Recently many chaotic systems’ circuits are designed to generate phenomenon of chaos signals. The ability to synchronize chaotic circuits opens a great number of ways to use them in application signals masking. In this paper, first a new nonlinear chaotic dynamical system had be design, analyze and build circuit. Second, using GYC, partial region stability theory is applied to adaptive control for two identical chaotic systems with uncertain parameters. The results of numerical simulation are performed to verify examples of the proposed nonlinear controllers.
1. Introduction
Chaotic systems provide a rich mechanism for signal design and generation, with potential applications to communications and signal processing. Because chaotic signals are typically broadband, noise-like, and difficult to predict, they can be used in various contexts for masking information bearing waveforms. They can also be used as modulating waveforms in spread spectrum systems.
This can be useful in many practical circumstances like securing communication channels [1], masking signals [2], spreading data sequence, or for generating random signals [3]. Of course, chaos is an unwanted phenomenon in many situations. The essential problem which is faced here is how to distinguish such motion from long transient behavior. A spectrum called Lyapunov exponents (LE) is often used as a quantifier [4] of chaotic motion or regular motion. These real numbers measure the average ratio of the exponential separation of the two neighborhood trajectories. For chaos it is necessary to have at least one positive of Lyapunov exponents; the second one represents a direction of the flow and must converge to zero. The last one must be negative with the largest absolute value, since the flow is dissipative. Circuits of nonlinear dynamic system provide an excellent tool for the study of chaotic behavior. Some of these circuits treat time as a discrete variable, employing sample-and-hold subcircuits and analog multipliers to model iterated maps such as the logistic map [5, 6].
In this paper, an adaptive generalized synchronization of the four-dimensional Lorenz-Stenflo system [7–12] with uncertain chaotic parameters strategy by GYC partial region stability theory is proposed [13–16]. By using the GYC partial region stability theory the Lyapunov function is a simple linear homogeneous function of states and the structure of controllers are simpler and have less simulation error because they are in lower order than that of traditional controllers design.
This paper is organized as follows. In Section 2, the four-dimensional Lorenz-Stenflo system analyzes characteristics of nonlinear dynamic behavior, construction realization circuit of the four-dimensional Lorenz-Stenflo system. In Section 3, adaptive generalized chaos synchronization with uncertain parameters strategy by GCY partial region stability theory is proposed. In Section 4, numerical results of adaptive generalized synchronization of the four-dimensional Lorenz-Stenflo system with uncertain chaotic parameters using GYC partial region stability. Finally, some concluding remarks are given in Section 5.
2. Chaotic Dynamic Analysis of a Four-Dimensional Lorenz-Stenflo System2.1. Phase Portraits and Poincaré Map
The Lorenz-Stenflo system is described as [7, 8]
(1)x˙1=ax2-ax1+bx4x˙2=cx1-x1x3-x2x˙3=x1x2-dx3x˙4=-x1-ax4,
where a is Prandtl number, b is rotation number, c is Rayleigh number, and d is geometric parameter.
The bx4 of Lorenz-Stenflo system from first item changes to third item, new four-dimensional Lorenz-Stenflo system.
The Lorenz-Stenflo system has many dynamical behaviors that are one chaotic motion and six different periodic motions as shown in Figure 1. To have a better understanding of the four-dimensional Lorenz-Stenflo system, we use an innovative technique to present the three-dimensional phase portraits shown in Figure 2. These figures plot three dimensions phase portraits and their projection simultaneously. Thus, we can comprehend the four-dimensional Lorenz-Stenflo system much easier.
Phase portraits and Poincaré maps for (a) chaotic, (b) period-1, (c) period-2, (d) period-4, (e) period-3, (f) period-6, and (g) period-12, respectively, of the four-dimensional Lorenz-Stenflo system.
Three dimensions phase portrait of the four-dimensional Lorenz-Stenflo system and its projection.
2.2. Equilibrium Analysis
The equilibrium points of the system are shown as follows:
(3)E0(0,0,0,0),c-1,b+b2-4a2d+4a2cd2a,c-1,E1(b-b2-4a2d+4a2cd2a,b-b2-4a2d+4a2cd2a,c-1,-b+b2-4a2d+4a2cd2a2,b-b2-4a2d+4a2cd2a,c-1,)E2(b+b2-4a2d+4a2cd2a,b+b2-4a2d+4a2cd2a,c-1,b+b2-4a2d+4a2cd2a2,b+b2-4a2d+4a2cd2a,c-1,).
