization Scheme

There are two different nonlinear chaotic systems. The partner A controls the partner B partially. The partner A is given by (3)x˙=f(t,x,A(t)), where x=[x1,x2,,xn]TRn is a state vector, A(t)=[A1(t),A2(t),,AM(t)]TRM is a vector of uncertain coefficients in f, and f is a vector function.

The partner B is given by(4a)y˙=g(t,y,B(t)), where y=[y1,y2,,yn]TRn is a state vector, B(t)=[B1(t),B2(t),,Bm(t)]TRm is a vector of uncertain coefficients in g, and g is a vector function different from f.

After a controller u(t) is added, partner B becomes (4b)y˙=g(t,y,B(t))+u(t),

where u(t)=[u1(t),u2(t),,un(t)]TRn is 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(t,x˙,y˙,x,y)+F(t), a given function H(t,x˙,y˙,x,y) plus a given vector function F(t)=[F1(t),F2(t),,Fn(t)]T which is a regular or a chaotic function. Define error vector e(t)=[e1,e2,,en]T: (5)e=H(t,x˙,y˙,x,y)-y+F(t),(6)limte=0 is demanded.

From (5), it is obtained that (7)e˙=Ht+HTΨ˙-y˙+F˙(t), where Ψ˙=[x¨y¨x˙y˙]T.

Using (3), (4a), and (4b), (7) can be rewritten as (8)e˙=Ht+Hx˙x¨+Hy˙y¨+Hxf(t,x,A(t))+Hyg(t,y,B(t))-g(t,y,B(t))-u(t)+F˙(t).

Proof.

A positive definite Lyapunov function V(e) is chosen [14, 15] as (9)V(e)=12eTe.

Its derivative along any solution of (8) is (10)V˙(e)=eT{HtHt+Hx˙x¨+Hy˙y¨+HxeT×f(t,x,A(t))+Hyg(t,y,B(t))eT-g(t,y,B(t))+F˙(t)-u(t)Ht}. In (10), the u(t) is designed so that V˙(e)=eTCn×ne, where Cn×n is a diagonal negative definite matrix. The V˙ is a negative definite function of e.

Remark 1.

Note that e approaches zero when time approaches infinitly, according to Lyapunov theorem of asymptotical stability. The enhanced symplectic synchronization is obtained [12, 13, 1619].

3. Numerical Results for the Enhanced Symplectic Chaos Synchronization of Chen System with Uncertain Parameters and Hyperchaotic Lorenz System

To further illustrate the effectiveness of the controller, we select hyperchaotic Chen system and hyperchaotic Lorenz system as the master system and the slave system, respectively. Consider (11)x˙1=a(x2-x1)+x4,x˙2=dx1+cx2-x1x3,x˙3=-bx3+x1x2,x˙4=rx4+x2x3,(12)y˙1=a1(y2-y1)+y4,y˙2=b1y1-y2-y1y3,y˙3=-c1y3+y1y2,y˙4=d1y4-y1y3, where a,b,c,d,r,a1,b1,c1, and d1 are parameters. The parameters of master system and slave system are chosen as a=31,b=3.5,c=11,d=7.7,r=0.1,a1=11,b1=28,c1=2.8, and d1=1.2.

The controllers u1,u2,u3, and u4 are added to the four equations of (12), respectively as follows: (13)y˙1=a1(y2-y1)+y4+u1,y˙2=b1y1-y2-y1y3+u2,y˙3=-c1y3+y1y2+u3,y˙4=d1y4-y1y3+u4.

The initial values of the states of the Chen system and of the Lorenz system are taken as x1(0)=11,  x2(0)=13,   x3(0)=12,  x4(0)=12,  y1(0)=-11,  y2(0)=-13,  y3(0)=-12, and y4(0)=-12.

Case 1 (a symplectic synchronization).

