Dynamics and synchronization of the complex simplified Lorenz system

In this paper, the complex simplified Lorenz system is proposed. It is the complex extension of the simplified Lorenz system. Dynamics of the proposed system is investigated by theoretical analysis as well as numerical simulation, including bifurcation diagram, Lyapunov exponent spectrum, phase portraits, Poincaré section, and basins of attraction. The results show that the complex simplified Lorenz system has non-trivial circular equilibria and displays abundant and complicated dynamical behaviors. Particularly, the coexistence of infinitely many attractors, i.e., extreme multistability, is discovered in the proposed system. Furthermore, the adaptive complex generalized function projective synchronization between two complex simplified Lorenz systems with unknown parameter is achieved. Based on Lyapunov stability theory, the corresponding adaptive controllers and parameter update law are designed. The numerical simulation results demonstrate the effectiveness and feasibility of the proposed synchronization scheme. It provides a theoretical and experimental basis for the applications of the complex simplified Lorenz system.


Introduction
Since Lorenz discovered the first canonical chaotic attractor in 1963 [1], chaos has been extensively investigated over the last few decades [2][3][4][5][6] due to its great importance and broad applications. Particularly, in 1982, Fowler et al. [7] introduced the complex Lorenz equations as a generalization of the real Lorenz system, which opened the prelude to the study of complex chaos. However, compared with the limited studies of chaos in complex domain, there are a large quantity of articles of chaos in real domain. As a matter of fact, in the real world, many physical phenomena can be described by complex-variable nonlinear dynamical equations, such as thermal convection of liquid flows, detuned lasers, and complex nonlinear oscillators [8][9][10][11][12]. The complex chaotic system, i.e., the chaotic system with complex variables, is a kind of typical complex dynamical system, which is obtained by extending the chaotic system from real domain to complex domain. Due to the existence of complex variables, dynamical behaviors of the complex chaotic system tend to be more abundant and complicated, which indicates the wide scope of applications, especially for secure communication and digital encryption [13][14][15][16][17].
Apart from the above-mentioned complex Lorenz system, some other complex chaotic systems have been proposed and investigated theoretically and numerically, such as the complex Chen and Lü systems [18], the hyperchaotic complex Lorenz system [19], and the hyperchaotic complex Lü system [20]. On the other hand, it is well known that the synchronization of chaotic systems plays an important role in chaotic secure communication. Therefore, some scholars also focused their attention on the study of chaos control and synchronization of complex chaotic systems in recent years, and a great number of synchronization schemes have been proposed, such as complete synchronization [21], lag synchronization [22], complex modified projective synchronization [23], adaptive impulsive synchronization [24], adaptive complex modified projective synchronization [25], combination complex synchronization [26], adaptive complex modified hybrid function projective synchronization [27], and complex modified function projective synchronization [28]. However, the investigation of generalized synchronization of complex chaotic systems, where the complex drive and response systems synchronize with respect to a given functional relationship, was rarely reported. Only in [29], Wang et al. discussed the adaptive complex generalized synchronization of a memristor-based hyperchaotic complex Lü system. Furthermore, to better enhance the anti-attack and anti-translated performance of the transmitted signals in chaotic secure communication, it is significant and meaningful to investigate the adaptive complex generalized function projective synchronization of complex chaotic systems with unknown parameters. To the best of our knowledge, although the generalized function projective synchronization of real-variable chaotic systems has been studied [30,31], the adaptive complex generalized function projective synchronization of complex chaotic systems with unknown parameters was seldom discussed.
Recently, the coexistence of infinitely many attractors, which is defined as extreme multistability [32], has become another research hotspot. In fact, the extreme multistability is an intrinsic property of many nonlinear dynamical systems. It means the coexistence of infinitely many stable states for a given set of parameters and the final state mainly depends on initial conditions. At present, the phenomenon of extreme multistability is found in some nonlinear circuits and dynamical systems [32][33][34][35][36]. Particularly, Bao et al. [37] discovered the hidden extreme multistability in a memristive hyperchaotic system. Lai et al. [38] established a simple three-dimensional chaotic system with infinitely many coexisting attractors based on the Sprott B system [39]. Furthermore, Li et al. [40] successfully doubled the number of the coexisting attractors by apply-ing the offset boosting technique in combination with the absolute-value function. Generally, the system with multiple attractors is considered to be more complicated than the system with single attractor, and thus has more potential values in practical applications [41]. However, the coexistence of infinitely many attractors (extreme multistability) in complex chaotic systems has not been effectively explored so far.
Motivated by the above discussion, in this paper, by extending the state variables from real domain to complex domain, the complex simplified Lorenz system is proposed. It is the complex extension of the simplified Lorenz system [42]. Dynamics and synchronization of the proposed system are investigated theoretically and numerically. The results show that the complex simplified Lorenz system has non-trivial circular equilibria, and exhibits abundant and complicated dynamical behaviors. Particularly, the coexistence of infinitely many attractors, i.e., extreme multistability, is found in this system. Furthermore, the adaptive complex generalized function projective synchronization between two complex simplified Lorenz systems with unknown parameter is achieved. Based on Lyapunov stability theory, the adaptive controllers and parameter update law are designed. The numerical simulation results demonstrate the effectiveness and feasibility of the proposed synchronization scheme.
The rest of this paper is organized as follows. In Sect. 2, the complex simplified Lorenz system is proposed, and basic dynamics of the proposed system is investigated theoretically and numerically. In Sect. 3, the adaptive complex generalized function projective synchronization between two complex simplified Lorenz systems with unknown parameter is discussed. The numerical simulation results are presented to verify the effectiveness and feasibility of the proposed synchronization scheme. Finally, the conclusions are summarized in Sect. 4.

The complex simplified Lorenz system and its basic dynamics
In 2009, Sun and Sprott [42] simplified the classic Lorenz system, and proposed the simplified Lorenz system with only one bifurcation parameter. This system has similar dynamical properties to the Lorenz system, but its algebraic form is simpler. It is suitable for the study of chaos control, synchronization and circuit implementation, where serving as a simple model of chaos in real domain. The simplified Lorenz system is described by where x 1 , x 2 , x 3 ∈ R are real state variables, and c ∈ R is the bifurcation parameter of system (1). This system can be rewritten in the following canonical form [43]: where the sign of a 12 a 21 distinguishes non-equivalent topologies. For the simplified Lorenz system, a 12 a 21 > 0 when c < 6; a 12 a 21 = 0 when c = 6; and a 12 a 21 < 0 when c > 6. Hence, the system includes three different topological structures and has abundant dynamical behaviors. Here, we extend the real state variables x 1 , x 2 from real domain to complex domain, and obtain the complex simplified Lorenz system, which is written in the following form where x 1 , x 2 ∈ C are complex state variables, x 3 ∈ R is a real state variable, and c ∈ R is the bifurcation parameter of system (3). In addition,x 1 ,x 2 stands for the complex conjugate variables of x 1 , x 2 , respectively. For the convenience of dynamics analysis, setting , and equating the real and imaginary parts of system (3), we obtain where u 1 , u 2 , u 3 , u 4 , u 5 ∈ R are real state variables, and c ∈ R is the bifurcation parameter of system (4).

Symmetry and invariance
System (4) is symmetric and invariant under the coordinates transformation: , reflection about u 5axis. In addition, this symmetry and invariance are accurate for all values of the parameter c.

Dissipation and the existence of attractors
The rate of volume contraction of system (4) is given by the Lie derivative which can be solved to yield If p < 0, i.e., c < 34/3, then system (4) is dissipative. It means all the trajectories of system (4) contract at an exponential rate p onto an attractor of zero volume that can be a fixed point, a limit cycle or a strange attractor when t → +∞.
To study the stability of E 0 , we consider the Jacobian matrix of system (4) at E 0 : and its eigenvalues satisfy the following characteristic equation thus we have μ 1 = −8/3, and the other four eigenvalues satisfy the following equation According to the Routh-Hurwitz criterion, if 10 − c > 0 and 30c − 240 > 0, i.e., c ∈ (8, 10), then E 0 is stable, otherwise it is unstable. In addition, to study the stability of E θ , we consider the Jacobian matrix of system (4) at E θ : and find that their eigenvalues satisfy the same characteristic equation, which is written as thus we have μ 1 = 0, μ 2 = c − 10, and the other three eigenvalues satisfy the following equation According to the Routh-Hurwitz criterion, if c ∈ [−1.59, 7.75], then there exist a negative real eigenvalue and a pair of conjugate complex eigenvalues with positive real part. For example, if c = 2, then the corresponding eigenvalues are μ 1 = 0, μ 2 = −8, μ 3 = −11.7318, μ 4 = 0.5326 + 9.0302 j, and μ 5 = 0.5326−9.0302 j, which indicates that E θ are unstable and saddle-foci with index two.

Bifurcation and chaos with different system parameter c
As it is well known, bifurcation diagram and Lyapunov exponent spectrum are two of the major methods for analyzing dynamical behaviors of a nonlinear system. Here, the classic fourth-order Runge-Kutta method is applied for the calculation of system (4). When the initial condition u 0 = (1, −1, 1, −1, 1), the bifurcation diagram and Lyapunov exponents of system (4) are shown in Fig. 1a  The figure also illustrates that the transition to chaos is apparently different at the two extremes of the system parameter c. When c increases from −∞ to −1.67, system (4) enters into chaos by a boundary crisis bifurcation at c ∼ = −1.67, whereas a decrease in c from +∞ causes system (4)  In particular, we set c = 2 with the initial condition u 0 = (1, −1, 1, −1, 1), then all the five Lyapunov exponents of system (4) can be calculated as λ 1 = 0.8580, λ 2 = 0, λ 3 = 0, λ 4 = −7.9995, and λ 5 = −11.5260. Therefore, the Lyapunov dimension of system (4) is fractional, which is defined as where m is the largest integer such that m k=1 λ k > 0 and m+1 k=1 λ k < 0. Furthermore, the phase portrait on (u 1 , u 5 ) plane and its corresponding Poincaré section generated in the hyperplane u 5 = 18 (which passes through the non-trivial equilibria) are plotted in Fig.  2a and b, respectively. Obviously, as observed in Fig.  2, system (4) is chaotic and there exists a standard butterfly-shaped attractor in this case.

Coexistence of infinitely many attractors
The coexistence of infinitely many attractors, which is defined as extreme multistability, exhibits a rich diversity of stable states of a nonlinear dynamical system and makes such system offer a great flexibility without changing the system parameters. The coexis-tence of infinitely many attractors depending on initial conditions is found in system (4). For instance, the phase portraits of five coexisting chaotic attractors (c = 6.39) as well as five coexisting periodic attractors (c = 7.50) of system (4) are shown in Fig. 3a and b, respectively. As shown in Fig. 3, different colors represent different attractors with different initial conditions, where the blue attractors match (1, 2, 3, 4, 5), the red attractors match (2, 2, 3, 4, 5), the green attractors match (3, 2, 3, 4, 5), the magenta attractors match (4, 2, 3, 4, 5), and the cyan attractors match (5, 2, 3, 4, 5). Moreover, it is well known that the basin of attraction is an effective method to detect the coexistence of attractors in a nonlinear system. In particular, when the initial condition of system (4) is given as u 0 = (u 1 , u 2 , 3, 4, 5), the basins of attraction with respect to different initial conditions at c = 6.39 and c = 7.50 are shown in Fig. 4a and b, respectively, where u 1 ∈ [−10, 10], u 2 ∈ [−10, 10], and different colors represent different attractors. Obviously, due to the coexistence of infinitely many attractors, system (4) has the property of extreme multistability, which is closely related to its infinite non-trivial circular equilibria. The phenomenon of extreme multistability reveals the abundant and complicated dynamical behaviors of the complex simplified Lorenz system and indicates its great potential values in practical applications.

