Hopf Bifurcation and Control of Lorenz 84 System

Transcription

1 ISSN print), online) International Journal of Nonlinear Science Vol.63) No.,pp.5-3 Hopf Bifurcation and Control of Loren 8 Sstem Xuedi Wang, Kaihua Shi, Yang Zhou Nonlinear Scientific Research Center, Facult of Science Jiangsu Universit, Zhenjiang, Jiangsu, PR China Received 5 March 3, accepted 7 September 3) Abstract: In this paper, we consider Hopf bifurcation in a new Loren 8 sstem when a parameter G =. Firstl, we analse the stabilit of the limit ccle in previous sstem. And then, through designing a control law based on Hopf bifurcation theorem, we achieve the goal to increase the amplitude of the limit ccle. Finall, numerical simulations are provided to verif the feasibilit and effectiveness of the theoretical analsis. Kewords: Hopf bifurcation; the limit ccle; amplitude; stabilit Introduction In the 9s, the phenomenons of bifurcation were found since Van der Pol made the research on the tube oscillations and imitated the cardiac impulse of a person. After that, the research of bifurcation is mainl done in the field of application for a long. In the late of 98s, one of the research branches about Theor of Control is that the theor of dnamical sstems is applied in researching the nonlinear control sstem. The heart of that issue is about bifurcation control in the nonlinear sstem, so the bifurcation research in nonlinear sstem is one of the important issues in the field of nonlinear science. At the same, it has been widel applied in man fields, especiall in controlling science [], dnamics, information science [], biolog [3-6], medical science [7, 8], etc. Especiall in engineering actual application, the amplitude control of the limit ccle produced b Hopf bifurcation has become the focus. Aiming at the amplitude control of the limit ccle, decreasing or increasing the amplitude is full utilied b people. In recent ears, we can find a lot of research results in the amplitude control of the limit ccle [9, ]. Cui. et.al Ge have sstematicall studied the amplitude control of the limit ccle of Langford sstem and obtained effective control results []. Some authors adopted the washout filter aided linear or nonlinear feedback controllers and achieved the amplitude control of the limit ccle of that sstem [, 3]. Zhang analsed the bifurcation of Loren 8 sstem []. After that, Yang and Zhang investigated globall eponentiall attractive set of Loren 8 sstem and its application [5]. In this paper, we make an analsis on the limit ccle of Loren 8 sstem when a parameter G =. Apart from this, we also discuss a change on the amplitude of the limit ccle b designing a controller and also achieve the result that we want. This paper is organied as follows. In the net section, we shortl discuss the equilibrium point of a new Loren 8 sstem. In section 3, stabilit of the limit ccle in the previous sstem is studied. We change successfull the amplitude of the limit ccle through designing a controller in section. In section 5, some numerical simulations are performed to illustrate the analtical results. Statement of the sstem and the equilibrium point A new Loren 8 sstem [] is as follows: ẋ = a + af ẏ = b + G ż = b +. ) Corresponding author. address: shikaihuadeai@63.com,wd959@ujs.edu.cn Copright c World Academic Press, World Academic Union IJNS.3..5/75

2 6 International Journal of Nonlinear Science, Vol.63), No., pp. 5-3 where,, are state variables, a, b, F are positive parameters. In the followings, we mainl discuss when G = in sstem ), i.e. ẋ = a + af ẏ = b ) ż = b + It has an equilibrium pointf,, ), and the Jacobian of sstem ) at F,, ) is J = F a bf. bf F So the characteristic polnomial of the sstem ) at F,, ) It is clear that 3) has the characteristic roots λ + a){[λ F )] + b F } =. 3) λ, = F ± ib, λ 3 = a In [], when the conditions of G = and F = are satisfied, sstem ) occurs the Hopf bifurcation at F,, ). 3 Stabilit of the limit ccle in the previous sstem Coefficient of curvature of limit ccle is an important measurement for determining the stabilit of limit ccle in a sstem. The object in this section is to anale the stabilit of limit ccle in the original sstem. In the following, we decide to move equilibrium point of the sstem) from F,, ) to,, ) through the transformations = + F, =, =, In terms of the new variables, the sstem ) can be rewritten as the following form: ẋ = a ẏ = F ) bf + b ) ż = bf + F ) + b + where the matri of the nonlinear part is Q = Q Q Q 3 = b b + Through the nonlinear part above, we can calculate the characteristic quantit in the followings: g = Q Q + Q + i Q Q Q )) = g = Q + Q + i Q + Q )) = G = Q + Q + i Q Q )) = bi G = Q Q + i Q + Q )) = bi W = λ 3 a ) Q 3 + Q 3 ) = IJNS for contribution: editor@nonlinearscience.org.uk

