_{1}

^{*}

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.

In 1990, Ott, Grebogi and Yorke presented the OGY method to control chaos [

Liu system [

When

Considering the voltage restraint of practical electronic components, let

Then in the new coordinate system, system (1) will be described as

System (3) can be seemed as a reduced Liu system and the equilibriums are

In _{1}, C_{2}, C_{3} are used as variables. The relevant function can be described as

When we choose R_{1} = 10 kΩ, R_{2} = 20 kΩ, R_{3} = 10 kΩ, R_{4} = 100 kΩ, R_{5} = 10 kΩ, R_{6} = 20 kΩ, R_{7} = 10 kΩ, R_{8} = 80 kΩ, R_{9} = 40 kΩ, R_{10} = 100 kΩ, R_{11} = 10 kΩ, R_{12} = 10 kΩ, R_{13} = 100 kΩ, R_{14} = 10 kΩ, R_{15} = 10 kΩ, R_{16} = 40 kΩ, C_{1} = C_{2} = C_{3} = 1 μF, the circuit system (4) is equivalent to system (3). The supplies of all active devices are ±18 V and the initial voltages of C_{1}, C_{2}, C_{3} are random, we obtain the experiment observations of system (4) as

Comparing

Suppose we want to stabilize Liu system at equilibrium

where

In order to study the relation between

We obtain the above conclusions by numerical calculation. In fact, the accurate range for

Suppose

where

According to Routh-Hurwitz criterion, when

As for the reduced Liu system, it’s easy to obtain the relevant controlled system:

Obviously the above conclusions about

When we choose

Substitute the value of

Choose typical value

From

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.

The work was supported by Doctor Specific Funds of Dalian University.

MingjunWang, (2016) Controlling Liu Chaotic System with Feedback Method and Its Circuit Realization. International Journal of Modern Nonlinear Theory and Application,05,40-47. doi: 10.4236/ijmnta.2016.51004