A Fractional-Order Chaotic System with an Infinite Number of Equilibrium Points

Abstract

A new 4D fractional-order chaotic system, which has an infinite number of equilibrium points, is introduced. There is no-chaotic behavior for its corresponded integer-order system. We obtain that the largest Lyapunov exponent of this 4D fractional-order chaotic system is 0.8939 and yield the chaotic attractor. A chaotic synchronization scheme is presented for this 4D fractional-order chaotic system. Numerical simulations is verified the effectiveness of the proposed scheme.

1. Introduction

Because the chaotic (hyperchaotic) signal can be used in electrical engineering, telecommunications, information processing, material engineering, and so forth much attention has been paid to effectively generating chaotic and hyperchaotic systems. Many chaotic (hyperchaotic) systems and fractional-order chaotic (hyperchaotic) systems are reported in recent years [1–14], such as Lorenz chaotic (hyperchaotic) system and its corresponded fractional-order system, and integer-order and fractional-order Chen chaotic (hyperchaotic) system, integer-order and fractional-order Lü chaotic (hyperchaotic) system.

However, for all the previous integer-order and fractional-order chaotic (hyperchaotic), many systems have a finite number of equilibrium points. For example, some chaotic systems have one equilibrium point [15–17], some chaotic systems have two equilibrium points [18], and some chaotic systems have three equilibrium points [1, 2, 5, 6, 9, 10], so a natural and interesting question is can we construct a chaotic (hyperchaotic) system which has an infinite number of equilibrium points? Moreover, many fractional-order chaotic and hyperchaotic systems also possess chaotic attractor for its corresponded integer-order system, so the other question is as follows: are the fractional-order chaotic and hyperchaotic systems no-chaotic behavior for its corresponded integer-order system? To the best of our knowledge, few results on the above mentioned two questions are reported.

Motivated by the above discussions, a new 4D fractional-order chaotic system is presented in this paper. This new 4D fractional-order chaotic system has an infinite number of equilibrium points, and no-chaotic behavior for its corresponded integer-order system. The largest Lyapunov exponent and chaotic attractor are yielded for the new 4D fractional-order chaotic system. A chaotic synchronization scheme is presented for this new 4D fractional-order chaotic system.

2. A New 4D Fractional-Order Chaotic System

Now, a new 4D fractional-order chaotic system is constructed, which is described as follows: 𝑑^𝑞𝑥₁𝑑𝑡^𝑞=10𝑥₂−𝑥₁+𝑥₄𝑑^𝑞𝑥₂𝑑𝑡^𝑞=15𝑥₁−𝑥₁𝑥₃𝑑^𝑞𝑥₃𝑑𝑡^𝑞=−2.5𝑥₃+4𝑥₁²𝑑^𝑞𝑥₄𝑑𝑡^𝑞=−10𝑥₂−𝑥₄, (1) where 𝑞=0.95 is the fractional-order, and 𝑥_𝑖(𝑖=1,2,3,4) are real state variables.

The real equilibrium points of system (1) is calculated by 10𝑥₂−𝑥₁+𝑥₄=015𝑥₁−𝑥₁𝑥₃=0−2.5𝑥₃+4𝑥₁²=0−10𝑥₂−𝑥₄=0. (2)

Obviously, (𝑥₁, 𝑥₂, 𝑥₃, 𝑥₄) = (0, 𝑥₂, 0, −10𝑥₂) is the real equilibrium points of system (1), where 𝑥₂ is a any real numbers, so system (1) has an infinite number of real equilibrium points. To the best of our knowledge, this result is different from all the previous fractional-order chaotic and hyperchaotic systems. It implies that we yield a new 4D fractional-order system, which has an infinite number of real equilibrium points.

The Jacobian 𝐽 at all equilibrium points is 𝐽=−1010011500000−2.500−100−1 (3) and its eigenvalues are 𝜆₁=−18.548, 𝜆₂=−2.5, 𝜆₃=0, and 𝜆₄=7.548 for all 𝑥₂. Therefore, all the equilibrium points in system (1) are unstable.

The dynamical behaviors of system (1) for its corresponded integer-order system (𝑞=1) can be characterized by its Lyapunov exponents. The Lyapunov exponents for its corresponded integer-order system are 0, 0, −0.779, and −12.724, respectively. Therefore, the fractional-order system (1) no-chaotic behaviors for 𝑞=1, and which is periodic orbit for its corresponded integer-order system. Figure 1 shows the periodic orbit of fractional-order system (1) for its corresponded integer-order system (𝑞=1).

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Figure 1 

The periodic orbit of fractional-order system (1) for its corresponded integer-order system (𝑞=1).