The adaptive complex generalized function projective synchronization of the complex simplified Lorenz system with unknown parameter
In this section, based on Lyapunov stability theory, the adaptive complex generalized function projective synchronization between two complex simplified Lorenz systems with unknown parameter is investigated.

Mathematical model and problem description
Firstly, we consider the following n-dimensional complex drive systeṁ and the n-dimensional complex response system with adaptive controller is depicted aṡ U(x, y,Â,B), (17) where x = (x 1 , x 2 , . . . , x n ) T ∈ C n and y = (y 1 , y 2 , . . . , y n ) T ∈ C n are complex state vectors of the complex drive system (16) and complex response system (17), respectively. A = (a 1 , a 2 , . . . , a p ) T ∈ R p and B = (b 1 , b 2 , . . . , b q ) T ∈ R q are real vectors of unknown parameters of the complex drive system (16) and complex response system (17), respectively. F : C n×1 → C n× p is a n × p complex function matrix and f : C n×1 → C n×1 is a n × 1 complex function vector. G : C n×1 → C n×q is a n × q complex function matrix and g : C n×1 → C n×1 is a n × 1 complex function vector. U(x, y,Â,B) is the undetermined adaptive controller.Â,B are parameter estimations of the unknown parameters A and B. The adaptive complex generalized function projective synchronization error vector e(t) is defined as

y = G( y)B + g( y) +
where (t) is a complex function transformation matrix. Therefore, the adaptive complex generalized function projective synchronization error dynamical system is written aṡ + U(x, y,Â,B). (19) Next, we introduce the definition of adaptive complex generalized function projective synchronization of complex chaotic systems with unknown parameters.
Definition 1 For the complex drive system (16) and complex response system (17), if there exists an adaptive controller U(x, y,Â,B), such that lim t→+∞ y(t)− (t)x(t) = 0, then the adaptive complex generalized function projective synchronization between the complex drive system (16) and complex response system (17) is achieved asymptotically with respect to a complex function transformation matrix (t) when t → +∞, and the parameter estimationsÂ,B converge to the true values of the unknown parameters A and B, respectively.
As mentioned in [21,[27][28][29], the well-known adaptive control [44][45][46] is an effective method to achieve synchronization and parameter identification of complex chaotic systems with unknown parameters. Here, we propose the adaptive complex generalized function projective synchronization scheme of complex chaotic systems with unknown parameters based on Lyapunov stability theory and adaptive control. x(0) and y(0), if the adaptive controller U(x, y,Â,B) is designed as   U(x, y,Â,B)

Theorem 1 For the given complex function transformation matrix (t), along with the initial conditions
where K = diag[k 1 , k 2 , . . . , k n ] is the feedback gain matrix, whose diagonal elements are negative real constants, i.e., k l < 0 (l = 1, 2, . . . , n), and the corre-sponding parameter update laws are selected as then the adaptive complex generalized function projective synchronization between the complex drive system (16) and complex response system (17) is achieved asymptotically when t → +∞, andÂ,B converge to the true values of A and B, respectively.

Hypothesis For the given complex function transformation matrix (t), along with the complex function matrices F(x) and G( y), the complex function vector group, consisting of all the columns of (t)F(x) and −G( y), is linear independence.
Proof Substituting Eq. (20) into Eq. (19), we havė and its derivative with respect to time iṡ V e r , e i ,Ã,B = e r T K e r + e i T K e i .
Consequently, the adaptive complex generalized function projective synchronization between the complex drive system (16) and complex response system (17) is achieved asymptotically with respect to a complex function transformation matrix (t) by using the adaptive controller (20) and parameter adaptive laws (21). Besides, according to the LaSalle's invariance principle [48], the maximal invariant set can be written as Therefore, we have (t)F(x)Ã−G( y)B = 0, and due to the linear independence hypothesis, we can conclude thatÃ = 0 andB = 0, i.e., the unknown parameters A and B can be correctly identified. The proof is completed.
where K = diag[k 1 , k 2 , . . . , k n ] is the feedback gain matrix, whose diagonal elements are negative real constants, i.e., k l < 0 (l = 1, 2, . . . , n), and the corresponding parameter update law is selected aṡ Therefore, the adaptive complex generalized function projective synchronization between two identical complex chaotic systems with unknown parameters can also be achieved asymptotically when t → +∞, andÂ converges to the true value of A.

Numerical simulations and discussion
To investigate the adaptive complex generalized function projective synchronization between two complex simplified Lorenz systems with unknown parameter, we define the complex drive system as (28) and the complex response system with adaptive controllers is depicted as where v 1 , v 2 , v 3 are undetermined adaptive controllers, and c is the unknown parameter. If the complex function transformation matrix (t) is chosen as then the errors between the complex drive system (28) and complex response system (29) are expressed by therefore, the adaptive complex generalized function projective synchronization error dynamical system is written as − 10(x 2 − x 1 )exp(2 jt) + v 2 , e 3 = − 8y 3 /3 + 1/2(ȳ 1 y 2 + y 1ȳ2 ) + 3x 3 sin 3t whereĉ is the estimation of the unknown parameter c, and its parameter update law is selected aṡ then the adaptive complex generalized function projective synchronization between the complex drive system (28) and response system (29) with unknown parame-ter c is achieved asymptotically when t → +∞, andĉ converges to the true value of c.
In the numerical simulations, let c = 2. In this case, the complex simplified Lorenz system behaves chaotically. The initial conditions are randomly selected as x(0) = (1 + 2 j, 3 + 4 j, 5) and y(0) = (6 + 7 j, 8 + 9 j, 10). The initial value of the parameter estimationĉ and the feedback gain matrix K are set asĉ(0) = 2.5 and K = diag[−5, −5, −5], respectively. Time evolutions of the synchronization error and parameter estimation are plotted in Fig. 5, where |e| = |e 1 | 2 + |e 2 | 2 + |e 3 | 2 . As observed in Fig. 5, the numerical simulation results illustrate that the adaptive complex generalized function projective synchronization between the complex drive system (28) and complex response system (29) has been achieved almost within two seconds, and the unknown parameter c is successfully identified. Additionally, the feedback gain matrix K has a great effect on the performance of the proposed synchronization scheme. Time evolutions of the synchronization error and parameter estimation with different feedback gain matrices are plotted in Fig. 6 Obviously, as shown in Fig. 6, the adaptive com-plex generalized function projective synchronization between the complex drive system (28) and complex response system (29) is achieved more rapidly with a stronger feedback gain matrix. Meanwhile, the unknown parameter c is also identified more quickly.

Conclusions
In this paper, by extending the state variables from real domain to complex domain, the complex simplified Lorenz system is proposed. Dynamics and synchronization of the proposed system are investigated theoretically and numerically. The results show that the complex simplified Lorenz system has non-trivial circular equilibria and displays abundant and complicated dynamical behaviors. Particularly, the coexistence of infinitely many attractors, i.e., extreme multistability, is revealed by phase portraits and basins of attraction. Furthermore, the adaptive complex generalized function projective synchronization between two complex simplified Lorenz systems with unknown parameter is achieved. Based on Lyapunov stability theory, the corresponding adaptive controllers and parameter update law are designed. The numerical simulation results demonstrate the effectiveness and feasibility of the proposed synchronization scheme. Additionally, it is found that the feedback gain matrix has a great effect on the performance of the proposed synchronization scheme. With a stronger feedback gain matrix, the synchronization between the complex drive system and complex response system is achieved more rapidly, and the unknown parameter is also identified more quickly. In chaotic secure communication, using complex variables (doubling the number of variables) increases the content and security of the transmitted information. The proposed synchronization scheme can be applied to secure communication for higher transmission efficiency and security performance due to the complex variables, unknown parameters, and unpredictable complex function transformation matrix. Next, we will try to study the hardware implementation and engineering applications of the complex simplified Lorenz system.