3 Xuedi Wang, Kaihua Shi, Yang Zhou: Hopf bifurcation and control of Loren 8 sstem 7 W Q 3 = i λ 3 a )) Q 3 i Q 3 ib ) = i + a) G = 8 3 Q Q + 3 Q + 3 Q 3 + i 3 Q Q 3 Q 3 Q 3 )) = Then the coefficient of curvature of limit ccle of sstem ) is σ = Re{ g g i + G W + G + G W ab } = 8 a <, a >, b > ) 5) + 6) From the above 5), we can see that the coefficient of curvature σ <, so the limit ccle of sstem ) is stable. Then, the characteristic roots of sstem ) at F,, ) is a pair of imaginar roots, the real part of these roots is F, namel Re λ, ) = F. Whats more, we derivate F of Reλ, ), then we can get the following result: α ) = Re [ λ, F ) ] = 6) We can put 5) and 6) into the approimate analtic solution of amplitude of the limit ccle r = α ) F F ) 7) σ So that we can obtain the approimate analtic solution of amplitude of the limit ccle of sstem ) 8 a + 6) r = ab F ) 8) when the conditions of F < F = and F F are satisfied. For convenience, we substitute some special values into8) for numerical simulations. So we put a =, b = into 8), then we will get r = F ) 9) Controlling the amplitude of the limit ccle Using the nonlinear feedback controller U = u u u 3 = k ) + k + ) The controlled sstem is ẋ = a + k + ) ẏ = F ) bf + b + k + ) ż = b + F ) + b + ) At the same, the equilibrium point of sstem ) is,,). Now, using the transformation matri = T. where T = Imτ, Reτ, τ 3 ) = τ, τ are the eigenvectors of J corresponding to λ = ib, λ 3 = a respectivel. Then the sstem ) can be changed to the following form, and ẋ = F ) bf F + + ũ ẏ = F ) + bf + + b + ũ ż = a + ũ 3 ) IJNS homepage:

4 8 International Journal of Nonlinear Science, Vol.63), No., pp. 5-3 This is the chosen controller ũ = k + ) ũ = k + ) ũ 3 = Therefore, the controlled sstem ) is simplified to the following normal form when F =. ẋ = b + Q + k + ) ẏ = b + Q + k + ) ) ż = a + Q 3 Q = + where Q = Q = + b Q 3 = are nonlinear terms of the sstem ). In the followings, we calculate the coefficient of curvature of limit ccle of the controlled sstem ). We can know these coefficients as follows: C p ij =, D = k, D = k, D = k, D = k. Then and ψ = W g = g + C C + C + ic C C)) = g = g + C + C + ic + C)) = G = G + C 3 + C3 + ic3 C3)) = bi G = G + C 3 C3 + ic3 + C3)) = bi G = G + 3D + D + D + 3D) + i3d + D D 3D)) = k W = W λ 3 a ) C3 + C) 3 = W = W + i λ 3 a )) C3 C 3 ic) 3 = ib i+a) C3 C3) + W g +g 8 C ic ) + g C3 + C3)i + g 8 C + ic) g 8 C ic) G λ 3a ) C3 + C) 3 + G λ 3 a )+i)c 3 C3 )+G λ 3 a )i )C 3 6+λ 3 a = )) ϕ = 8 3D + D + D + 3D) + W C3 + C3) + 8 C C CC ) + 6 C C + C) CC + C)) λ 3 a ) C3 + C)C C3) λ 3a )C 3 +C3 C3 ) 86+λ 3 a C )) 3 + C3) + C 3 +λ3a)c3 C3 ) 86+λ 3 a )) C3 C3) = k So the coefficient of curvature of limit ccle is The amplitude of the limit ccle is r σ = σ + Reψ) + ϕ = k α ) σ F ) = F ) k 3) The coefficient of curvature of limit ccle σ is an important factor to determine the amplitude r. From the theor of the stabilit, we can gain that the limit ccle is stable when k <. 5 Numerical simulation In uncontrolled sstem ), when a >, b >,the point,, ) is a Hopf bifurcation point. Fig..a-d) The limit ccle of uncontrolled sstem is shown in Fig..d. After adding the controller, in the controlled sstem ),,,) remains a Hopf bifurcation point. Fig..a-d) The limit ccle of controlled sstem is shown in Fig..d. According to the analsis in section, we can obtain the amplitude of the limit ccle is larger than the one which is in the uncontrolled sstem. Fig.3) IJNS for contribution: editor@nonlinearscience.org.uk

