Hopf Bifurcation and Limit Cycle Analysis of the Rikitake System

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1 ISSN (print), (online) International Journal of Nonlinear Science Vol.4(0) No.,pp.-5 Hopf Bifurcation and Limit Cycle Analysis of the Rikitake System Xuedi Wang, Tianyu Yang, Wei Xu Nonlinear Scientific Research Center, Jiangsu University, Zhenjiang, Jiangsu 03, P.R. China (Received 8 June 0, accepted 5 September 0) Abstract: In this paper we will investigate some important properties of the Rikitake system in a new perspective. Firstly, we use a more convenient way to determine the stability of the positive equilibrium. And then, we study the existence of Hopf bifurcation at the positive equilibrium point, after that,we investigate the stability of periodic orbit which is generated by this point, both of them have not been involved in the past years. And last, we verify the results are effective by numerical simulation. Keywords: Rikitake system; Hopf bifurcation; stability; coefficient of curvature Introduction The Rikitake system (see e.g.[]) has been widely investigated in the past years. This system is a mathematical model obtained from a simple mechanical system used by Rikitake [] to study the reversals of the Earth s magnetic field in a two-disc dynamo model. Among the studied topics related with the Rikitake system, we recall a few of them together with a partial list of references, namely: the stability of the equilibrium points [], the chaotic behavior [], integrals and invariant manifolds [3,4], KCC theory [5], Hamiltonian dynamics [6,7], and many others. In this paper we will use some more convenient ways to study the properties of the Rikitake system and analyze some properties which are not studied in the previous. More exactly, in the third section, we will use Routh-Hurwitz criterion to verify the stability of the positive equilibrium point, which is more convenient than the previous method. Moreover, we exploit the Hurwitz determinant to verify the existence of Hopf bifurcation, and in the fourth section, we take advantage of the coefficient of curvature of limit cycle to judge the stability of limit cycle, both of them have not been studied in the previous. Furthermore, in the fifth section, we verify the results which are established by numerical simulation. Finally, we will present the conclusion in the section sixth. Statement of the system and the equilibrium points Consider the Rikitake system ẋ = µx yz ẏ = y (z a)x ż = xy () (x, y, z) R 3 are the state variables and a > 0, µ > 0 are parameters. Note that system () is a quadratic system in R 3. The choice of the parameters a > 0 and µ > 0 reflects a physical meaning in the Rikitake model. It is well known that (see []) system () has two equilibrium points E = (x 0, y 0, z 0 ), E = ( x 0, y 0, z 0 ) a a x 0 = 4µ, y 0 = a a 4µ, z 0 = a a 4µ () Corresponding author. address: wxd959@ujs.edu.cn Copyright c World Academic Press, World Academic Union IJNS.0.0.5/657

2 International Journal of Nonlinear Science, Vol.4(0), No., pp. -5 In order to study the stability of E ±, it is only sufficient to study the stability of the equilibrium point E. Then, the Jacobian of system () at the equilibrium point E is G(E, µ) = µ z 0 y 0 z 0 a µ x 0 (3) y 0 x 0 0 So, the characteristic polynomial of the system () at the equilibrium point E is λ 3 Aλ Bλ C = 0 (4) A =, B = µ x 0 y 0 z 0 az 0, C = µx 0 µy 0 x 0 y 0 z 0 ay 0 z 0 (5) So the characteristic values of system () are λ, = ±iω 0, λ 3 = µ (6) ω 0 = 4 a 4µ µ 3 Stability of equilibrium points and existence of Hopf bifurcation It follows that the equilibrium point E is non-hyperbolic for all a > 0 and µ > 0. From the Center Manifold Theorem, at the equilibrium E a two-dimensional center manifold is well-defined and it is invariant under the flow generated by () (see [8], p. 5). This center manifold is normally attracted since λ 3 < 0. So it is enough to study the stability of E for the flow of system () which is restricted to the family of parameter dependent center manifolds. In general, to decide the stability of a non-hyperbolic equilibrium point of a system in R 3 is very difficult even for quadratic systems. As far as we know, the stability of E was analyzed by Rikitake [] from the numerical point of view (see []), while its analytical study has not been achieved yet. The study carried out in the present note may contribute to understand the stability of the equilibrium point E of system (). More precisely, in this paper we prove the following theorem. Lemma. Consider the Rikitake system (). The equilibrium E of system () is unstable and Hopf bifurcation exists for all the values of the parameters a > 0 and µ > 0. Proof. According to Routh-Hurwitz criterion, When to meet the conditions of A > 0, B > 0, AB c > 0, the equilibrium E is stable; but when it is suffice for the conditions of A > 0, B > 0, AB c = 0, the equilibrium E is unstable. By (5), we can know that A =, B = a a 4µ C = a a 4µ a a 4µ (7) a a 4µ (8) then we can easily draw A > 0, B > 0, because of the parameters a > 0 and µ > 0. Following we will proof the relationship between AB and C. AB C = ( a a 4µ a a 4µ (a a 4µ a a 4µ ) = 0 (9) so the equilibrium E is unstable. After that, we try to proof the existence of Hopf bifurcation. Suppose (µ) = A(µ) =, (µ) = A(µ)B(µ) C(µ) (0) i (i =, ) is the Hurwitz s determinant[9] of (4). Then, the conditions for the occurrence of Hopf bifurcation are A(µ) > 0, B(µ) > 0, C(µ) > 0, (µ) > 0, (µ) = 0, d( (µ)) dµ 0 () IJNS for contribution: editor@nonlinearscience.org.uk

3 X. Wang, T. Yang, W. Xu: Hopf Bifurcation and Limit Cycle Analysis of the Rikitake System 3 i.e. > 0, a a 4µ a a 4µ > 0, ( a a 4µ a a 4µ a a 4µ > 0, a a 4µ (a a 4µ a a 4µ ) = 0, µ a 4µ a a 4µ a a 4µ a 4µ so, the Hopf bifurcation exists for all the values of the parameters a > 0 and µ > 0. Theorem is proved. (a ) a 4µ a 4µ 0 () 4 Stability of the limit cycle in the previous system Coefficient of curvature of limit cycle is an important measurement for determining the stability of a system. The object in this section is to analyze the stability of limit cycle in the original system separately. Follow the above analysis, we obtain that E is a Hopf bifurcation point. In order to facilitate the operation, we assume that µ = = µ 0, a = Then λ = i, λ = i, λ 3 = 4. Move the original system For simplicity, move the system () to equilibrium point form E 0 (x 0, y 0, z 0 ) to (0, 0, 0). Suppose x = x x 0, y = y y 0, z = z z 0 then Now, we put (3) in system () that x = x x 0, y = y y 0, z = z z 0 (3) ζ = Gζ f(ζ ) (4) then ζ = x y z, G = , f(ζ ) = y z x z x y (5) ẋ = x 3.73y 0.576z y z ẏ = 0.680x y.939z x z ż = 0.576x.939y x y (6) At this, the equilibrium point of system (6) is (0, 0, 0). IJNS homepage:

4 4 International Journal of Nonlinear Science, Vol.4(0), No., pp Coefficient of curvature and stability of the limit cycle Where Now, using the transformation matrix ζ = T ζ (7) ζ = [ x y z ] T, T = [ Imτ Reτ τ 3 ] with τ is an Characteristic vector corresponding to λ, τ 3 is an Characteristic vector corresponding to λ 3, we can put the system (7) into ζ = J(µ)ζ Q(ζ, µ) (8) the coefficient matrix of the linear part is J(µ 0 ) = (9) When the parameter µ =, we can put system (7) into ẋ = y x 5.73y 0.576z y z ẏ = x.73x y.939z x z (0) ż = z 0.576x.939y z x y Obviously, the equilibrium point of system (0) is (0, 0, 0) and the coefficient matrix of the nonlinear part is Q = Q Q = x 5.73y 0.576z y z.73x y.939z x z () Q 0.576x.939y z x y So, the coefficient of curvature of limit cycle is g 0 = 4 ( Q x σ = Re{ g 0g 4 Q y i G 0 W G G 0 W 0 } = 68 () Q i( Q x y x Q y Q )) = 0 (3) x y W 0 = g = 4 ( Q x Q y i( Q x Q y )) = 0 (4) G 0 = ( Q x z Q y z i( Q x z Q y z )) = 0 (5) G 0 = ( Q x z Q y z i( Q x z Q y z )) = i (6) W = 4λ 3 (µ 0 ) ( Q 3 x Q 3 y ) = 0 (7) Q 3 4(iω(µ 0 ) λ 3 (µ 0 )) ( x Q 3 y i Q 3 ) = i x y 8i G = Q 8 ( 3 x 3 3 Q y 3 3 Q x y 3 Q x y i( 3 Q x 3 3 Q y 3 3 Q x y 3 Q x y )) = 0 (9) By (), we can see that the coefficient of curvature σ < 0, so the limit cycle of system () is stable. 5 Numerical simulation In system (), when a = 0.3, µ = 0., (x 0, y 0, z 0 ) is a Hopf bifurcation point. (Figure., Figure. ) (8) IJNS for contribution: editor@nonlinearscience.org.uk

5 X. Wang, T. Yang, W. Xu: Hopf Bifurcation and Limit Cycle Analysis of the Rikitake System x. y Figure : The series of x(t), y(t) z.3 z x y.5 Figure : The series of z(t) and the stable limit cycle. 6 Conclusions In this paper we mainly discuss some natures of Rikitake system. Firstly, we obtaine the equilibrium points and the characteristic polynomial of Rikitake system by calculating. After that, we use Routh-Hurwitz criterion to determine the stability of equilibrium points and the existence of Hopf bifurcation. At last, we judge the stability of limit cycle by calculating the coefficient of curvature of limit cycle. References [] Denis de Carvalho Braga and Fabio Scalco Dias and Luis Fernando Mello. On the stability of the equilibria of the Rikitake system. Physics Letters, 374(00): [] Y. X.Chang and X. J. Liu and X. F. Li. Chaos and chaos control of the Rikitake two-disk dynamo. Liaoning Norm. Univ. Nat. Sci, 9(006): [3] J. Llibre and X. Zhang. Invariant algebraic surfaces of the Rikitake system. Phys, 33(000): [4] C. Valls. Rikitake system: analytic and Darbouxian integrals. Proc. Roy. Soc. Edinburgh Sect, 35(005): [5] T. Yajima and H. Nagahama. KCC-theory and geometry of the Rikitake system. Phys, 40(007): [6] R. M. Tudoran and A. Aron and S. Nicoara. On a Hamiltonian version of the Rikitake system. SIAM J. Appl. Dyn. Syst, 8(009): [7] R. M. Tudoran and A. Girban. A Hamiltonian look at the Rikitake two-disk dynamo system. Nonlinear Anal. Real World Appl, (00): [8] Y.A. Kuznetsov. Elements of Applied Bifurcation Theory. Springer-Verlag, New York [9] Shu Zhongzhou and Zhang Jiye and Cao Dengqing Stability of movement. Chinese Rail way Press, China. 00 IJNS homepage: