A Memristor-Based Complex Lorenz System and Its Modified Projective Synchronization

Abstract

The aim of this paper is to introduce and investigate a novel complex Lorenz system with a flux-controlled memristor, and to realize its synchronization. The system has an infinite number of stable and unstable equilibrium points, and can generate abundant dynamical behaviors with different parameters and initial conditions, such as limit cycle, torus, chaos, transient phenomena, etc., which are explored by means of time-domain waveforms, phase portraits, bifurcation diagrams, and Lyapunov exponents. Furthermore, an active controller is designed to achieve modified projective synchronization (MPS) of this system based on Lyapunov stability theory. The corresponding numerical simulations agree well with the theoretical analysis, and demonstrate that the response system is asymptotically synchronized with the drive system within a short time.

1. Introduction

Memristor is a passive two-terminal fundamental circuit element with the characteristics of nonlinearity, non-volatility, low power consumption, etc. It has widely potential applications in electronic circuits and systems, neural networks, computer memory, nonlinear science, and so on. In recent years, generation and nonlinear analyses of memristor-based chaotic systems have attracted more and more attention, and become a research hotspot. By replacing Chua’s diodes with memristors, adding memristors to existing nonlinear systems, and designing new systems with memristors, some new chaotic and hyperchaotic systems have been constructed, such as a memristor-based Chua’s circuit [1,2,3,4,5,6], memristor-based van der Pol oscillator [7], memristor-based Lorenz system [8,9,10], memristor-based Chen system [11], memristor-based Lü system [12], etc. Theoretical analyses, numerical simulations and circuit experiments consistently indicate that such memristor-based systems can exhibit complex dynamical behaviours including various bifurcation [1,2,3,4,5,6,7,8,9,10,11,12], chaos [1,2,3,4,5,6,7,8,9,10,11,12], hyperchaos [12,13,14], coexistence of attractors [15], transient dynamical behaviors [16], and initial state dependent dynamical behaviors [17]. Moreover, several fractional-order memristor-based systems and their dynamical behaviors are presented in [9,18,19]. However, the above-mentioned systems are all real systems, and to our best knowledge, there are no reports of memristor-based complex systems. Complex systems are usually constructed to describe and simulate the physics of detuned lasers, rotating fluids, disk dynamos, electronic circuits, etc. Several chaotic and hyperchaotic complex systems have been proposed in [20,21,22]. The studies have revealed more abundant dynamical behaviors of such complex systems, and have concluded that chaotic secure communication with complex systems can increase transmission contents, and enhance anti-attack ability. Due to the advantages of memristor and complex system, it is meaningful and challenging for researchers to construct memristor-based complex systems and to investigate their synchronization for secure communication.

Chaos synchronization plays an important role in secure communication, digital cryptography, image encryption, signal and control processing, etc. [22,23,24,25,26,27,28,29], and it has been a popular research area. Recently, a number of control schemes have been developed to achieve various synchronization of memristor-based systems and complex systems. The adopted synchronization schemes include adaptive control [30,31,32], backstepping control [33], fuzzy control [8], impulsive control [34,35], etc. The realized synchronization types include complete synchronization [36,37], anti-synchronization [32], projective synchronization [23,38], lag synchronization [24,39], combination synchronization [40,41], compound synchronization [10], etc. However, to our best knowledge, memristor-based complex systems are not involved in the existing literature, unfortunately.

Motivated by the above discussions, this paper aims to construct a novel complex Lorenz system with a flux-controlled memristor, and to realize modified projective synchronization (MPS) of two identical memristor-based complex Lorenz systems. The system is generated from a 4D (four-dimensional) memristor-based real Lorenz system by replacing two real variables to complex variables, which can be expressed as a 6D (six-dimensional) real ordinary differential equations (ODEs). Symmetry and invariance, dissipation, equilibria and stability of the system are analyzed theoretically. Dynamical behaviors are numerically investigated by means of bifurcation diagram, Lyapunov exponent, time-domain waveform, and phase portrait. Fixed points, limit cycles, torus, chaos, and transient behaviors are revealed in the system with different parameters and initial conditions. Furthermore, MPS is designed to synchronize the memristor-based Lorenz systems quickly and efficiently, which is verified by numerical simulations.

The rest of this paper is organized as follows. In Section 2, a novel memristor-based complex Lorenz system is proposed, and its properties are analyzed theoretically. In Section 3, dynamical behaviors of the system are explored numerically. In Section 4, an active controller is designed to realize MPS for identical memristor-based complex Lorenz systems, and numerical simulations are presented to illustrate the effectiveness of the proposed scheme. Finally, some conclusions are drawn in Section 5.

2. A Novel Memristor-Based Complex Lorenz System and Its Properties

In this section, we construct a novel memristor-based complex Lorenz system and analyze its properties including symmetry, dissipation, equilibria and stability.

