Chaos in the Shimizu-Morioka Model With Fractional Order

The investigation of dynamical behaviors for fractional-order chaotic systems is a new trend recently. This article is numerically concerned with the Shimizu-Morioka model with a fractional order. We find that chaos exists in the fractional-order model with order less than three by utilizing the fractional calculus techniques, and some phase diagrams are also constructed.


INTRODUCTION
In the past twenty years, many scientists paid their attention on the fractional-order chaotic dynamical systems (see Genesio-Tesi system [1], Rabinovich system [2], and Lü system [3] et al.). They presented chaotic attractors indeed occur in the fractional-order model with order less than 3. Sheu and Chen [4] found that the lowest order of the fractional-order Newton-Leipnik system is 2.82. In 2004, Li and Peng [5] discovered the rich dynamical behavior displayed in the fractional-order Chen system such as the fixed points, limit cycles, periodic motions, and chaotic motions.
The original Shimizu-Morioka model [6] is described by the following ordinary differential equation: x P x, y, z y _ y Q x, y, z x −βy − xz _ z R x, y, z −αz + x 2 (1.1) where (x, y, z) ∈ R 3 are the state variables andα,β are positive real parameters. This model has been proposed as a simplified and an alternative model for studying the dynamics of the well-known Lorenz system [7] for large Rayleigh numbers (Ra), in which the complex behavior of the trajectories has been discovered by means of computer simulation. As in the Lorenz model, the Shimizu-Morioka model is invariant, with respect to the substitution (x, y, z) → (−x, −y, z). The model received much attention due to its stability to describe bifurcation of the associated Lorenz-like strange attractors [8], for example, takingα 0.45 andβ 0.75 ( Figure 1). Intrigued by the above interesting work, many researchers [9,10] focused their study on the dynamical behavior analysis of the Shimizu-Morioka model. In particular, articles [11,12] use feedback control laws and the delay feedback control method [13,14] to study the local and global stabilization and bifurcation of the Shimizu-Morioka chaotic model. If the dynamical system of Eq. 1.1 follows then the system is known to be a dissipative one. In 1992, British scholar Rucklideg studied two-dimensional convection problems of solute gradients and magnetic fields, and introduced the following chaotic system [15]: where (x , y , z ) ∈ R 3 are the state variables and a, b are the positive real parameters. Whenα ≠ 0, by transformation: Therefore, system Eq. 1.5 and system Eq. 1.3 are equivalent when a −βα − 1 , b 1/α 2 , andα ≠ 0.
In this article, Section 2 provides a brief review of the fractional-order operator and discretization fractional-order Shimizu-Morioka model using numerical algorithm. In Section 3, the complex dynamical behaviors of the Shimizu-Morioka model with a fractional order are studied numerically in four cases. Finally, conclusions are given in Section 4.

FRACTIONAL ORDER OPERATOR AND NUMERICAL ALGORITHM
In this section, we first give out the fractional-order differential operator and the Shimizu-Morioka model with a fractional order. Furthermore, we use the predictor-correctors scheme to discrete the fractional-order Shimizu-Morioka model. Last, we discuss the necessary condition for the existence of chaotic attractors.
where m [α] is the first integer which is not less than α, y m is the ordinary m-order derivative, J β is the β-order Riemann-Liouville integral operator defined by where Γ(β) is the gamma function. The classical Riemann-Liouville fractional derivative is defined by which requires the homogeneous initial conditions. The main reason why we chose the Caputo-type fractional derivative is that the inhomogeneous initial conditions are also permitted. The integer-order Shimizu-Morioka model Eq. 1.1 has been extended to the fractional-order Shimizu-Morioka model, which could describe the memory and hereditary properties of the model better. The fractional-order Shimizu-Morioka model is described as follows-in which the standard derivative will be replaced by the fractional-order derivative.
Applying the above formula, system Eq. 2.9 can be discretized as follows: The fractional-order Shimizu-Morioka model of system Eq. 2.9 discretes to system Eq. 2.12. Now, we discuss the necessary condition for the existence of chaotic attractors in the fractional-order Shimizu-Morioka model. Set d q1 x/dt q1 0, d q2 y/dt q2 0, d q3 z/dt q3 0, we get the following equilibrium points of system Eq. 2.9.
Suppose λ is the unstable eigenvalue of the saddle points, then the necessary condition for the fractional-order system Eq. 2.9 to remain chaotic is keeping the eigenvalue λ in the unstable region. By [21], if the eigenvalue λ is in the unstable region, then the following condition is satisfied.
where arg λ i denotes the argument of the eigenvalue λ. That is,

