In the last few decades, chaos, bifurcation and stability in dynamic systems have been extensively studied, since they are the main sources leading to hidden attractors, hyperchaos or other poor performance of a system.^[1–5]

Hidden attractors represent a new interesting topic and attract considerable attention. They are important in engineering applications because they allow unexpected and potentially disastrous responses to perturbations in a structure like a bridge or an airplane wing.^[6] They exist in systems with a stable equilibrium,^[7–9] a line equilibrium^[10] or no equilibria.^[5,6] Wei studied hidden hyperchaotic attractors with no equilibria^[6] and with only one stable equilibrium.^[7] Jafari and Sprott studied chaotic attractors of three-dimensional (3D) systems with a line equilibrium.^[10]

Theoretically analyzing local and global characteristics is essential for understanding of what is meant by chaos and hyperchaos, such as pitchfork bifurcation, Hopf bifurcation, ultimate bound and some other complex dynamical behaviors, which also show that these systems have rich nonlinear dynamics and are of significance in practical application.^[11]

Four or higher dimensional chaotic and hyperchaotic dynamical systems modeling a natural phenomenon have been discovered to present nature more explicitly than the 3D systems.^[5] Identifying and locating coexisting attractors are also crucial for studying the behavior of nonlinear systems, which become more difficult for hidden or rare states. Based on the segmented disc dynamo,^[12] we introduce a 4D self-exciting segmented disc dynamo with hidden hyperchaotic attractors. The research on the novel system will feed into more studies of high dimensional systems, and also help to identify the geometrical characteristics of lower dimensional chaotic attractors. We hope that the work will shed light on more systematic studies of the segmented disc dynamo.

The proposed 4D self-exciting segmented disc dynamo possesses coexisting hidden attractors with only one stable equilibrium or a line equilibrium when parameters vary. As far as we know, it is new. It also satisfies two of the criteria introduced by Sprott:^[13] (i) The system should credibly model some important unsolved problems in nature and shed insight on that problem; (ii) The system should exhibit some behavior previously unobserved.

Then by using the center manifold theorem and bifurcation theory,^[14,15] we analyze the Hopf bifurcation and pitchfork bifurcation of the novel system. The ultimate bound of the system is estimated by combining the unified approach and the optimization idea.^[16] Numerical investigations are also used to demonstrate the corresponding theoretical results for the two bifurcations and the ultimate boundary region.

The paper is organized as follows. We introduce a new system with only one stable equilibrium or a line equilibrium in Section 2. The Hopf bifurcation is investigated in Section 3 and in Section 4 we analyze the pitchfork bifurcation. Section 5 studies the ultimate bound. Finally, we conclude the paper in Section 6.

The segmented disc dynamo is introduced by Moffatt as follows:^[12] Based on the above system, we propose a 4D segmented disc dynamo 1 where .

System (1) is invariant under the transformation: Namely, it has rotation symmetry around the z axis. The divergence of the system is , and it is dissipative. Figure 1 shows the spectrum of Lyapunov exponents and the corresponding bifurcation diagram versus .

Fig. 1.

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	Fig. 1. (color online) Parameters and , initial condition (10, 200, 30 230); (a) Lyapunov exponents spectrum of system (1); (b) Bifurcation diagram of system (1).

2.2. Coexistence of hidden hyperchaotic and chaotic attractors with only one stable equilibrium

When equation (1) only has one stable equilibrium and the eigenvalues at are i)

For initial condition (0.5268, 7.3786, 2.6912, 4.2284), the Lyapunov exponents of the system (1) are = 0.0182, = 0.0023, = −0.0228, = −2.0297, and the fractional dimension is 2.8991. A hyperchaotic attractor with a stable equilibrium (0,0,0,0) can be obtained (see Fig. 2(a)). Figure 2(b) shows the Poincare map.

ii)

For initial condition (0, 7.3786, 2.6912, 4.2284), the trajectories of the system (1) converge on a chaotic attractor. The Lyapunov exponents of the system are = 0.0172, = −0.0076, = −0.0074, = −2.0342 and the fractional dimension is 3.0011.

Fig. 2.

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	Fig. 2. (color online) Parameters = (0.01, 0.021, 12, 12, 100, 0.001), initial condition (0.5268, 7.3786, 2.6912, 4.2284); (a) hyperchaotic attractor of system (1); (b) Poincaré map on the y–z plane.

2.3. Coexistence of hidden hyperchaotic and chaotic attractors with a line equilibrium

When , the system (1) only has a line equilibrium (0, 0, z, 0). i)

For initial condition , the Lyapunov exponents of the system (1) are found to be , , , and the fractional dimension is 3.2518. A hyperchaotic attractor can be obtained.

ii)

For initial condition (3, 0, 0, 0), the Lyapunov exponents of the system are and the fractional dimension is 1.0014. The trajectories of the system (1) converge on a chaotic attractor (as shown in Fig. 3).

Fig. 3

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	Fig. 3 (color online) Chaotic attractor of the system (1) with initial condition (3,0,0,0) for .

The section utilizes the projection method described in Ref. [14] to calculate the first Lyapunov coefficient associated with the Hopf bifurcation.

For , the system (1) has an equilibrium . The corresponding characteristic equation at is Eq. (2).

Let

When , the system (1) possesses two negative real roots and conjugate purely imaginary roots at .

Substituting into and differentiating with respect to give Considering that are the roots of , we have Hence the transversality condition 4 is also satisfied, and Hopf bifurcation at occurs. The stability of the transversal Hopf point depends on the value of the first Lyapunov coefficient, as stated in the following theorem.

Fig. 4.

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	Fig. 4. (color online) A unstable limit cycle of the system (1) with initial condition for .

We employ the center manifold theorem^[15] to study the pitchfork bifurcation of the system (1).

Let

For , the system (1) only has an equilibrium O[0,0,0,0]. By Eq. (2), the eigenvalues at O are O[0,0,0,0] is nonhyperbolic, and then we can obtain the following theorem.

Fig. 5.

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	Fig. 5. The pitchfork bifurcation diagram in the system (1) near .

Now we utilize the unified approach^[16] to derive the ultimate bound of the system (1).

Table 1 shows numerically actual and estimated bounds of the hyperchaotic attractor. In order to illustrate the bound vividly, figure 6 shows the phase portraits of the hyperchaotic attractor.

Fig. 6

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	Fig. 6 (color online) The hyperchaotic attractor of the system (1) with and its ultimate bound estimated with in the (x, y,z) space and w = 0.

Table 1.

Table 1.

Table 1. The estimated bound of the hyperchaotic attractor. . Variable Actual bound Estimated bound set x (-9.7007, 8.7668) (-48.9084, 48.9084) y (-30.3310, 28.9109) (-48.4744, 48.4744) z (-84.5560, −49.274) (-145.3364, −0.0036) w (-33.8435, 33.2820) (-51.6069, 51.6069)

Table 1. The estimated bound of the hyperchaotic attractor. .

The 4D segmented disc dynamo is very interesting and reported rarely, in that it possesses coexisting hidden attractors with only one stable equilibrium or a line equilibrium. In addition, by choosing an appropriate bifurcation parameter, the paper proves that the Hopf bifurcation and pitchfork bifurcation occur in the system. Also, the ellipsoidal ultimate bound is obtained. Numerical simulations are done to demonstrate the emergence of the two bifurcations and show the ultimate boundary region.

It is hoped that the investigation of this paper can be quite beneficial to further studies of the dynamically rich segmented disc dynamo, which is very desirable for magnetic field generation and reversals in astrophysical bodies, and will shed some light leading to final revealing the true geometrical structure of the attractor of the amazing original system.