We take F1(t)=x43(t),  F2(t)=x13(t),F3(t)=x23(t), and F4(t)=x33(t). They are chaotic functions of time. Hi(x,y,t)=-xi2yi(i=1,2,3,4) are given. By (6), we have (14)limtei=limt  (-xi2yi-yi+xj3)=0,i=1,2,3,4;j={4,i=1,i-1,i1. From (7), we have (15)e˙i=-2x˙ixiyi-xi2y˙i-y˙i+3x˙jxj2,i=1,2,3,4;j={4,i=1,i-1,i1. Equation (8) can be expressed as (16)e˙1=-2y1x1[a(x2-x1)+x4]-(1+x12)×[a1(y2-y1)+y4]-u1+3x42(rx4+x2x3),e˙2=-2y2x2(dx1+cx2-x1x3)-(1+x22)×(b1y1-y2-y1y3)-u2+3x12[a(x2-x1)+x4],e˙3=-2y3x3(-bx3+x1x2)-(1+x32)×(-c1y3+y1y2)-u3+3x22(dx1+cx2-x1x3),e˙4=-2y4x4(rx4+x2x3)-(1+x42)×(d1y4-y1y3)-u4+3x32(-bx3+x1x2), where e1=-x12y1-y1+x43, e2=-x22y2-y2+x13, e3=-x32y3-y3+x23, and e4=-x42y4-y4+x33.

Choose a positive definite Lyapunov function as (17)V(e1,e2,e3,e4)=12(e12+e22+e32+e42). Its time derivative along any solution of (16) is (18)V˙=e1{(1+x12)-2y1x1[a(x2-x1)+x4]e1-(1+x12)×[a1(y2-y1)+y4]-u1e1+3x42(rx4+x2x3)(1+x12)}+e2{(1+x12)-2y2x2(dx1+cx2-x1x3)+e2-(1+x22)×(b1y1-y2-y1y3)+e2-u2+3x12[a(x2-x1)+x4](1+x12)}+e3{(1+x12)-2y3x3(-bx3+x1x2)+e3-(1+x32)×(-c1y3+y1y2)-u3+e3+3x22(dx1+cx2-x1x3)(1+x12)}+e4{(1+x12)-2y4x4(rx4+x2x3)+e3-(1+x42)×(d1y4-y1y3)+e3-u4+3x32(-bx3+x1x2)(1+x12)}.

According to (10), we get the controller (19)u1=-2y1x1[a(x2-x1)+x4]+3x42(rx4+x2x3)-(1+x12)×[a1(y2-y1)+y4]+e1,u2=-2y2x2(dx1+cx2-x1x3)+3x12[a(x2-x1)+x4]-(1+x22)×(b1y1-y2-y1y3)+e2,u3=-2y3x3(-bx3+x1x2)+3x22(dx1+cx2-x1x3)+(1+x32)×(c1y3-y1y2)+e3,u4=-2y4x4(rx4+x2x3)+3x32(-bx3+x1x2)-(1+x42)×(d1y4-y1y3)+e4. Equation (18) becomes (20)V˙=-(e12+e22+e32+e42)<0, which is negative definite. The Lyapunov asymptotical stability theorem is satisfied. The symplectic synchronization of the Chen system and the Lorenz system is achieved. The numerical results are shown in Figures 1, 2, and 3. After 1 second, the motion trajectories enter a chaotic attractor.

Projections of phase portrait for master system (11) and slave system (12).

Projections of the phase portrait for chaotic system (13) of Case 1.

Time histories of states, state errors, F1,F2,F3,F4,H1,H2,H3, and H4 for Case 1.

Case 2 (a symplectic synchronization with uncertain parameters).

The master Chen system with uncertain variable parameters is (21)x˙1=a(t)(x2-x1)+x4,x˙2=d(t)x1+c(t)x2-x1x3,x˙3=-b(t)x3+x1x2,x˙4=r(t)x4+x2x3, where a(t),b(t),c(t),d(t), and r(t) are uncertain parameters. In simulation, we take (22)a(t)=a(1+k1sinω1t),b(t)=b(1+k2sinω2t),c(t)=c(1+k3sinω3t),d(t)=d(1+k4sinω4t),r(t)=r(1+k5sinω5t), where k1,k2,k3,k4,k5,ω1,ω2,ω3,ω4, and ω5 are constants. Take k1=0.3,  k2=0.5,  k3=0.2,k4=0.4,  k5=0.6,  ω1=13,  ω2=17,  ω3=19,  ω4=23, and ω5=29. So, (21) is chaotic system, shown in Figure 4.

We take F1(t)=x43(t),F2(t)=x13(t),F3(t)=x23(t), and F4(t)=x33(t). They are chaotic functions of time. Hi(x,y,t)=-xi2yi(i=1,2,3,4) are given. By (6), we have (23)limtei=limt(-xi2yi-yi+xj3)=0,i=1,2,3,4;j={4,i=1,i-1,i1. From (7), we have (24)e˙i=-2x˙ixiyi-xi2y˙i-y˙i+3x˙jxj2,i=1,2,3,4;j={4,i=1,i-1,i1.