NUMERICAL SIMULATIONS
In what follows, some numerical simulations of system Eq. 2.12 will be studied. We chose the parametersα 0.45, β 0.75, and the initial value (x 0 , y 0 , z 0 ) (1, 1, 2). The phase portraits and time histories are used to research the dynamical behaviors of system Eq. 2.9. Four cases are considered as follows.
3.1 Commensurate Order q 1 q 2 q 3 α System Eq. 2.9 is calculated numerically against α ∈ [0.89, 0.99], while the incremental value of α is 0.01. Figure 2 shows the phase portraits in the x − y space at q i (i 1, 2, 3) 0.99, 0.92, 0.912, and 0.89, respectively. We find that system Eq. 2.9 behaves chaotically when α ∈ [0.92, 0.99] is greater than 0.9152; when α 0.912 is less than 0.9152, system Eq. 2.9 exhibits periodic motion; and when α 0.89, the chaotic motions disappear and the system stabilizes to the fixed point. The numerical simulation results coincide with the necessary conditions for the existence of chaotic attractors that were observed in the last section. The lowest order to yield chaos is 2.76.

q 1 q 3 1 and Let q 2 Vary Less Than one
System Eq. 2.9 is calculated numerically against α ∈ [0.61, 0.97], while the incremental value of α is 0.01. Figure 3 shows the phase portraits in the x − y space at q i (i 1, 3) 1, q 2 0.97, 0.73, 0.71, and 0.61, respectively. We find that system Eq. 2.9 behaves chaotically when α ∈ [0.73, 0.97]; when α 0.71, system Eq. 2.9 exhibits periodic motion; and when α 0.61, the chaotic motions disappear and the system stabilizes to the fixed point.

q 1 q 2 1 and Let q 3 Vary Less Than one
Simulations of system Eq. 2.9 are performed against α ∈ [0.74, 0.99], while the incremental value of α is 0.01. Figure 4 shows the phase portraits in the x − y space at q i (i 1, 2) 1, q 3 0.99, 0.779, 0.77, and 0.74, respectively. We find that system Eq. 2.9 behaves chaotically when

Chaos Control
We will apply feedback control and the fractional Routh-Hurwitz criterion to suppress the three-dimensional fractional Shimizu-Morioka chaotic system. The threedimensional fractional Shimizu-Morioka chaotic controlled system is described as follows: where q ∈ (0, 1), k 1 , k 2 , k 3 are control parameters. E (x, y, z) is the equilibrium of system Eq. 4.13. We will apply linear feedback to stabilize the equilibrium E 0 (0, 0, 0) of system Eq. 4.13. Wheñ α 0.45,β 0.75, the Jacobian matrix of system Eq. 4.17 at E 0 is The corresponding characteristic equation at E 0 is and the discriminant is the equilibrium E 0 is locally asymptotic stability.

CONCLUSION
This article mainly discussed the dynamical behaviors of the fractional-order Shimizu-Morioka model. We find that chaos does exist in the fractional-order model with order less than 3. Future work that requires further consideration regarding this topic includes theoretical analysis of system Eq. 2.9, the largest Lyapunov exponent in the state space, the linear and nonlinear feedback controller, synchronization of this kind of system, and in-depth studies on chaos control for the fractional state.

DATA AVAILABILITY STATEMENT
The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author/s.