The parameters of four-dimensional Lorenz-Stenflo system are a=3.7, b=1.5, c=26, and d=0.7, so the equilibrium points become
(4)E0(0,0,0,0)E1(-3.98551,-3.98551,25,1.07716)E2(4.39091,4.39091,25,-1.18673).
For the equilibrium point E0, the Jacobian matrix of the system is
(5)J0=[-aa00c-10000-db-100-a]=[-3.73.70026-10000-0.71.5-100-3.7].
The eigenvalues of the Jacobian matrix below are obvious:
(6)λ1=3.7,λ2=-1,λ3=-0.7,λ4=-3.7.
Then we linearize the system about the nonzero equilibrium points and obtaine the Jacobian matrix:
(7)Jnz=[-aa00c-x3-1-x10x2x1-db-100-a].
The Jnz eigenvalues are
(8)det|Jnz-λI|=[-a-λa00c-x3-1-λ-x10x2x1-d-λb-100-a-λ]=λ4+(2a+d+1)λ3+(a(2+2d+x3-c)+a2+d+x12)λ2+(a2+2ad-a2c+a2d-acd+2ax12+a2x3+abx3+ax1x2)λ+a2(x12+x1x2+dx3)-abx1+a2d-a2cd.
By letting det|Jnz-λI|=0, the eigenvalues of the matrix at each equilibrium point can be obtained as shown in Table 1. The eigenvalues of each equilibrium point are the same. And all the equilibrium points are unstable, since at least one eigenvalue has positive real part for each equilibrium point.
The divergence represents the volume density of the outward flux of a vector field. The divergence of a system implies whether this system dissipative or not. For the four-dimensional Lorenz-Stenflo system, its divergence is
(9)divΦ=∇·Φ=∂Φ∂x1+∂Φ∂x2+∂Φ∂x3+∂Φ∂x4=-a-1-d-a,
where Φ=(x˙1,x˙2,x˙3,x˙4).
The parameters are taken as a=3.7, b=1.5, c=26, and d=0.7; hence, the divergence of the system becomes
(10)divΦ=3.7-1+0.7-3.7=-9.1.
So, (2) is dissipative, with an exponential contraction rate:
(11)dΦdt=e-(2a+d+1).
So, a volume element Π0 is contracted by the flow into a volume element Φ0e-(2a+d+1) in time. All orbits of (2) are ultimately confined to a set having zero volume and asymptotic motion settles onto an attractor. The four-dimensional Lorenz-Stenflo system has bund system.
2.4. Bifurcation Diagram
For numerical analysis, all of the initial conditions are x1(0)=20, x2(0)=20, x3(0)=20, and x4(0)=20, in this and next sections. The bifurcation diagrams of the four-dimensional Lorenz-Stenflo system are plotted in Figure 3(a) to Figure 6(a). The parameters a, b, c, and d in (2) will be varied, respectively, for plotting the bifurcation diagrams. From the bifurcation diagrams, we can learn more about the behaviors of the four-dimensional Lorenz-Stenflo system.
Lyapunov exponents and bifurcation diagrams of the four-dimensional Lorenz-Stenflo system with parameters b=1.5, c=26, and d=0.7.
Lyapunov exponents and bifurcation diagrams of the four-dimensional Lorenz-Stenflo system with parameters a=3.7, c=26, and d=0.7.
Lyapunov exponents and bifurcation diagrams of the four-dimensional Lorenz-Stenflo system with parameters a=3.7, b=1.5, and d=0.7.
Lyapunov exponents and bifurcation diagrams of the four-dimensional Lorenz-Stenflo system with parameters a=3.7, b=1.5, and c=26.
2.5. Lyapunov Exponent and Lyapunov Dimension
The Lyapunov exponent may be used to measure the sensitive dependence on initial conditions. Once the Lyapunov exponent is greater than zero at specific value of parameters, it is said to be chaos, otherwise, periodic solution. The Lyapunov exponents diagrams of the four-dimensional Lorenz-Stenflo system are shown in Figure 3(b) to Figure 6(b). In order to verify the chaotic behavior of the system, Lyapunov exponents diagrams and bifurcation diagrams can be compared together.
