CONTROLLING LIU SYSTEM WITH DIFFERENT METHODS

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M.-J. Wang & X.-Y. Wang
Fig. 1.
The chaotic attractor of Liu system.
Fig. 2.
The projections of Liu attractor.
2. Description of Liu System Liu system15 is described as ˙ = a(y − x) x y ˙ = bx − kxz . z ˙ = −cz + hx2
1. 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 methods7 – 11 are used to stabilize the unstable periodic orbit of chaotic systems by feeding back their states. Nonfeedback methods11 – 14 are adopted to change chaotic behaviors by applying perturbations to some parameters or variables. In this paper, we use these two methods to control the dynamic behavior of Liu system. With the direct feedback method, Liu system can be stabilized at equilibrium point or limit cycle around its equilibrium. With the adaptive time-delayed feedback method, Liu system can be stabilized at its original unstable periodic orbit. In the nonfeedback method, periodic parametric perturbation with different frequencies can control Liu system to not only periodic motion but also hyperchaos. Numerical simulations show the effectiveness of these methods.
(1)
When a = 10, b = 40, c = 2.5, h = 4 and k = 1, system (1) exhibits a chaotic behavior. Its attractor is shown in Fig. 1. The projections of system (1)’s attractor are shown in Fig. 2. In fact, Liu system is closely related to the generalized Lorenz system family. Next, some results in the existing references will be cited to clarify this point. The following nonlinear system of ordinary differential equations is called the generalized Lorenz system (GLS):16 0 0 0 A 0 a11 a12 x ˙ = z + x1 0 0 −1 x, , (2) A= a21 a22 0 λ3 0 1 0 where x = [x1 , x2 , x3 ]T , λ3 ∈ R, and A has eigenvalues λ1 , λ2 ∈ R, λ1 > 0, λ2,3 < 0. This GLS is said to be nontrivial if it has at least one solution that goes neither to zero nor to infinity nor to a limit cycle.
The following results, clarifying the relation between GLS and GLCF, were presented in Ref. 17: For the nontrivial GLS (2), there exists a nonsingular oneparameter linear transformation of coordinates, z = Tτ x, which takes GLS (2) into GLCF (3) with τ > −1. Liu system has been proven to be a special case of the proposed generalized Lorenz system family.21 Actually, GLCF (3) with τ = −1 is equivalent to Liu system according to Ref. 21. Via the following linear transformation of coordinates, √ x = x√ hk (5) y = y hk , z = kz system (1) with a, b, c, k, h > 0 is equivalent to ˙ x = a(y − x ) ˙ = bx − x z . y ˙ z = −cz + x 2
Modern Physics Letters B, Vol. 23, No. 14 (2009) 1805–1818 c World Scientific Publishing Company
CONTROLLING LIU SYSTEM WITH DIFFERENT METHODS
MINGJUN WANG∗ and XINGYUAN WANG† School of Electronic & Information Engineering, Dalian University of Technology, Dalian 116024, China ∗ wmjhome@ † wangxy@ Received 31 August 2008 Revised 10 September 2008 In this paper, two different kinds of methods are adopted to control Liu system — feedback method and nonfeedback method. On the one hand, direct feedback and adaptive time-delayed feedback are taken as examples for the study of feedback control. In the direct feedback method, Liu system can be stabilized at one equilibrium point or a limit cycle surrounding its equilibrium. In the adaptive time-delayed feedback method, feedback coefficient and delay time can be adjusted adaptively to stabilize Liu system at its original unstable periodic orbit. On the other hand, periodic parametric perturbation is used to control chaos in Liu system as a typical nonfeedback method. By changing the frequency of the perturbation signal, Liu system can be guided to not only periodic motion but also hyperchaos. Numerical simulations show the effectiveness of our methods. Keywords : Liu system; chaotic control; equilibrium point; limit cycle; hyperchaos.
(6)
In the same way, for any λ1 , λ2 , λ3 ∈ R, λ1 > 0, λ2,3 < 0, GLCF (3) with τ = −1 is equivalent to Eq. (6) via λ2 − λ1 x = (z1 − z2 ) λ2 + λ1 λ2 − λ1 λ2 z1 − λ1 z2 (7) y = λ + λ λ + λ 2 1 2 1 z = z3 λ2 − λ1 λ2 + λ1