Stabilization of a Class of Complex Chaotic Systems by the Dynamic Feedback Control

)e stabilization problem of the complex chaotic system is investigated in this paper. First, a systematic method is proposed, by which a given complex chaotic system can be transformed into its equivalent real chaotic system. )en, both simple and physical controller is designed for the corresponding real chaotic system by the dynamic feedback control method, thereby the controller for the original complex chaotic system is obtained. Especially, for some complex system, the controller is obtained by the linear feedback control method. Finally, two illustrative examples with numerical simulations are used to verify the validity and effectiveness of the theoretical results.


Introduction
It is well known that the first chaotic system was proposed by Lorenz in 1963. From then on, many works have been done about both theoretical results and applications, see Refs. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] and the references therein. Complex chaotic system whose state variables belong to complex space is another important type of chaotic dynamical system, which has been widely investigated in both theorem and applications and has become a hot topic in recent years, for details see Refs. [16][17][18][19][20][21][22][23][24]. Especially, the encryption effect is better due to the fact that the complex chaotic system is composed of real and imaginary numbers. Since the dynamic behavior of the complex system is more complicated than that of the real chaotic system, the control problems of such system is very difficult. Many researchers usually adopted this strategy; they firstly transfer the complex chaotic system into its corresponding real chaotic system by separating the real parts and imaginary parts of the complex state variables and then they investigate the control method of the real chaotic system. Ultimately, the control problems of such complex system were realized. However, on one hand, there is lack of a systematic method in the first step, i.e., for a specific complex chaotic system, a specific method is applied to transform it into its equivalent real chaotic system. How to find a systematic method by which the complex chaotic system can be transformed into its equivalent real chaotic system is not only important in theory but also significant in applications; thus, it stimulates our work in this paper.
On the other hand, most of the controllers designed in the aforementioned existing results are complicated; thereby, they are hard to be performed in real applications. As a matter of fact, how to design a both simple and physical controller to realize the control problems of the complex chaotic systems is also important both in theory and applications. Among the existing methods, the dynamic feedback control method and the linear feedback control method are widely applied, and thus these two methods are adopted by this study.
Motivated by the above conclusions, the stabilization problem of the complex chaotic system is studied by the dynamic feedback control method. e main contributions of this paper are given as follows: (1) A systematic method is proposed, which can be used to transform a given complex chaotic system into its equivalent real chaotic system (2) Both simple and physical controller is designed for the original complex chaotic system is obtained by the dynamic feedback control method and the linear feedback control method, respectively, and numerical simulations are performed to verify the above theoretical results Before ending this section, we present some notations used in this paper. R n is the n dimensional Euclidean space, C n is the n dimensional complex space, I n denotes the n × n identity matrix, M m×n denotes the set of m × n real matrices, ⊗ is the Kronecker product, and ∝ is the semitensor product (STP), i.e., let M m×n ∈ M m×n , N p×q ∈ N p×q , and t � lcm n, p be the least common multiple of n and p. e STP of M and N is defined as where Re(·) and Im(·) represent the real part and the imaginary part of (·), respectively, and i is the imaginary unit, i.e., i 2 � − 1.

Preliminary
Consider the following controlled chaotic system: where w ∈ R n is the state, G(w) ∈ R n is continuous function with G(0) � 0, b ∈ R n×r is a constant matrix, and v ∈ R r is the controller to be designed.
Lemma 1 (see [18]). Consider system (3). If (G(w), b) can be stabilized, then the designed controller v is of the following form: where K � k(t)b T , and the feedback gain k(t) is updated by the following equation:

Problem Formulation
Consider the following controlled complex chaotic system: where z ∈ C m and x ∈ R n− m are the state, m ≥ 1, where z is the conjugate of z, M(x) ∈ R m×m , H(i) ∈ C m×(n− m) is a complex constant matrix, and N(x, z, z) ∈ R n− m , b ∈ R n×r , u ∈ C r is the designed controller, i.e., where that is, u z ∈ C k , u x ∈ R r− k , and 1 ≤ k < r.
Remark 1. e complex chaotic system of equation (6) is very common, which covers a lot of complex chaotic systems, such as complex Lorenz system and complex hyperchaotic Lorenz system.
2 Complexity e goal of this paper is to investigate the stabilization of system (6), i.e., how to design a controller u to guarantee lim t⟶∞ p(t) � 0. (11)

A Systematic Method which is Used to Transform a
Complex System into its Equivalent Real System. In this section, a systematic method proposed, by which a complex system can be transformed into its equivalent real system.

Theorem 1.
Consider the complex chaotic system (6). Its equivalent real system is described as the following form: □ Remark 2. Since the complex chaotic system (6) is equivalent to the corresponding real system (12). us, the stabilization problem of system (12) is investigated and the controller U is designed. Moreover, the controller u of system (6) is obtained by Remark 3. For a given general controlled complex chaotic system of the following form: where z ∈ C n is the state, f(z) ∈ C n is continuous function with f(0) � 0, and b ∈ R n×r and u ∈ C r are the designed controller, r ≥ 1. By eorem 1, its equivalent real system is obtained as follows: where y ∈ R 2n is the state, F(y) ∈ R 2n is continuous function with F(0) � 0, B ∈ R 2n×2r is a constant matrix, and U ∈ R 2r is the controller to be designed, i.e., If the complex chaotic system in which there exist model uncertainty and external disturbance, we present the following result.

Corollary 1.
Consider the following complex chaotic system with both model uncertainty and external disturbance: where p, f(p), b, and u are given in equations (7)- (10), respectively, where Δf(p) ∈ C n stands for the model uncertainty and d(t) ∈ R n is the external disturbance. Its equivalent real system is described as the following form: where y, F(y), B, and U are the same as those in eorem 1, and

Stabilization of the Complex Chaotic System.
In this section, the stabilization of the complex chaotic system is investigated by the dynamic feedback control method and the linear feedback control method, respectively, and the conclusions are presented.
Theorem 2. Consider system (12). If (F(y), B) can be stabilized, then the controller U is designed of the following form: where K � k(t)B T , and k(t) is updated by Remark 4. According to equation (14), the controller u for system (6) is obtained; thus, the stabilization of system (6) is realized.
If system (12) has some special structure, the stabilization problem of such system can be realized by the linear feedback control method, and the result is proposed.

Theorem 3. Consider system (12) of the following form:
i.e., with is globally asymptotically stable. If (F(y), B) is controllable, then the designed controller U is of the following form: where K(X) meets the matrix B Y K(X) + A(X) is Hurwitz whatever X is.
Proof. Since (F(y), B) is controllable, thus (A(X), B Y ) is also controllable whatever X is. According to the pole assignment theory, the controller U given in (27) is as requested, which completes the proof.

Illustrative Examples with Numerical Simulations
In this section, we shall take two complex chaotic systems for example to show how to apply the obtained theoretical results, and then numerical simulations are performed to verify the effectiveness and the validity of the aforementioned theoretical results.

Complexity 5
According to eorem 3, the controller U is designed as follows: where us, Numerical simulation is carried out with the initial conditions: y(0) � [− 5, 3, − 2, − 6, 7]. Figure 4 shows y 1 , y 2 , and y 3 are asymptotically stable, and Figure 5 shows y 4 and y 5 are asymptotically stable, which implies the z(t) and x(t) are stabilized.

Conclusions
In conclusion, the stabilization problem of the complex chaotic system has been studied in this paper. First, a systematic method has been proposed, which is applied to transform the complex chaotic system into its equivalent real chaotic system. en, both simple and physical controllers have been designed for the complex chaotic system by the dynamic feedback control method and the linear feedback control method, respectively. Finally, two illustrative examples with numerical simulations have been performed to verify the validity and effectiveness of the theoretical results.

Data Availability
No data were used in this paper.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.