The Lyapunov dimension has been proposed by Frederickson et al. [17] to measure the complexity of the attractor. The relation between Lyapunov dimension and Lyapunov exponent is
(12)dL=j+∑i=1jλi|λj+1|,
where λi is the largest Lyapunov exponent and j is defined by the condition as
(13)∑i=1jλi≥0,∑i=1j+1λi<0.
The Lyapunov dimension diagrams are plotted in Figure 7.
Lyapunov dimension diagrams of the four-dimensional Lorenz-Stenflo system.
2.6. Electronic Circuit Implementation
The four-dimensional Lorenz-Stenflo system can be modeled by an electronic circuit [18–21]. The main electronic components in the circuit are integrated amplifiers and inverting amplifiers. In this circuit simulation, we use electronic circuit simulation software named Multisim. All the parameters and initial conditions are the same as numerical simulation. But here we modify the value of capacitors to decrease the output voltage within a range from −12.0 to 12.0, and all the state values reduced tenfold in circuit simulation. The chaotic circuit configuration diagram is shown in Figure 8, and the governing integral equation of the circuit can be written as
(14)x˙1=1R1C1x2-1R2C1x1,x˙2=1R3C2x1-1R4C2x1x3-1R5C2x2,x˙3=1R7C3x1x2+1R6C3x4-1R8C3x3,x˙4=-1R9C4x1-1R10C4x4.
Chaotic circuit of the four-dimensional Lorenz-Stenflo system.
Components of the chaotic system circuit are chosen to be R1=R2=R10=270kΩ, R3=38.461kΩ, R6=667kΩ, R4=R5=R7=R9=1MΩ, R8=1429kΩ, R11=R12=R13=R14=R15=R16=R17=R18=100kΩ, and C1=C2=C3=C4=1μF. The voltage output signals of the chaotic circuit simulated in Multisim software are shown in Figure 9. In addition, the real circuit of the four-dimensional Lorenz-Stenflo system was constructed on a breadboard shown in Figure 13. The apparatus and the components used in the circuit are listed in Table 2. The experimental setup is shown in Figure 11 and the experimental results are shown in Figure 12.
List of apparatus and components in the circuit experiment.
Apparatus/Component
Model
Quantity
Multiplier
AD633
2
OP-Amplifier
LF412
4
Capacitor
2000 pF~0.47 uF
4
Resister
1 k~1.2 M
19
270 kΩ
3
38.461 kΩ
1
667 kΩ
1
1429 kΩ
1
100 kΩ
8
1 MΩ
4
Variable resistor
100 k
2
DC power supply
GPS-3303
1
Oscilloscope graphics for voltage signals outputs of the four-dimensional Lorenz-Stenflo system: (a) Vx1-Vx2, (b) Vx1-Vx3, (c) Vx1-Vx4, (d) Vx2-Vx3, (e) Vx2-Vx4, and (f) Vx3-Vx4.
In order to have a better concept about chaotic motions and periodic motions, we present a four-dimensional diagram, which is a combination of the phase portraits of the circuits simulation and the bifurcation diagram of variable parameter a, plotted in Figure 10.
Bifurcation diagram with phase portraits of the four-dimensional Lorenz-Stenflo system.
Experimental setup for realization of the four-dimensional Lorenz-Stenflo system.
Realization and simulation circuit compare results for voltage signals outputs of the four-dimensional Lorenz-Stenflo system: (a) Vx1-Vx2, (b) Vx1-Vx3, (c) Vx1-Vx4, (d) Vx2-Vx3, (e) Vx2-Vx4, and (f) Vx3-Vx4.
Realization circuit of the four-dimensional Lorenz-Stenflo system.
3. Adaptive Generalized Synchronization of the Four-Dimensional Lorenz-Stenflo System with Uncertain Chaotic Parameters via GYC Partial Region Stability Theory Strategy
Consider the master system
(15)x˙=f(t,x,A(t)),
and the slave system
(16)y˙=f(t,y,A^(t))+u,
where x=[x1,x2,…,xn]T∈Rn and y=[y1,y2,…,yn]T∈Rn denote the master state vector and slave state vector, respectively, f is nonlinear vector functions, A(t) is uncertain chaotic parameter in f, A^(t) is estimates of uncertain chaotic parameter in f, and u=[u1,u2,…,un]T∈Rn is a control input vector.
Our goal is to design a controller u(t) so that the state vector of the slave system (16) asymptotically approaches the state vector of the master system (15) plus a given vector function F(t)=[F1(t),F2(t),…,Fn(t)]T which is either a regular or a chaotic function of time, and finally the synchronization will be accomplished in the sense that the limit of the error vector e(t)=[e1,e2,…,en]T approaches zero:
(17)limt→∞e(t)=0,
where ei=xi-yi+Fi(t)+Kei,i=1,2,…,n. The Kei are positive constants which make the error dynamics always happens in the first quadrant.
From (17) we have
(18)e˙i=x˙i-y˙i+F˙i(t),i=1,2,…,n.
Introducing (15) and (16) in (18), we get
(19)e˙=f(t,x,A(t))-f(t,y,A^(t))+F˙(t)-u(t).
A Lyapunov function V(e,A~) is chosen as a positive definite function in the first quadrant.
Consider(20)V(e,A~)=e+A~,
where A~j=Aj(t)-A^j(t)+Kpi,(j=1,2,…,m). The Kpi are positive constants which make the update laws always happen in the first quadrant.
Its derivative along the solution of (20) is
(21)V˙(e,A~)=f(t,x,A(t))-f(t,y,A^(t))+F˙(t)-u(t)+A~˙,
where u(t) and A~˙ are chosen so that V˙=Ce+DA~,C and D are negative constants, and V˙ is a negative definite function of e1,e2,…,en and A~1,A~2,…,A~m. When
(22)limt→∞e=0,limt→∞A~=0,
the generalized synchronization is obtained.
By using the GYC partial region stability theory, the Lyapunov function is easier to find, since the terms of first degree can be used to construct the definite Lyapunov function and the controller can be designed in lower order.
4. Numerical Results of Adaptive Generalized Synchronization of the Four-Dimensional Lorenz-Stenflo System with Uncertain Chaotic Parameters via GYC Partial Region Stability Theory
The master system is
(23)x˙1=A1(t)(x2-x1)x˙2=A3(t)x1-x1x3-x2x˙3=A2(t)x3-A4(t)x3+x1x2x˙4=-A1(t)x4-x1,
where A1(t),A2(t),A3(t), and A4(t) are uncertain chaotic parameters. Equation (23) is called four-dimensional Lorenz-Stenflo system with uncertain chaotic parameters.
Consider
(24)A1(t)=a(1+f1z1)A2(t)=b(1+f2z2)A3(t)=c(1+f3z3)A4(t)=d(1+f4z4),
where f1,f2,f3, and f4 are arbitrary positive constant and f1=f2=f3=f4=0.005.
The chaotic signals system is
(25)z˙1=-z2z3+az1+bz4z˙2=z1z3+cz2z˙3=13z1z2+dz3z˙4=z2z3+rz4,
where a=3.7, b=1.5, c=26, and d=0.7.
The slave system is
(26)y˙1=A^1(y2-y1)y˙2=A^3y1-y1y3-y2y˙3=A^2y4-A^4y3+y1y2y˙4=-A^1y4-y1,
where A^1(t),A^2(t),A^3(t), and A^4(t) are estimates of uncertain chaotic parameters, respectively.
The nonlinear controllers, u1,u2,u3, and u4, are added to each equation in (26), respectively.
The given functional system for generalized synchronization is also the four-dimensional Lorenz-Stenflo system but with different initial conditions.
Consider
(28)w˙1=a(w2-w1)w˙2=cw1-w1w3-w2w˙3=bw4-dw3+w1w2w˙4=-aw4-w1.
The initial value of the states of the master system, of the slave system, of the chaotic signals system, and of the given functional system are taken as x1(0)=x2(0)=x3(0)=x4(0)=0.2, y1(0)=y2(0)=y3(0)=y4(0)=0.3, z1(0)=z2(0)=z3(0)=z4(0)=0.7, and w1(0)=w2(0)=w3(0)=w4(0)=0.5, respectively.
The generalized synchronization error functions are
(29)ei=xi-yi+wi+Kei,(i=1,2,3,4).
The addition of constant Kei makes the error states always greater than zero. As Figure 15 shows, we can confirm that the error states are always greater than zero. From (29), we get the following error dynamics:
(30)e˙1=A^1(e2-e1-w2+w1)+A~1(x2-x1)+a(w2-w1)-u1e˙2=A^3(e1-w1-K)+A~3x1+cw1-x1x3-w1w3+(x1+w1-e1+K)×(x3+w3-e3+K)-(e2-K)-u2e˙3=A^2(e4-w4-K)+A~2x4+bw4-A^4(e3-w3-K)-A~4x3-dw3+x1x3+w1w3-(x1+w1-e1+K)×(x2+w2-e2+K)-u3e˙4=-A^1(e4-w4-K)-A~1x4-aw4-(e1-K)-u4,
where e1=x1-y1+w1,e2=x2-y2+w2,e3=x3-y3+w3,e4=x4-y4+w4,A~1=A1-A^1+Kp1, A~2=A2-A^2+Kp2,A~3=A3-A^3+Kp3, and A~4=A4-A^4+Kp4. The addition of constant Kpi(i=1,2,3,4) makes the update laws always greater than zero. As Figure 17 shows, we can confirm that the update laws are always greater than zero.
Choose a Lyapunov function in the form of a positive definite function:
(31)V(e1,e2,e3,e4,A~1,A~2,A~3,A~4)=(e1+e2+e3+e4+A~1+A~2+A~3+A~4)
and its time derivative is
(32)V˙(e1,e2,e3,e4,A~1,A~2,A~3,A~4)=e˙1+e˙2+e˙3+e˙4+a~˙+b~˙+c~˙+d~˙=(A^1(e2-e1-w2+w1)+A~1(x2-x1)+a(w2-w1)-u1A^1)+(A^3(e1-w1-K)+A~3x1+cw1-x1x3-w1w3+(x1+w1-e1+K)(x3+w3-e3+K)-(e2-K)-u2A^3)+(A^2(e4-w4-K)+A~2x4+bw4-A^4(e3-w3-K)-A~4x3-dw3+x1x3+w1w3-(x1+w1-e1+K)(x2+w2-e2+K)-u3A^2)+(-A^1(e4-w4-K)-A~1x4-aw4-(e1-K)-u4)+(-a^˙)+(-b^˙)+(-c^˙)+(-d^˙)+(-r^˙).
We choose the update laws for those uncertain parameters as
(33)A~˙1=A˙1-A^˙1=-A~1e1-A~1A~˙2=A˙2-A^˙2=-A~2e2-A~2A~˙3=A˙3-A^˙3=-A~3e3-A~3A~˙4=A˙4-A^˙4=-A~4e4-A~4.
The initial values of estimates for uncertain parameters are A^1(0)=A^2(0)=A^3(0)=A^4(0)=0. Through (32) and (33), the appropriate controllers can be designed as
(34)u1=A^1(e2-e1-w2+w1)+A~1(x2-x1-x4-e1)+a(w2-w1)+e1u2=A^3(e1-w1-K)+A~2(x4-e2)+cw1-x1x3-w1w3+(x1-e1+w1+K)(x3-e3+w3+K)+e2u3=A^2(e4-w4-K)+bw4-A^4(e3-w3-K)-dw3+A~3(x1-e3)+x1x2+w1w2-(x1-e1+w1+K)(x2-e2+w2+K)+e3u4=-A^1(e4-w4-K)-aw4+A~4(-x3-e4)-(e1-K)+e4.
Substituting (33) and (34) into (32), we obtain
(35)V˙=-e1-e2-e3-e4-A~1-A~2-A~3-A~4<0
which is a negative-definite function of e1,e2,e3,e4,A~1,A~2,A~3,andA~4. The numerical simulation results are shown in Figures 14, 15, 16, and 17.
Time histories of the master system and slave system.
Time histories of error states.
Time histories of estimated parameters.
Time histories of parameters differences.
5. Conclusions
In summary, the paper has studied the nonlinear dynamical behaviors of the new chaotic attractor, including some basic dynamical properties, bifurcations, Lyapunov exponents, Lyapunov dimension, periodic motion, and routes to chaos motion. Also, the four-dimensional Lorenz-Stenflo system based realization circuit has been constructed to implement various wing butterfly attractors. Furthermore, by electronic circuit implementation of the proposed four-dimensional Lorenz-Stenflo system, it is shown that the chaotic attractors do physically exist. A new strategy, the GYC partial region stability theory, is proposed to achieve adaptive chaos synchronization with uncertain chaotic parameters.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This research was supported by the National Science Council, Republic of China, under Grant no. NSC 102-2221-E-011-034.
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