EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 4, 2017, 858-870 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global New derived systems of Hide’s coupled dynamo model Ali Allahem Department of Mathematics, College of Sciences, Qassim University, Saudi Arabia Abstract. In this paper, we present the results of a preliminary analytical and numerical study of new derived system of single and double coupled self-exciting Faraday disk homopolar dynamos by Hide et al [3, 7]. Also, well-known systems including, Rikitake and Bullard systems have been reached by using Tikhonov theorem [6, 10] and eliminating method. 2010 Mathematics Subject Classifications: 37C45, 37C25 Key Words and Phrases: Dynamo model, Dynamical system, Bifurcations, Tikhonov’s theorem, Stability 1. Introduction Hide et al [3] have studied the novel autonomous sets of dimensionless nonlinear ordi- nary differential equations (ODEs) ẋ = x(y − 1)− βz, ẏ = α(1− x2)− κy, ż = x− λz, where ẋ = dx/dτ, etc. (1) These equations govern the behaviour of self-exciting homopolar dynamo system. The independent variable τ denotes time t. The dependent variables are x(τ), y(τ) and z(τ) such that x(τ) is the rescaled electric current in the dynamo, y(τ) is the angular rotation rate of the disk and z(τ) measures the angular speed of rotation of the motor. Also, we have four parameters (α, β, κ, λ) which are the system dependents on. These four parameters must be positive because they are physically unrealistic otherwise. Parameters represent where α measures the applied couple; β measures the inverse moment of inertia of the armature; κ measures the mechanical friction in the disk and λ measures the mechanical friction in the motor. Hide [7] has introduced a system of N self-exciting Faraday disk homopolar dynamos, symmetrically coupled, arranged in a ring. Each unit has an electric motor and is con- nected in series with a coil and a disk, being driven into motion by the dynamo. The Email address: a.allahem@qu.edu.sa (A. Allahem) http://www.ejpam.com 858 c© 2017 EJPAM All rights reserved. A. Allahem / Eur. J. Pure Appl. Math, 10 (4) (2017), 858-870 859 system (1) study the case when N = 1 (one coupled dynamo). Hide [2] has extended the above study and consider the N = 2 case (two coupled dynamos) with their six dependent variables and thirteen dimensionless parameters in general. The system where N = 2 is given by the following set of nonlinear ordinary differential equations [7]: ẋ1 = mx2y1 − x1 − βz1, ẏ1 = α(1−mx2x1)− κy1, ż1 = x1 − λz1. ẋ2 = l−1[x1y2 − rx2 − hβz2], ẏ2 = a−1[α(g − x2x1)− kκy2], ż2 = b−1[hx2 − dλz2]. (2) In this paper, we derive new systems from N = 2 coupled dynamo models by applying some restrictions in the parameters. These restrictions allow us to simplify the system (2) and derive new reduced systems, which may find interested systems or lead to one of the well-known systems. We set all addition parameters to unity unless a, b and l. In other words, we consider some constraints that being used k = r = m = h = d = g = 1. (3) Note that when a = b = l = 1, we turn to Hide’s single-disk homopolar dynamo (1). Also, we apply an important theorem in perturbation theory that so-called Tikhonovs Theorem which reduces the dimension of system. The idea of the theorem is based on singular perturbation problem [8]. Consider a system of differential equations of the form ẋ = f(x, z, t), µż = g(x, z, t), 0 < µ� 1 (4) where f and g are sufficiently differentiable. The functions f, g and the initial values x(0), z(0) may depend smoothly on µ. For simplicity of notation we suppress this depen- dence. The corresponding equation for µ = 0, ẋ = f(x, z, t), 0 = g(x, z, t), (5) is the reduced problem. The function f is supposed to possess an