5 Xuedi Wang, Kaihua Shi, Yang Zhou: Hopf bifurcation and control of Loren 8 sstem a) The series of t) before controlling b) The series of t) before controlling c) The series of t) before controlling d) The limit ccle of uncontrolled sstem,initial point.7, 3, ) Figure : The uncontrolled sstem a) The series of t) after controlling b) The series of t) after controlling IJNS homepage:

6 3 International Journal of Nonlinear Science, Vol.63), No., pp c) The series of t) after controlling d) d:the limit ccle of controlled ssterm,initial point.5,, ) Figure : The controlled sstem Figure 3: The limit ccles in Hopf bifurcation in original sstem and controlled sstem for k =.. the inner ccle: original sstem; the outer ccle: the controlled sstem. 6 Conclusions In this paper, we have made an analsis on the limit ccle in a new Loren 8 sstem when a parameter. Through designing a nonlinear controller based on Hopf bifurcation theorem, we successfull obtain the conclusion that the amplitude of the limit ccle is enlarged. Numerical simulations have also been provided to show the effectiveness and feasibilit of our theoretical analsis. References [] Zhou Yufei, Jiang Xuedong, Chen Junning. Analsis of comple intermittenc in boost converter from a bifurcation control viewpoint. Sci China Ser F-Inf Sci/Dec, 8):35-9. [] Zhang Wenming, Wang Bohua, Li Haibin, He Minjia, Liu Shuang. Hopf bifurcation control for some nonlinear torsional vibration sstem with delaed feedbacks. In Prodeeding of the 3st Chinese Control Conference, 7):-7. [3] Zhang Qingling, Chang Yugen, Niu Hong, Liu Chao. Stabilit analsis and bifurcation control of generalied phtoplankton-ooplankton sstem. In Proceedings of the Chinese Control and Decision ConferenceCCDC, ): IJNS for contribution: editor@nonlinearscience.org.uk

7 Xuedi Wang, Kaihua Shi, Yang Zhou: Hopf bifurcation and control of Loren 8 sstem 3 [] Yaoao Lan, Qing-Guo Li. Control of Hopf bifurcation in a simple plankton population model with a non-integer eponent of closure. Applied Mathematics and Computation,8):-3. [5] Guodong Zhang, Lulu Zhu, Boshan Chen. Hopf bifurcation and stabilit for a differential-algebraic biological economic sstem. Applied Mathematics and Computation, 7): [6] Meng Xinou, Huo Haifeng, Zhang Xiaobing. Stabilit and global Hopf bifurcation in a delaed food web consisting of a pre and two predators. Commun Nonlinear Sci Numer Simulat, 6): [7] Tailei Zhang, Junli Liu, Zhidong Teng. Stabilit of Hopf bifurcation of a delaed SIRS epidemic model with stage structure. Nonlinear Analsis: Real World Applications, ): [8] Zhiping Wang, Rui Xu. Stabilit and Hopf bifurcation in a viral infection model with nonlinear incidence rate and delaed immune response. Commun Nonlinear Sci Numer Simulat, 7)): [9] Wang Xuedi, Zhang Wenli, Chen Wenwen. Hopf bifurcation analsis and amplitude control of modified Loren sstem. Journal of Jiangsu Universit : Natural Science Edition, 33):-. [] Ouang Kejian, Tang Jiashi, Liang Cuiiang. Amplitude control of limit ccle in a van der Pol-Duffing sstem. Chinese Phsics B, 8)9): [] Cui Yan, Liu Suhua, Ge Xiaoling. Amplitude control of limit ccle from Hopf bifurcation in the Langford sstem. Acta Phs Sin, 6)):--9. [] Liu Suhua, Zhao Chenggang, Tang Jiashi, Yang Xianlin. Hopf bifurcation analsis and amplitude control of limit ccle in the Qi sstem. Journal of Dnamics and Control, 6)8):-5. [3] Zhang Rongan, Li Jifu, Sun Guiling. Hopf bifurcation control of L sstem via washout filter. In Proceedings of the Chinese Control and Decision ConferenceCCDC, ):-3. [] Zhang Meihua. Bifurcation and numerical analsis of Loren-8 sstem. Science Technolog and Engineering, 3)): [5] Yang Hongliang, Zhang Fuchen. Globall eponentiall attractive set of Loren 8 chaotic sstem and its application. Computer Engineering and Applications, 86)):-3. IJNS homepage: