COMPLEX DYNAMICS IN THE SEGMENTED DISC DYNAMO

. The present work is devoted to giving new insights into the segmented disc dynamo. The integrability of the system is studied. The paper provides its ﬁrst integrals for the parameter r = 0. For r > 0, the system has neither polynomial ﬁrst integrals nor exponential factors, and it is also further proved not to be Darboux integrable. In addition, by choosing an appropriate bifurcation parameter, the paper proves that Hopf bifurcations occur in the system and presents the formulae for determining the direction of the Hopf bifurcations and the stability of bifurcating periodic solutions.


1.
Introduction. Since the discovery of the Lorenz chaotic system, chaos has been developed and intensively studied in the past four decades. This study about chaos has concentrated on not only proposing new and interesting chaotic systems, but also enhancing complex dynamics and topological structure based on the existing chaotic systems [5,8,13].
In order to understand magnetic field generation and reversals in astrophysical bodies, model dynamos have been extensively investigated during the past decades [1,4,6,10,11]. The self-exciting disc dynamo has frequently been invoked as a simple prototype of dynamo action, analogous to the dynamo process that is believed to operate in the liquid conducting core of the Earth and in the convective envelope of the Sun. Considering that the conventional treatment of the simplest such model, the self-exciting homopolar disc dynamo, was not self-consistent, Moffatt introduced a segmented disc dynamo in which the current associated with the radial diffusion of the magnetic field could be included in a simple way [10]. The dynamo is described by the following set of ordinary differential equations: x (t) = r (y − x) = P (x, y, z) , y (t) = mx − (1 + m) y + xz = Q (x, y, z) , z (t) = g mx 2 + 1 − (1 + m) xy = R (x, y, z) , where x(t) and y(t) denote the magnetic fluxes due to radial and azimuthal current distributions, z(t) is the angular velocity of the disc; g measures the applied torque, and r and m are positive constants that depend on the electrical properties of the circuit [6,10].
The system (1.1) is not identical to the Lorenz system [10]. Knobloch added the term − vz to the right side of the third equation of system (1.1) and system (1.1) becomes the Lorenz system [6].
The integrability of systems of differential equations is one of central topics in the theory of ordinary differential equations. The Darboux theory of integrability plays a central role in the integrability of the polynomial differential models.We can compute the Darboux first integrals by knowing a sufficient number of algebraic invariant surfaces (the Darboux polynomials) and of the exponential factors (see [9,12] and the references therein).
Here, we shall use the Darboux theory of integrability to study the Darboux integrability of the the segmented disc dynamo (1.1). We further contribute to the understanding of the complexity, or more precisely of the topological structure of the dynamics of system (1.1) by studying its integrability. Obviously, system (1.1) is integrable for r = 0, and we provide its first integrals. For r > 0, the system has neither polynomial first integrals nor exponential factors, and it is also further proved not to be Darboux integrable.
On the other hand, by choosing an appropriate bifurcation parameter, the paper proves that Hopf bifurcations occur in system (1.1) when the bifurcation parameter exceeds a critical value and presents the formulae for determining the direction of the Hopf bifurcations and the stability of bifurcating periodic solutions by applying the normal form theory [3,7].
The paper is organized as follows. Sect. 2 investigates the dynamical behaviors of the segmented disc dynamo. Sect. 3 investigates the integrable of the system. In Sect. 4, by using the normal form theory, the direction of Hopf bifurcations and the stability of bifurcating periodic solutions are analyzed in detail. In Sect. 5, some numerical simulations are presented to illustrate the theoretical analysis. And Sect. 6 concludes the paper.
2. Dynamical behaviors of system (1.1). This system is invariant under the transformation (x, y, z) → (−x, −y, z) . Namely, the system has rotation symmetry around the z-axis. The divergence of system (1.1) is ∇ · V = − (1 + m + r), and the system is dissipative.
The characteristic equation about the equilibrium P + [1, 1, 1] is the same as that of According to Ref. [10], the two equilibria P + and P − are both asymptotically stable when the following conditions are met.
For r = (m+1) 2 1−m , the Jacobian matrix has a pair of purely imaginary eigenvalues and a nonzero real eigenvalue. Therefore, P + and P − are both not hyperbolic but weak repelling focus.

Coexistence of chaotic attractors with different types of equilibria.
Generally, it is difficult to analytically specify parametric regions of a chaotic system. Therefore, certain numerical indices for identifying chaotic properties of system orbits are verified. There are three kinds of cases in system (1.1) as follows.

Case 2. Chaotic attractor coexists with two saddle-foci.
When parameters (r, m, g)=(5, 0.5, 11.6), the chaotic attractor can be observed (see Figure 2(a)). The Poincaré map is shown in Figure 2 It is noted that the chaotic attractor is a bit different from Case 1, because the equilibria P + and P − are both unstable. Figure 1(b) and Figure 2(b) also show sheets of the chaotic attractors visualized on the Poincaré maps. It is clear that some sheets are folded. In addition, system (1.1) has rotation symmetry around the z-axis, so the scatter of points of the Poincaré maps on the x − z plane and y − z plane should be symmetrical, as shown in Figure 1(b) and Figure 2(b).

Preliminary results.
Let R [x, y, z] be the ring of the real polynomials in the variables x, y and z. We say that for some polynomial L f , called the cofactor of f (x, y, z). If f (x, y, z) is a Darboux polynomial, then the surface f (x, y, z) = 0 is an invariant algebraic surface, which means that if an orbit of system (1.1) has a point on the surface f (x, y, z) = 0, then the whole orbit is contained in it. Let f, g ∈ R [x, y, z] be coprime. We say that a nonconstant function E = e g f is an exponential factor of system (1.1) if E satisfies    . Parameters values (m,g)=(0.5,11.6), Lyapunov exponent spectrum of system (1.1) for r ∈ [5,7] for some polynomial L e ∈ R [x, y, z], called the cofactor of E and having degree at For a geometrical and algebraic meaning of the exponential factors see [2]. A first integral G of system (1.1) is of Darboux type if it has the form where f 1 , · · · , f p are Darboux polynomials, E 1 , · · · , E q are exponential factors and λ j , µ k ∈ R for j = 1, · · · , p and k = 1, · · · , q.

Darboux integrability.
We claim that the degree k of the cofactor L f is less than or equal to 1. The claim follows from the fact that in (1.1) Therefore, without loss of generality, we can assume that the cofactor is of the form and then (3.1) becomes Theorem 3.1. If r = 0, system (1.1) is integrable with the first integrals

JIANGHONG BAO
It is straightforward to verify that H 1 and H 2 in the statement of the theorem are first integrals of system (1.1). Therefore the proof of Theorem 3.1 will be omitted and from now on we consider the cases in which r = 0.
Theorem 3.2. The following statements hold for system (1.1) with r = 0.
(1) It has no Darboux polynomials with non-zero cofactor.
(2) It has no polynomial first integrals.
where each f i = f i (x, y, z) is a homogeneous polynomial of degree i. Without loss of generality we can assume that f n = 0 and n = 0.
Computing the terms of degree n + 1 in (3.5) yields Solving the equation gets where F is an arbitrary continuously differentiable function and k (x, y, z) = arctan g (1 + m) (mx − (1 + m)y) (1 + m) z .
Since f n (x, y, z) is a polynomial of degree n, it forces that b 1 = 0, b 2 = 0, and b 3 = 0. Let where p is a nonnegative integer and p ≤ n 2 . Computing the terms of degree n in (3.5) yields where F is an arbitrary continuously differentiable function. Since f n−1 (x, y, z) is a polynomial, it forces that b 0 = 0. And the case is impossible.
(2) As long as we , the proof is similar to (1).
(3) Considering that system (1.1) has no Darboux polynomials with non-zero cofactor, we assume that E = e h with h ∈ R[x, y, z] is an exponential factor of system (1.1) and we shall reach a contradiction. Without loss of generality, we assume that the cofactor L is of the form Clearly, by (3.3), h satisfies ∂h ∂x P + ∂h ∂y Q + ∂h ∂z h i (x, y, z), (3.12) where each h i = h i (x, y, z) is a homogeneous polynomial of degree i. Without loss of generality we assume that h n = 0 and n = 0. Computing the terms of degree n + 1 in (3.11) yields xz ∂h n ∂y + g mx 2 − (1 + m) xy ∂h n ∂z = 0.
Solving the equation gets where F is an arbitrary continuously differentiable function. Let where p is a nonnegative integer and p ≤ n 2 . Computing the terms of degree n in (3.11) yields xz ∂f n−1 ∂y + g mx 2 − (1 + m) xy ∂f n−1 ∂z + r (y − x) ∂f n ∂x + (mx − my − y) ∂f n ∂y = 0.
where F is an arbitrary continuously differentiable function. Since f n−1 (x, y, z) is a polynomial, it forces that n = 0, which contradicts with the assumption n = 0. And this case is impossible.
Case 2. p > 0. Solving the equation (3.13) yields where F is an arbitrary continuously differentiable function, and Since f n−1 (x, y, z) is a polynomial, it forces that p = 0 and n = 0, which contradicts with the assumption n = 0. And this case is impossible.
4. Hopf bifurcations in system (1.1). In this section, we apply the normal form theory [3,7] to study the direction, stability and period of bifurcating periodic solutions for system (1.1). We first consider the Hopf bifurcation of system (1.1) at P + [1, 1, 1]. When m < 1 and r = r 0 = (1+m) 2 1−m , Eq. (2.1) possesses a negative real root 2(m+1) m−1 and a pair of conjugate purely imaginary roots ± g (m + 1) i. Under this condition, the transversality condition is also satisfied. Accordingly, Hopf bifurcation at P + occurs. The above analysis is summarized as follows: The direction, stability and period of bifurcating periodic solutions for system (1.1) are as follows.