isolated solution z = ξ(x, t). The substituting of z into the first equation yields to ẋ = f(x, ξ(x, t), t), x(0) = x0. (6) The paper is structured as follows. In the section 2, we provide an overview of Single- Disk homopolar dynamo that was given by Hide et al [3]. We also present The Double- Disk homopolar dynamo and study. In Double-Disk homopolar dynamo [7], we have case of study, when κ = 0 in section 3. Throughout this case, we attempt to reduce the dimension of system which make it simple and comparable with well-known system. Numerical analysis including numerical integration, bifurcations study and linear stability are included. We draw the conclusion in section 4. A. Allahem / Eur. J. Pure Appl. Math, 10 (4) (2017), 858-870 860 2. Theory 2.1. Single-Disk homopolar dynamo: Theory Hide et al. [3] proposed a model for self-exciting dynamo action in which a Faraday disk and coil are arranged in series with either a capacitor or a motor. The system (1) contains a steady equilibrium solution either (x, y, z) = (0, α/κ, 0) or (x, y, z) = (± √ 1− κ/α(β/λ+ 1), β/λ+ 1, x/λ). A linear stability analysis of (1) about the steady equilibrium solution shows that the eigenvalues of Jacobian matrix are −κ, 1/2{α/κ− 1− λ± √ (α/κ− 1 + λ)2 − 4β} Steady bifurcations occur along the line α/κ = β/λ+ 1, where symmetry breaking bifurcation occur and two lines of Hopf bifurcations along α/κ = λ+ 1, provided β ≥ λ2 and α/κ = [(2β − κλ− λ2)/2(κ− β/λ) + 3β/2λ+ 1]. In bifurcation diagram ( see Fig.5 in [3]), there is a Taken-Bogdanov (double zero eigen- value type) bifurcations occur at the point P where P = (α/κ, β) = (λ+ 1, λ2), Note that, all lines of steady bifurcations and two lines of Hopf bifurcations meet in at point P with reflection symmetry. There is a global bifurcation occurs and it emerges from the Taken-Bogdanov point, label it P in bifurcation diagram. In addition Hide et al. [3] have shown that dynamo action occurs when the steady equilibrium solution (x, y, z) = (0, α/λ, 0) is unstable, namely when α/κ > min(1 + β/λ, 1 + λ), but not otherwise. 2.2. Double-disk homopolar dynamo: Theory The set of equations for general coupled dynamo (2) is reduced by a constrains (3) to: ẋ1 = x2y1 − x1 − βz1, ẏ1 = α(1− x2x1)− κy1, ż1 = x1 − λz1. ẋ2 = l−1[x1y2 − x2 − βz2], ẏ2 = a−1[α(1− x2x1)− κy2], ż2 = b−1[x2 − λz2]. (7) A. Allahem / Eur. J. Pure Appl. Math, 10 (4) (2017), 858-870 861 Similar to the single homopolar dynamo, the equations (7) have a reflection symmetry: they are unchanged and invariant under the transformation (x1, y1, z1, x2, y2, z2)→ (−x1, y1,−z1,−x2, y2,−z2). In other word, the y1, y2 are invariant. The equilibrium states can be found in Ref. [4]. 3. Case Study The case is when we have nonzero λ and zero κ. The models reduction here are based on two kind of reduction methods. First of all, eliminating the differential equation. Second of all, we use an important theorem in perturbation theory [6, 8, 10] that so-called Tikhonov’sTheorem. We start the problem in the system (7)with κ = 0, and hence ẋ1 = x2y1 − x1 − βz1, ẏ1 = α(1− x2x1), ż1 = x1 − λz1. ẋ2 = l−1[x1y2 − x2 − βz2], ẏ2 = a−1[α(1− x2x1)], ż2 = b−1[x2 − λz2]. (8) First of all, we reduce the six dimensions of system (8) to five dimensions by eliminating the equation of ẏ2 with ẏ1. Suppose ẏ1 = f(x1, x2), and so, ẏ2 = a−1f(x1, x2). Let v̇ = f(x1, x2), and hence ẏ2 = a−1v̇, y2 = a−1v + c, Thus, the relation between y1 and y2 is y2 = a−1y1 + c. (9) From the equation (9), we can find the simple conditions on parameters such that ay2(t)→ y1(t) as t → ∞, if κ = 0 and the new variable c = 0. It is clear that if we differentiate the equation (9), we get ċ = 0. In other word, if the solution of y2 in six dimensional system is stable (unstable), then the solution