Controlling Liu Chaotic System with Feedback Method and Its Circuit Realization

In the paper, the Liu system with a feedback controller is discussed. The influence of the feedback coefficient of the controlled system is studied through Lyapunov exponents spectrum and bifurcation diagram. Various attractors are demonstrated not only by numerical simulations but also by circuit experiments. Only one feedback channel is used in our study, which is useful in communication. The circuit experiments show that our study has significance in practical applications.


Introduction
In 1990, Ott, Grebogi and Yorke presented the OGY method to control chaos [1].After their pioneering work, chaotic control has become a focus in nonlinear problems and a lot of work has been done in the field [2]- [4].Nowadays, many methods have been proposed to control chaos [5] [6].Generally speaking, there are two kinds of control ways: feedback control and nonfeedback control.Feedback methods [7]- [11] are used to stabilize the unstable periodic orbit of chaotic systems by feeding back their states.Nonfeedback methods [11]- [14] are adopted to change chaotic behaviors by applying perturbations to some parameters or variables.In the paper, we use feedback method to control the dynamic behavior of Liu system.By adjusting feedback coefficient, Liu system can be stabilized at equilibrium point or limit cycle around its equilibrium.Lyapunov exponents spectrum and bifurcation diagram are adopted to analyze the dynamic behavior of the controlled system.Numerical simulations and circuit experiments show the effectiveness of this method.

The Description of Liu System
Liu system [15] is described as x x y y z z Then in the new coordinate system, system (1) will be described as x a y x y bx kx z z cz hx System (3) can be seemed as a reduced Liu system and the equilibriums are       Comparing Figure 2 and Figure 4, we can know that a reduced Liu system has been realized by circuit experiment.Next, we will add a feedback controller to this circuit to control chaos.Various attractors will be demonstrated not only by numerical simulations but also by the circuit experiment observations.

Feedback Control of Liu System
Suppose we want to stabilize Liu system at equilibrium 1 S and the limit cycle surrounding 1 S respectively.For convenience, choose x as feedback variable, this feedback can be added to any of the three functions of Liu system.Applying the controller to the second function, then the controlled Liu system is described as x a y x where γ is feedback coefficient.
In order to study the relation between γ and system (5)'s behavior, we make the bifurcation diagram of sys- tem (5) with 0 20 γ ≤ ≤ in Figure 5. max X stands for the largest x in every unsteady period or steady period.When system ( 5) is stabilized at fixed point or system (5)'s behavior is periodic, max X has only one value or numbered values with certain γ ; When system (5)'s behavior is chaotic, max X will have numberless values with certain γ .According to the method presented by Ramasubramanian et al. [16], we obtain the Lyapunov exponents spectrum of system (5) with 0 20 γ ≤ ≤ in Figure 6.When the largest Lyapunov exponent 1 0 λ > , system (5)'s behavior is chaotic; When 1 0 λ = , system (5)'s behavior is periodic; When 1 0 λ < , system ( 5) is stabilized at fixed point.From Figure 5 and Figure 6, we have the following conclusions: when 4.9 γ < , system (5) is chaotic (except a very narrow zone near 4.3 γ = , where system (5) may be periodic); when 4.9 16.9 γ ≤ < , system (5) is periodic; when 16.9 γ ≥ , system ( 5) is stabilized at 1 S .We obtain the above conclusions by numerical calculation.In fact, the accurate range for γ to stabilize sys- tem (5) at 1 S can be obtained by theoretical calculation.Substitute the values of parameters and equilibriums, we obtain the Jacobian matrix of system (5) at Suppose λ as eigenvalue, then the characteristic equation of Equation ( 6) is    6) are negative, then system (5) will be stabilized at

Numerical Simulations and Circuit Realization
As for the reduced Liu system, it's easy to obtain controlled system: x a y x Obviously the above conclusions about γ are still available to system (8).Next we will use system (8) for numerical simulations and circuit experiments.The circuit diagram for system (8) is shown in Figure 7.The relevant function can be described as When we choose 17 , all other cognominal electronic components are defined as the above, then circuit system (9) is equivalent to system (8) and we can adjust 19 R to obtain proper feedback coefficient.
Substitute the value of 17 18 9 10 2 1 , , , , , R R R R C V , we have    ), the reduced Liu system is stabilized at ( ) 0.5, 0.5, 4 lastly.These results accord with the conclusions in Section 3.

Conclusion
We study the chaotic control of Liu system with feedback method in the paper.Liu chaotic system and its control are realized not only by numerical simulations but also by circuit experiments.Computer simulation and circuit experiment results show the effectiveness of our method.Moreover, our control needs only one communication channel, which is significant in practical applications.

Figure 1 .
system (1) exhibits a chaotic behavior.Its attractor is shown in The projections of system (1)'s attractor are shown in Figure2.System (1) has three equilibriums: realization of Equation (3) is shown in Figure 3.In Figure 3, the voltages of C 1 , C 2 , C 3 are used as variables.The relevant function can be described as