Now, we discuss the numerical solution of fractional differential equations. It is well known that there are direct time domain approximation (the improved version of Adams-Bashforth-Moulton numerical algorithm) and frequency domain approximation for nonlinear fractional-order system [6]. However, frequency domain approximation may result in wrong consequences [19], so the direct time domain approximation [6] numerical simulation is used to solve the fractional-order system in this paper. Let ℎ=𝑇/𝑁, 𝑡_𝑛=𝑛ℎ(𝑛=0,1,2…,𝑁), and initial condition (𝑥₁(0),𝑥₂(0),𝑥₃(0),𝑥₄(0)), so the fractional-order chaotic system (1) can be discretized as follows: 𝑥₁(𝑛+1)=𝑥₁(0)+ℎ^𝑞Γ×𝑥𝑞+210^𝑝₂(𝑛+1)−𝑥^𝑝₁(𝑛+1)+𝑥^𝑝₄()+𝑛+1_𝑛𝑗=0𝛼_{1,𝑗,𝑛+1}×10𝑥₂𝑗−𝑥₁𝑗+𝑥₄𝑗𝑥₂(𝑛+1)=𝑥₂(0)+ℎ^𝑞Γ×𝑞+215𝑥^𝑝₁(𝑛+1)−𝑥^𝑝₁(𝑛+1)𝑥^𝑝₃()+𝑛+1_𝑛𝑗=0𝛼_{2,𝑗,𝑛+1}15𝑥₁𝑗−𝑥₁𝑗𝑥₃𝑗𝑥₃(𝑛+1)=𝑥₃(0)+ℎ^𝑞Γ×4𝑥𝑞+2_𝑝1(𝑛+1)²−2.5𝑥^𝑝₃()+𝑛+1_𝑛𝑗=0𝛼_{3,𝑗,𝑛+1}4𝑥₁𝑗²−2.5𝑥₃𝑗𝑥₄(𝑛+1)=𝑥₄(0)+ℎ^𝑞Γ×𝑞+2−10𝑥^𝑝₂(𝑛+1)−𝑥^𝑝₄()+𝑛+1_𝑛𝑗=0𝛼_{4,𝑗,𝑛+1}×−10𝑥₂𝑗−𝑥₄𝑗, (4) where 𝑥^𝑝₁(𝑛+1)=𝑥₁(0)+1Γ𝑞×^𝑛_𝑗=0𝑏_{1,𝑗,𝑛+1}×10𝑥₂𝑗−𝑥₁𝑗+𝑥₄𝑗𝑥^𝑝₂(𝑛+1)=𝑥₂(0)+1Γ𝑞×^𝑛_𝑗=0𝑏_{2,𝑗,𝑛+1}15𝑥₁𝑗−𝑥₁𝑗𝑥₃𝑗𝑥^𝑝₃(𝑛+1)=𝑥₃(0)+1Γ𝑞×^𝑛_𝑗=0𝑏_{3,𝑗,𝑛+1}4𝑥₁𝑗²−2.5𝑥₃𝑗𝑥^𝑝₄(𝑛+1)=𝑥₄(0)+1Γ𝑞×^𝑛_𝑗=0𝑏_{4,𝑗,𝑛+1}×−10𝑥₂𝑗−𝑥₄𝑗, (5) and for 𝑖=1,2,3,4𝛼_{𝑖,𝑗,𝑛+1}=𝑛^𝑞+1−𝑛−𝑞(𝑛+1)^𝑞,𝑗=0𝑛−𝑗+2^𝑞+1+𝑛−𝑗^𝑞+1−2𝑛−𝑗+1^𝑞+1𝑏,1≤𝑗≤𝑛1,𝑗=𝑛+1,_{𝑖,𝑗,𝑛+1}=ℎ^𝑞𝑞𝑛−𝑗+1^𝑞−𝑛−𝑗^𝑞,0≤𝑗≤𝑛 (6)

The error of this approximation is described as follows: 𝑥_𝑖𝑡_𝑛−𝑥_𝑖(𝑛)=𝑜ℎ^𝑝,𝑝=min2,1+𝑞. (7)

The dynamical behaviors of 4D fractional-order system (1) can be characterized by its largest Lyapunov exponent. By computer simulation, we can obtain that the largest Lyapunov exponent of fractional-order system (1) is 0.8939, so the 4D fractional-order system (1) is chaotic. The chaotic attractor is shown in Figure 2.

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Figure 2 

The chaotic attractor of 4D fractional-order system (1).

According to the above mentioned, we obtain a new 4D fractional-order chaotic system, which has an infinite number of real equilibrium points. Moreover, the 4D fractional-order chaotic system is no-chaotic behaviors for its corresponded integer-order system (𝑞=1). The result in our paper is different from all the previous fractional-order chaotic and hyperchaotic systems.

3. Chaotic Synchronization for the New 4D Fractional-Order Chaotic System

In this section, the chaotic synchronization for the new 4D fractional-order chaotic system (1) is considered. Based on the stability theory of nonlinear fractional-order systems [20–24], one synchronization scheme is proposed, and some numerical simulations are performed.

Now, the response fractional-order chaotic system is considered as 𝐷^𝑞𝑦₁=10𝑦₂−𝑦₁+𝑦₄+𝑢₁𝐷^𝑞𝑦₂=15𝑦₁−𝑦₁𝑦₃+𝑢₂𝐷^𝑞𝑦₃=−2.5𝑦₃+4𝑦₁²+𝑢₃𝐷^𝑞𝑦₄=−10𝑦₂−𝑦₄+𝑢₄, (8) where 𝑢_𝑖(𝑖=1,2,3,4) is the feedback controller, and 𝑦_𝑖(𝑖=1,2,3,4) are real state variables. Our goal is to choose suitable 𝑢_𝑖(𝑖=1,2,3,4) such that drive system (1) and response system (8) can be achieved with chaotic synchronization.

Definition the synchronization errors are 𝑒_𝑖=𝑦_𝑖−𝑥_𝑖(𝑖=1,2,3,4). The following Theorem 1 is given in order to achieve the chaotic synchronization between the fractional-order chaotic system (1) and the fractional-order chaotic system (8).

Theorem 1. If the feedback controllers are 𝑢₁=𝑦₃−25𝑒₂−4𝑦₁+𝑥₁𝑒₃𝑢₂=10𝑒₄𝑢₃=𝑥₁𝑒₂𝑢₄=−𝑒₁, (9) then the chaotic synchronization between fractional-order chaotic system (1) and fractional-order chaotic system (8) can be arrived.

Proof. Combining the fractional-order chaotic system (1), fractional-order chaotic system (8), and the feedback controller (9), we can obtain the following error system 𝐷^𝑞𝑒₁𝐷^𝑞𝑒₂𝐷^𝑞𝑒₃𝐷^𝑞𝑒₄=−10−15+𝑦₃−4𝑦₁+𝑥₁115−𝑦₃0−𝑥₁4𝑦10₁+𝑥₁𝑥₁×𝑒−2.50−1−100−1₁𝑒₂𝑒₃₄𝑒_Δ𝑒=𝐴𝑥,𝑦¹𝑒₂𝑒₃𝑒₄, (10) where matrix 𝐴𝑥,𝑦=^Δ−10−15+𝑦₃−4𝑦₁+𝑥₁115−𝑦₃0−𝑥₁4𝑦10₁+𝑥₁𝑥₁−2.50−1−100−1, (11)x=(x₁,x₂,x₃,x₄)^T, y=(y₁,y₂,y₃,y₄)^T.
Let 𝜆 be one of the eigenvalues of 𝐴(𝑥,𝑦) and 𝜓 is the corresponding nonzero eigenvector, so 𝐴𝜓𝑥,𝑦𝜓=𝜆𝜓,_𝐇𝐴𝑥,𝑦^𝐇=𝜆𝜓^𝐇, (12) where H is conjugate transpose, and 𝜆 is the conjugate for eigenvalues 𝜆.
According to (12), one can obtain 𝜓^𝐇𝐴𝑥,𝑦𝜓+𝜓^𝐇𝐴𝑥,𝑦^𝐇𝜓=𝜓^𝐇𝜆𝜓+𝜆𝜓^𝐇𝜓. (13) Therefore 𝜆+𝜓𝜆=^𝐇diag(−20,0,−5,−2)𝜓𝜓^𝐇𝜓, (14) so 𝜆+𝜆≤0. (15) That is |𝜆|≥𝜋arg2>𝑞𝜋2. (16)
Using the stability theory of nonlinear fractional-order systems, one can yield that the error system (10) is asymptotically stable, so lim_𝑡→+∞𝑒_𝑖=0(𝑖=1,2,3,4). (17) Equation (17) indicates that the chaotic synchronization between fractional-order chaotic system (1) and fractional-order chaotic system (8) can be achieved. The proof is completed.

Now, numerical simulations are considered. The numerical results are shown as Figure 3, in which the initial conditions are 𝐱=(3,3,1,2)^𝐓 for drive system (1), and 𝐲=(8,7,4,6)^𝐓 for response system (8), respectively.

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Figure 3 

The synchronization errors between the drive systems (1) and the response system (8).

4. Conclusions

In this paper, we obtain a new 4D fractional-order chaotic system, which has an infinite number of equilibrium points and no-chaotic behavior for its corresponded integer-order system. We yield the largest Lyapunov exponent of the new 4D fractional-order system and the Lyapunov exponents for its corresponded integer-order system. The chaotic attractor for the new 4D fractional-order chaotic system and the periodic orbit for its corresponded integer-order system are given. Finally, we realize the chaotic synchronization for the new 4D fractional-order chaotic system, and some numerical simulations are performed.

Acknowledgments

This work was supported by the Foundation of Science and Technology Project of Chongqing Education Commission (KJ110525, KJ100513) and by the National Natural Science Foundation of China (61104150).