of y1 in five dimensional system is also stable (unstable), respectively. Thus the system (8) becomes ẋ1 = x2y1 − x1 − βz1, ẏ1 = α(1− x2x1), ż1 = x1 − λz1. ẋ2 = l−1[x1(a −1y1 + c)− x2 − βz2], ż2 = b−1[x2 − λz2]. (10) A. Allahem / Eur. J. Pure Appl. Math, 10 (4) (2017), 858-870 862 This system can be reduced further. We have five dimensional system (10), and by rescaling x1 to be x1 = λX1, the ż1 equation as result becomes ż1 = λX1 − λz1. Divide it by λ ż1/λ = X1 − z1. Now we apply Tikhonov Theorem by choosing λ large, and hence X1 = z1. (11) Similarly in ż2, we rescale x2 as x2 = λX2 and then divide the equation by λ, we get b/λż2 = X2 − z2. Set b = c1λ. The equation becomes c1ż2 = X2 − z2, then apply Tikhonov Theorem by choosing c1 to be small (c1 � 1). Thus X2 = z2. (12) However, this is not always true, it can be work in term of perturbation theory. From (11), (12) and already y2 has been eliminated with y1, the 5D system (10) reduces to 3D system as follows Ẋ1 = X2y1 − (1 + β/λ)X1, ẏ1 = α(1− λ2X1X2), Ẋ2 = l−1[X1(a −1y1 + c)− (1 + β/λ)X2], (13) Assume that (1 + β/λ) = µ as a new parameter and rewrite the system(13) in x, y and z coordinates ẋ = zy − µx, ẏ = α(1− λ2xz), ż = l−1[x(a−1y + c)− µz], (14) One of the interesting point that Rikitake disc dynamo [9] is involved in the system (14) by setting the limit of parameters as c = −γ and l = a = α = λ = 1. Hence we have the Rikitake system ẋ = zy − µx, ẏ = x(z − γ)− µy, ż = 1− xy, (15) A. Allahem / Eur. J. Pure Appl. Math, 10 (4) (2017), 858-870 863 At value of parameters µ = 1.1 and γ = 7, we have found that system has chaotic oscillations as shown in figures 1. In more details, Cook and Roberts [1] found the fixed points of the Rikitake system which are N and R, (±K,±K−1, µK2) where K is given by γ = µ(K2 −K−2), or K = √ γ ± √ γ2 + 4µ2 2µ . There are a stable and centre manifold through each of the fixed points [5]. Thus the dynamics take place on centre manifold, on which the fixed points are both unstable. The stability of the fixed points R,N can be determined by the method of the eigenvalues. In the following presentation we will consider the fixed point R that has positive sign but the same computation may be applied to N . If we compute the Jacobian of the system (15) and evaluate it at R, we get the matrix J(R) =  −µ µK2 K−1 µK2 − γ −µ K −K−1 −K 0  Compute the eigenvalues of the this matrix results in characteristic equation of the form (σ2 +K2 +K−2)(σ + 2µ) = 0, which has roots σ0 = −2µ, σ1,2 = ±(K2 +K−2)i. Since the system (15) has two purely imaginary roots then the eigenvalues method fails. There are two options available to study. Firstly, take the higher order Taylor approxima- tion of system ξ̇(t) = fµ(ξ(t) + u∗) for system u̇ = fµ(u) where µ parameter and u∗ state-solution. Moreover, ξ(t) is the function represent the distance between the state solution and some other solution as ξ(t) = u(t)− u∗ with u(0)− u∗ = ξ0, Note that the system that results from such a truncation is no longer linear. Thus it fails too. The other option, we can use the Lyapounov function. This approach is more applicable, to show that points R and N are both unstable for any value of parameters and K saves when K = 1 or µ = 0. A. Allahem / Eur. J. Pure Appl. Math, 10 (4) (2017), 858-870 864 In general, the Rikitake system does not possess closed form solution. For certain parameter values the analytical solutions are known. So if we choose the values as K = 1 and µ = 0 for the system (15), then we have the following system ẋ = zy, ẏ = 1− xz, ż = xy, If we consider those solution for which x = z and perform the substitution u = x = z, then we get the following system u̇ = zu, ẏ = 1− u2, (16) These equations are well-known which are called the Bullard model dynamo [2]. Thus, two well-known systems (Rikitake [9] and Bullard [2]) have been derived from original system (2). Back to the five dimensional system (10) to study the linear stability around it fixed points. By setting all differential equations in (10) equal to zero, we get the fixed point. Firstly, from third and fifth equation, it is easy to see that z1 = 1 λx1 and z2 = 1 λx2, respectively. Also in the second equation we find x1x2 = 1. (17) Secondly, the first and fourth equations can be studied as follows, in the first equation, we have x2y1 − (1 + β/λ)x1 = 0 x1 = x2y1 µ where µ = 1 + β/λ. (18) Also in the fourth equation, we have x2 = 1 µ (a−1x1y1 + cx1) Substitute (18) in x2 we get x2 = 1 µ ((aµ)−1x2y 2 1 + c µ x2y1) Ay21 +By1 − 1 = 0 where A = 1 aµ2 and B = c µ . Hence y1 = −B ± √ B2 − 4A 2A ⇒ y1 = −c µ ± √ c2 µ2 − 4 aµ2 2 aµ2 A. Allahem / Eur. J. Pure Appl. Math, 10 (4) (2017), 858-870 865 Rearrange the fraction of y1 y1 = −c± √ c2 − 4 a 2/aµ . However, if we multiply (18) by x1 and use the relation in (17), then it simplifies to become y1 = µx21 which means that y1 is always positive. Hence we neglect the negative sign of the value of y1. Now plug that in the equation (17), so we have x1 = ( −c± √ c2 − 4 a 2/aµ︸ ︷︷ ︸ H )x2, then substitute it in (17) Hx22 = 1 ⇒ x2 = ± 1√ H . Similarly x1 = ± √ H. As a result the fixed point states are given by: x1 = ± √ H, x2 = ± 1√ H , y1 = µH, z1 = √ H λ z2 = 1 λ √ H , (19) where H = ( −c± √ c2− 4 a 2/aµ ) and µ = 1 + β/λ. To study the stability of (10) we need to compute the Jacobian matrix of the system (10). Thus, we get J =  −1 x2 −β y1 0 −αx2 0 0 −αx1 0 1 0 −λ 0 0 l−1(a−1y1 + c) x1(al) −1 0 −l−1 −l−1β 0 0 0 b−1 −b−1λ  (20) Since we have seen a good result in Rikitake equations in figure 1 when µ = 1.1 and γ = 7, we are going to apply the same values of parameters to find the relationship between the system (10) and Rikitake systems. However the limit of these parameters values come form [7] where assumed K = 2 and a value of µ between 1 and 2 which Rikitake model would the observed reversals are most faithfully. The relation occurs when c = −γ and l = a = α = λ = 1 and keep the value of b as small value, to be 0.001. Since µ = 1.1 and µ = 1 + β/λ, then β = 0.1. Before substituting fixed points in the Jacobian matrix, we need to calculate the value of x1, x2, y1 which are ±2.55,±0.39 and 7.16, respectively. So A. Allahem / Eur. J. Pure Appl. Math, 10 (4) (2017), 858-870 866 the Jacobian matrix at the fixed points is J(fp) =  −1 ±0.39 −0.1 7.16 0 ∓0.39 0 0 ∓2.55 0 1 0 −1 0 0 0.16 ±2.55 0 −1 −0.1 0 0 0 1000 −1000  (21) Thus the eigenvalues are as follows -9.998998904039592e+002 5.863307789355593e-003 +2.578916992181417e+000i 5.863307789355593e-003 -2.578916992181417e+000i -2.030767974296189e+000 -1.081068237323492e+000 Since we have three negative eigenvalues and a pair of complex conjugate eigenvalues with positive real part, this gives an indication of having unstable centre (foci) equilibrium [5] as in Rikitake equilibrium points. The parameters values as those in Rikitake Γ = µ = 1.1 , c = −γ = −7 and l = a = α = 1 and keep the value of b as small value, to be 0.001 to show the phase portrait in system (10). It would be shown in figures 2 and 3, respectively. It shows that the solution of x2 and z2 is clearly having same behaviour. It also shows that phase portrait in the (x2, z2), (x1, x2), (z1, z2), (z1, z2). (x2, y1), (x1, z2) and (x2, z1) are indicated of chaos behaviour. The behaviours of (x1, y1) are similar to the behaviours of (x, y) in Rikitike system. Also, the behaviours of (x2, y1) or (z2, y1) are similar to the behaviours of (z, y) in Rikitike system. Moreover, the behaviours of (x1, x2) and (x1, z2) are similar. The behaviours of (z1, x2) and (z1, z2) are similar. Also, the behaviours of (y1, z2) and (y1, x2) are similar. 