Since Lorenz found the chaotic attractor by chance during experiments in 1963,^[1] many results have been obtained from the investigation of chaotic systems.^[2–11] Currently, we are not only concerned on discovering new chaotic systems, but also concerned on the chaotic and bifurcation characteristics of the system itself. Bifurcation control has become one of the important research fields.^[12–21] Nguyen et al. proposed a novel dynamic state-feedback control law.^[21] Analytical schemes to determine the control gains have been derived, according to the conditions for the emergence of Hopf bifurcation. Gan et al. analyze the stability and direction of the Hopf bifurcation for a food finite model by the feedback control method.^[22] Non-negative equilibrium point of the system is asymptotically stable by analyzing the eigenvalues. The Hopf bifurcation is unstable when the bifurcation parameter exceeds the critical value. Ding et al. investigate the problem of bifurcation control for complex networks in a small-world networks model with time delay,^[23] presenting a proportional-derivative (PD) feedback controller to control the Hopf bifurcation which inherently occurs due to the networks topology. Liu et al. discuss the stable of the wireless network model using the hybrid control.^[24] The center manifold theorem and normal form theorem are used to study the stable and direction of the bifurcation periodic solutions. Wu et al. carry out the effect of controller parameters on the Hopf bifurcation point, bifurcation type, and periodic solution amplitude by the washout filter for the Rössler system.^[25]

At present, there are still little researches on the Pan chaotic system. Pan et al. proposed a three-dimensional chaos system,^[26] which controls this new attractor by linear feedback functions. The trajectories of the chaotic attractor can be controlled to any periodic target orbits or points. Sundarapandian et al. constructed explicit state-feedback control laws for regulating the output of the three-dimensional Pan chaos system so as to track constant reference signals.^[27] He also derived new results for the anti-synchronization of identical Pan chaotic systems by active nonlinear control. This control method is effective and convenient, which regards to achieve anti-synchronization of the chaotic systems addressed. AL-Azzawi analyzed the dynamic behaviors of a new three-dimensional Pan chaos system,^[28] including some basic dynamic properties, stability, and Hopf bifurcation. The stability and Hopf bifurcation are investigated by the Routh–Hurwitz method and the Cardan method, and the analysis shows that both methods justified the same results.

In this paper, a new five-dimensional Pan-like chaotic system is proposed, in which two new state variables based on the Pan system are set. The system contains many state variables and the dynamic behaviors are very complex. Based on the Routh–Hurwtiz theory and high-dimensional Hopf bifurcation theory, the existence and stability of the Hopf bifurcation are proved. By the hybrid control method, a nonlinear controller is designed to the system for changing the Hopf bifurcation point and expanding the range of the Hopf bifurcation stability. The Hopf bifurcation emerges at an expected location with predicted properties. Meanwhile, the parameters of the nonlinear controller can change the stability of the system. The range of stability can be expanded by choosing appropriate parameter values. The purpose of Hopf bifurcation control and the stability control are realized.

The Pan-like chaotic system has the form

Fig. 1.

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	Fig. 1. (color online) Chaotic attractors of the system (1).

2.1. Hopf bifurcation of equilibrium points

The system has three equilibrium points, given by

Clearly, E₀ is an equilibrium point of the system for all values of the parameters a, b, c, d, e. The equilibrium points E₁ and E₂ of the system exist only when acde (be − 1) (ad + 1) > 0.

The equilibrium points E₁ and E₂ have symmetrical characteristics, so we only analyze the stability of equilibrium points E₁. The Jacobi matrix of system (1) at the equilibrium points E₁ is given by

The characteristic equation of system (1) is where

In the case of a = 7, b = 16, d = −1, e = −2, and c ∈ (0,1.9460), we have

Since the above conditions are met, the real parts of characteristic roots of the characteristic equation (3) are all negative. Based on the Routh–Hurwitz theory, the system is asymptotically stable at equilibrium points E₁.

Take c as the Hopf bifurcation parameter. When c = c₀ = 1.9460,

The matrix J_E1 has a pair of pure imaginary roots, that is The real parts of other characteristic roots are all negative, and Therefore, based on the Hopf bifurcation theory, the system (1) undergoes Hopf bifurcation at c = c₀ = 1.9460.

2.2. Stability of Hopf bifurcation

As shown in Subsection 2.1, the system undergoes Hopf bifurcation at the critical value of bifurcation parameter c = c₀ = 1.9460. The characteristic values of the Jacobi matrix are λ_1.2 = ±7.3644i, λ₃ = −9.0273, λ₄ = −2.0008, and λ₅ = −0.9177. By making a linear transformation of the system as (x₁, y₁, z₁, u_z, v₁)^T = P(x₂, y₂, z₂, u_x, v₂)^T, where

the system has the normal form Because of the complicated expression of F_i (i = 1, 2, 3, 4, 5), we have omitted them in this paper.

Based on the Hopf bifurcation theory, the indicators

can be obtained by detailed calculation, and the stability coefficient is The stability coefficient of system (1) is β₂ > 0. Therefore, the Hopf bifurcation solution of the system is unstable. The local bifurcation diagram which takes c as the Hopf bifurcation parameter is shown in Fig. 2.

Fig. 2.

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	Fig. 2. (color online) Local bifurcation diagram of system (1).

When c > 1.9460, the system is in a stable state which has a stable limit cycle. When c < 1.9460, the Hopf bifurcation appears, which is in an unstable and quasi-periodic state, showing chaotic character finally. The sequence and phase diagram at different values of parameter c are shown in Figs. 3–5.

Fig. 3.

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	Fig. 3. (color online) Sequence and phase diagram at c = 1.85.

Fig. 4.

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	Fig. 4. (color online) Sequence and phase diagram at c = 1.946.

Fig. 5.

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	Fig. 5. (color online) Sequence and phase diagram at c = 2.2.

From the diagram above, when c = 1.85, the system is in an unstable state. The system thus has an unstable periodic solution, showing the chaotic phenomenon in the end. When c = 1.946, the system is unstable. The bifurcation amplitude is gradually increasing. When c = 2.2, the system is in a stable state. This proves that the former theoretical analysis is correct.

We analyze the dynamic characteristics of a Pan-like chaotic system based on the Routh–Hurwtiz theory and the high-dimensional Hopf bifurcation theory. The condition under which the Hopf bifurcation emerges is investigated, and the critical value of bifurcation parameters is calculated. The hybrid control method is used to set a nonlinear controller to control the Hopf bifurcation and stability of the system. Through analysis, it shows that the parameters of the linear controller can change the emergence of Hopf bifurcation of the system, as long as parameters are appropriate. At the same time, the nonlinear part of the control parameter can change the stability of the system and expand the system’s bifurcation stability range. The accuracy of the theoretical analysis is verified by numerical simulation.