Equation (8) can be expressed as (25)e˙1=-2y1x1[a(t)(x2-x1)+x4]-(1+x12)×[a1(y2-y1)+y4+u1]+3x42(r(t)x4+x2x3),e˙2=-2y2x2(d(t)x1+c(t)x2-x1x3)-(1+x22)×(b1y1-y2-y1y3+u2)+3x12[a(t)(x2-x1)+x4],e˙3=-2y3x3(-b(t)x3+x1x2)-(1+x32)×(-c1y3+y1y2+u3)+3x22(d(t)x1+c(t)x2-x1x3),e˙4=-2y4x4(r(t)x4+x2x3)-(1+x42)×(d1y4-y1y3+u4)+3x32(-b(t)x3+x1x2), where e1=-x12y1-y1+x43, e2=-x22y2-y2+x13, e3=-x32y3-y3+x23, and e4=-x42y4-y4+x33.

Choose a positive definite Lyapunov function as (26)V(e1,e2,e3,e4)=12(e12+e22+e32+e42). Its time derivative along any solution of (25) is (27)V˙=e1{1+x12-2y1x1[a(t)(x2-x1)+x4]e1-(1+x12)×[a1(y2-y1)+y4]-u1e1+3x42(r(t)x4+x2x3)1+x12}+e2{1+x12-2y2x2(d(t)x1+c(t)x2-x1x3)e1-(1+x22)×(b1y1-y2-y1y3)-u2e1+3x12[a(t)(x2-x1)+x4]1+x12}+e3{1+x12-2y3x3(-b(t)x3+x1x2)e1-(1+x32)×(-c1y3+y1y2)-u3e1+3x22(d(t)x1+c(t)x2-x1x3)1+x12}+e4{1+x12-2y4x4(r(t)x4+x2x3)e1-(1+x42)×(d1y4-y1y3)e1-u4+3x32(-b(t)x3+x1x2)1+x12}. According to (10), we get the controller (28)u1=-2y1x1[a(t)(x2-x1)+x4]+3x42(r(t)x4+x2x3)-(1+x12)[a1(y2-y1)-y4]+e1,u2=-2y2x2(d(t)x1+c(t)x2-x1x3)+3x12[a(t)(x2-x1)+x4]-(1+x22)(b1y1-y2-y1y3)+e2,u3=-2y3x3(-b(t)x3+x1x2)+3x22(d(t)x1+c(t)x2-x1x3)+(1+x32)(c1y3-y1y2)+e3,u4=-2y4x4(r(t)x4+x2x3)+3x32(-b(t)x3+x1x2)-(1+x42)(d1y4-y1y3)+e4. Equation (27) becomes (29)V˙=-(e12+e22+e32+e42)<0, which is negative definite. The Lyapunov asymptotical stability theorem is satisfied. The symplectic synchronization of the Chen system with uncertain parameters and the Lorenz system is achieved. The numerical results are shown in Figures 5 and 6. After 1 second, the motion trajectories enter a chaotic attractor.

Projections of the phase portrait for chaotic system (21).

Projections of the phase portrait for chaotic system (13) of Case 2.

Time histories of states, state errors, F1,F2,F3,F4,H1,H2,H3, and H4 for Case 2.

Case 3 (an enhanced symplectic synchronization with uncertain parameters).

We take F1(t)=x43(t),  F2(t)=x13(t),F3(t)=x23(t), and F4(t)=x33(t). They are chaotic functions of time. Hi(x˙,y˙,x,y,t)=-xi2yi-x˙-Ky˙(i=1,2,3,4) are given. The K value is 0.0001. By (6), we have (30)limtei=limt(-xi2yi-x˙i-Ky˙i-yi+xj3)=0,i=1,2,3,4;j={4,i=1,i-1,i1. From (7) we have (31)e˙i=-2xix˙iyi-xi2y˙i-x¨i-Ky¨i-y˙i+3x˙jxj2,i=1,2,3,4;j={4,i=1,i-1,i1. Equation (8) can be expressed as (32)e˙1=-2y1x1[a(t)(x2-x1)+x4]-(1+x12)×[a1(y2-y1)+y4+u1]-x¨1-Ky¨1+3x42(r(t)x4+x2x3),e˙2=-2y2x2(d(t)x1+c(t)x2-x1x3)-(1+x22)×(b1y1-y2-y1y3+u2)-x¨2-Ky¨2+3x12[a(t)(x2-x1)+x4],e˙3=-2y3x3(-b(t)x3+x1x2)-(1+x32)×(-c1y3+y1y2+u3)-x¨3-Ky¨3+3x22(d(t)x1+c(t)x2-x1x3),e˙4=-2y4x4(r(t)x4+x2x3)-(1+x42)×(d1y4-y1y3+u4)-x¨4-Ky¨4+3x32(-b(t)x3+x1x2), where e1=-x12y1-y1-x˙1-Ky˙1+x43, e2=-x22y2-y2-x˙2-Ky˙2+x13, e3=-x32y3-y3-x˙3-Ky˙3+x23, and e4=-x42y4-y4-x˙4-Ky˙4+x33.

Choose a positive definite Lyapunov function as (33)V(e1,e2,e3,e4)=12(e12+e22+e32+e42). Its time derivative along any solution of (32) is (34)V˙=e1{1+x12-2y1x1[a(t)(x2-x1)+x4]e1-(1+x12)×[a1(y2-y1)+y4]+u1e1-x¨1-Ky¨1+3x42(r(t)x4+x2x3)1+x12}+e2{1+x12-2y2x2(d(t)x1+c(t)x2-x1x3)e1-(1+x22)×(b1y1-y2-y1y3)+u2e1-x¨2-Ky¨2+3x12[a(t)(x2-x1)+x4]1+x12}+e3{1+x12-2y3x3(-b(t)x3+x1x2)e1-(1+x32)×(-c1y3+y1y2)+u3e1-x¨3-Ky¨3+3x22(d(t)x1+c(t)x2-x1x3)1+x12}+e4{1+x12-2y4x4(r(t)x4+x2x3)e1-(1+x42)×(d1y4-y1y3)+u4e1-x¨4-Ky¨4+3x32(-b(t)x3+x1x2)1+x12}.

According to (10), we get the controller (35)u1=-2y1x1[a(t)(x2-x1)+x4]-x¨1-Ky¨1+3x42(r(t)x4+x2x3)-(1+x12)[a1(y2-y1)+y4]+e1,u2=-2y2x2(d(t)x1+c(t)x2-x1x3)-x¨2-Ky¨2+3x12[a(t)(x2-x1)+x4]-(1+x22)(b1y1-y2-y1y3)+e2,u3=-2y3x3(-b(t)x3+x1x2)-x¨3-Ky¨3+3x22(d(t)x1+c(t)x2-x1x3)+(1+x32)(c1y3-y1y2)+e3,u4=-2y4x4(r(t)x4+x2x3)-x¨4-Ky¨4+3x32(-b(t)x3+x1x2)-(1+x42)(d1y4-y1y3)+e4. Equation (34) becomes (36)V˙=-(e12+e22+e32+e42)<0, which is negative definite. The Lyapunov asymptotical stability theorem is satisfied. The enhanced symplectic synchronization of the Chen system with uncertain parameters and the Lorenz system is achieved. The numerical results are shown in Figures 7 and 8. After 1 second, the motion trajectories enter a chaotic attractor.

Projections of the phase portrait for chaotic system (13) of Case 3.

Time histories of states, state errors, F1,F2,F3,F4,H1,H2,H3, and H4 for Case 3.

4. Conclusions

We achieve the novel enhanced symplectic synchronization of a Chen system with uncertain parameters, and a Lorenz system is obtained by the Lyapunov asymptotical stability theorem. All the theoretical results are verified by numerical simulations to demonstrate the effectiveness of the three cases of proposed synchronization schemes. The enhanced symplectic synchronization of chaotic systems with uncertain parameters can be used to increase the security of secret communication.

Acknowledgment

This research was supported by the National Science Council, Taiwan, under Grant no. 98-2218-E-011-010.