4. Conclusion The system of a double disk dynamo with motors is valuable and many cases can be studied to see the beauty of the dynamical system including, Chaos, Bifurcations and periodic orbits. In our case, we show how well-known system can derived from the original system using the reduction method provided. A further cases can be studied be setting the parameters to investigate many possibility in the dynamical system theory. Acknowledgements The author is supported by Qassim University and the Ministry of Education of Saudi Arabia. REFERENCES 867 References [1] A.E. Cook and P.H. Roberts. The rikitake two-disc dynamo system. Proc. Cambridge Philos Soc., 68:547, 1970. [2] Bullard E.C. The stability of a homopolar dynamo. Proc. Cambridge Philos Soc., 51:744–760, 1955. [3] Skeldon A. C. Hide R. and Acheson D. A study of two novel self-exciting single-disk homopolar dynamos: theory. Proc. R. Soc. Lond, pages 1369–1395, 1996. [4] Andrew M. Soward Irene M. Moroz, Raymond Hide. On self-exciting coupled faraday disk homopolar dynamos driving series motors. Physica D, 117:128–144, 1998. [5] Guckenheimer J. and P. Holmes. Nonlinear Oscillations, Dynamical Sytems and Bifurcations of Vector Fields. Springer-Verlag, 1998. [6] Wlodzimierz Klonowski. Simplifying principles for chemical and enzyme reaction kinetics. Biophysical Chemistry, 18(2):73–87, 1983. [7] Hide R. The nonlinear differential equations governing a hierarchy of self-exciting coupled faraday-disk homopolar dynamos. Phys. Earth Plant. Int., 103:281–291, 1997. [8] P. Swinnton-Dyer and T. Wegenknecht. Some third order ordinary differential equa- tions. Bull London Math Soc, 40(5):725–748, 2008. [9] Rikitake T. Oscillations of a system of disk dynamos. Proc. Cambridge Philos Soc., 54:89–105, 1958. [10] A. N. Tikhonov. Systems of differential equations containing small parameters in the derivatives. Mat. Sb. (N.S.), 31(73)(3):575–586, 1952. REFERENCES 868 -6 -4 -2 0 2 4 6 -2 -1 0 1 2 xHtL yHtL -2 -1 0 1 2 6 7 8 9 yHtL zHtL -5 0 5 xHtL -2 -1 0 1 2 yHtL 6 7 8 9 zHtL 0 20 40 60 80 100 -6 -4 -2 0 2 4 6 t xHtL 0 20 40 60 80 100 -2 -1 0 1 2 t yHtL 0 20 40 60 80 100 5 6 7 8 9 t zHtL Figure 1: Rikitake system with parameters choice as µ = 1.1 and � = 7. 7 Figure 1: Rikitake system with parameters choice as µ = 1.1 and γ = 7. REFERENCES 869 0 20 40 60 80 100 -6 -4 -2 0 2 4 6 t x 1HtL 0 20 40 60 80 100 6 7 8 9 t y 1HtL 0 20 40 60 80 100 -4 -2 0 2 4 t z 1HtL 0 20 40 60 80 100 -2 -1 0 1 2 t x 2HtL 0 20 40 60 80 100 -3 -2 -1 0 1 2 3 t z 2HtL Figure 2: Five dimensional system with parameters choice as � = 7,↵ = 1, � = 0.0001, l = 1, c = �7, = 0.60,� = 0.65, b = 0.1, a = 1, and µ = 1.1. 12 Figure 2: Five dimensional system with parameters choice as γ = 7, α = 1, β = 0.0001, l = 1, c = −7, κ = 0.60, λ = 0.65, b = 0.1, a = 1, and µ = 1.1. REFERENCES 870 -6 -4 -2 0 2 4 6 -2 -1 0 1 2 x1HtL x2HtL -6 -4 -2 0 2 4 6 5 6 7 8 9 x1HtL y1HtL -6 -4 -2 0 2 4 6 -4 -2 0 2 4 x1HtL z1HtL -2 -1 0 1 2 6 7 8 9 x2HtL y1HtL -4 -2 0 2 4 6 7 8 9 z1HtL y1HtL -3 -2 -1 0 1 2 3 6 7 8 9 z2HtL y1HtL -2 -1 0 1 2 -4 -2 0 2 4 x2HtL z1HtL -3 -2 -1 0 1 2 3 -4 -2 0 2 4 z2HtL z1HtL -6 -4 -2 0 2 4 6 -3 -2 -1 0 1 2 3 x1HtL z2HtL -3 -2 -1 0 1 2 3 -2 -1 0 1 2 z2HtL x2HtL Figure 3: Continued five dimensional system with parameters choice as � = 7,↵ = 1, � = 0.0001, l = 1, c = �7, = 0.60,� = 0.65, b = 0.1, a = 1, and µ = 1.1. It shows that the solution of x2 and z2 is clearly having same behaviour. It also shows that phase portrait in the (x2, z2), (x1, x2), (z1, z2), (z1, z2). 13 Figure 3: Continued five dimensional system with parameters choice as γ = 7, α = 1, β = 0.0001, l = 1, c = −7, κ = 0.60, λ = 0.65, b = 0.1, a = 1, and µ = 1.1.