Ge Z. M. Yang C. H. The generalized synchronization of a Quantum-CNN chaotic oscillator with different order systems Chaos, Solitons and Fractals 2008 35 5 980 990 2-s2.0-35248868404 10.1016/j.chaos.2006.05.090 Ge Z.-M. Yang C.-H. Synchronization of complex chaotic systems in series expansion form Chaos, Solitons and Fractals 2007 34 5 1649 1658 10.1016/j.chaos.2006.04.072 MR2335412 ZBL1152.37314 Pecora L. M. Carroll T. L. Synchronization in chaotic systems Physical Review Letters 1990 64 8 821 824 10.1103/PhysRevLett.64.821 MR1038263 ZBL0938.37019 Ge Z. M. Leu W. Y. Anti-control of chaos of two-degrees-of-freedom loudspeaker system and chaos synchronization of different order systems Chaos, Solitons and Fractals 2004 20 3 503 521 2-s2.0-0242334262 10.1016/j.chaos.2003.07.001 Femat R. Alvarez-Ramírez J. Fernández-Anaya G. Adaptive synchronization of high-order chaotic systems: a feedback with lower-order parametrization Physica D 2000 139 3-4 231 246 10.1016/S0167-2789(99)00226-2 MR1753083 Ge Z. M. Chang C. M. Chaos synchronization and parameters identification of single time scale brushless DC motors Chaos, Solitons and Fractals 2004 20 4 883 903 2-s2.0-0344924833 10.1016/j.chaos.2003.10.005 An Z. Zhu H. Li X. Xu C. Xu Y. Li X. Nonidentical linear pulse-coupled oscillators model with application to time synchronization in wireless sensor networks IEEE Transactions on Industrial Electronics 2011 58 6 2205 2215 2-s2.0-79956267504 10.1109/TIE.2009.2038407 Chen C.-H. Sheu L.-J. Chen H.-K. Chen J.-H. Wang H.-C. Chao Y.-C. Lin Y.-K. A new hyper-chaotic system and its synchronization Nonlinear Analysis. Real World Applications 2009 10 4 2088 2096 10.1016/j.nonrwa.2008.03.015 MR2508418 ZBL1163.65337 Guo X. Wu W. Chen Z. Multiple-complex coefficient-filter-based phase-locked loop and synchronization technique for three-phase grid-interfaced converters in distributed utility networks IEEE Transactions on Industrial Electronics 2011 58 4 1194 1204 2-s2.0-79952659288 10.1109/TIE.2010.2041738 Huang Y. Wang Y. W. Xiao J. W. Generalized lag-synchronization of continuous chaotic system Chaos, Solitons and Fractals 2009 40 2 766 770 2-s2.0-65249092078 10.1016/j.chaos.2007.08.022 Ge Z. M. Yang C. H. Symplectic synchronization of different chaotic systems Chaos, Solitons and Fractals 2009 40 5 2532 2543 2-s2.0-67349228030 10.1016/j.chaos.2007.10.055 Yan Z. Controlling hyperchaos in the new hyperchaotic Chen system Applied Mathematics and Computation 2005 168 2 1239 1250 10.1016/j.amc.2004.10.016 MR2171776 ZBL1160.93384 Jia Q. Projective synchronization of a new hyperchaotic Lorenz system Physics Letters A 2007 370 40 45 Vidyasagar M. Nonlinear System Analysis 1993 2nd Upper Saddle River, NJ, USA Prentice Hall Khalil M. Nonlinear Systems 1996 2nd Upper Saddle River, NJ, USA Prentice Hall Yang C.-H. Li S.-Y. Tsen P.-C. Synchronization of chaotic system with uncertain variable parameters by linear coupling and pragmatical adaptive tracking Nonlinear Dynamics 2012 70 3 2187 2202 10.1007/s11071-012-0609-6 MR2992207 Ge Z.-M. Chen Y.-S. Synchronization of unidirectional coupled chaotic systems via partial stability Chaos, Solitons and Fractals 2004 21 1 101 111 10.1016/j.chaos.2003.10.004 MR2033687 ZBL1048.37027 Chen S. J. Synchronization of an uncertain unified chaotic system via adaptive control Chaos, Solitons and Fractals 2002 14 4 643 647 2-s2.0-0036722179 Yang C.-H. Chen T.-W. Li S.-Y. Chang C.-M. Ge Z.-M. Chaos generalized synchronization of an inertial tachometer with new Mathieu-Van der Pol systems as functional system by GYC partial region stability theory Communications in Nonlinear Science and Numerical Simulation 2012 17 3 1355 1371 10.1016/j.cnsns.2011.07.008 MR2843801 ZBL1250.34044