2.1. Memristor Model

Memristor, a nonlinear resistor with memory effect, is defined by $f(\phi ,q)=0$, which includes flux-controlled memristor and charge-controlled memristor. Equations (1) and (2) describe flux-controlled memristor and charge-controlled memristor respectively [42]:

$$W(\phi )=\frac{dq(\phi )}{d\phi },i(t)=W(\phi )v(t),v=\frac{d\phi (t)}{dt}$$

(1)

$$M(q)=\frac{d\phi (q)}{dq},v(t)=M(q)i(t),i=\frac{dq(t)}{dt}$$

(2)

where $\phi$ and $q$ denote magnetic flux and charge; $i$ and $v$ denote the device terminal current and voltage; $W(\phi )$ and $M(q)$ denote memductance and memristance which are characterized by a piece-wise linear function [1], smooth continuous cubic nonlinear function [13], Chebyshev polynomials [19], etc. In this paper, a flux-controlled memristor with cubic nonlinear characteristics is considered to be:

$$q(\phi )=\alpha \phi +\beta {\phi }^3,W(\phi )=\frac{dq(\phi )}{d\phi }=\alpha +3\beta {\phi }^2$$

(3)

where $\alpha$ and $\beta$ are positive parameters.

2.2. Memristor-Based Complex Lorenz System

In [8], a modified Lorenz system with a flux-controlled memristor is described by:

$$\{\begin{array}{l}{\dot{x}}_1=-a_1{x}_1-W(x_4)x_1+a_2{x}_2\hfill \\ {\dot{x}}_2=a_3{x}_1-x_2-x_1{x}_3\hfill \\ {\dot{x}}_3=x_1{x}_2-a_4{x}_3\hfill \\ {\dot{x}}_4=-x_1\hfill \end{array}$$

(4)

where $a_1,a_2,a_3,a_4$ are positive parameters, $W(x_4)$ is memductance characterized by $W(x_4)=\alpha +3\beta x_4^2$ in this paper. By replacing real variables $x_1,x_2$ with complex variables, we construct a memristor-based complex Lorenz system of the form:

$$\{\begin{array}{l}{\dot{x}}_1=-(a_1+\alpha )x_1+a_2{x}_2-3\beta x_1{x}_4^2\hfill \\ {\dot{x}}_2=a_3{x}_1-x_2-x_1{x}_3\hfill \\ {\dot{x}}_3=(1/2)(x_1{\overline{x}}_2+{\overline{x}}_1{x}_2)-a_4{x}_3\hfill \\ {\dot{x}}_4=-(1/2)({\overline{x}}_1+x_1)\hfill \end{array}$$

(5)

where $a_1,a_2,a_3,a_4,\alpha ,\beta$ are positive parameters, $x_1,x_2$ are complex variables, $x_3,x_4$ are real variables, ${\overline{x}}_1,{\overline{x}}_2$ denote the complex conjugate variables of $x_1,x_2$, ${\dot{x}}_j(j=1-4)$ denotes derivative with respect to time. If $x_1=u_1+i{u}_2,x_2=u_3+i{u}_4$, $x_3=u_5,x_4=u_6$, $i=\sqrt{-1}$, the system (5) can be rewritten in the form of six real first ODEs:

$$\{\begin{array}{l}\begin{array}{l}{\dot{u}}_1=-(a_1+\alpha )u_1+a_2{u}_3-3\beta u_6^2{u}_1\\ {\dot{u}}_2=-(a_1+\alpha )u_2+a_2{u}_4-3\beta u_6^2{u}_2\end{array}\hfill \\ \begin{array}{l}{\dot{u}}_3=a_3{u}_1-u_3-u_1{u}_5\\ {\dot{u}}_4=a_3{u}_2-u_4-u_2{u}_5\end{array}\hfill \\ {\dot{u}}_5=u_1{u}_3+u_2{u}_4-a_4{u}_5\hfill \\ {\dot{u}}_6=-u_1\hfill \end{array}$$

(6)

2.3. Properties of the Memristor-Based Complex Lorenz System

2.3.1. Symmetry and Invariance

Due to the invariance of Equation (6) under the transformation from $(u_1,u_2,u_3,u_4,u_5,u_6)$ to $(-u_1,-u_2,-u_3,-u_4,u_5,-u_6)$, the system is symmetrical about the $u_5$ axes. That is, if $(u_1,u_2,u_3,u_4,u_5,u_6)$ is a solution of system (6), then $(-u_1,-u_2,-u_3,-u_4,u_5,-u_6)$ is also a solution of the system.

2.3.2. Dissipation

The divergence of system (6) is:

$$\begin{array}{c}\nabla V={\displaystyle \sum _{j=1}^6\frac{\partial \dot{u}}{\partial u}}=-(a_1+\alpha )-3\beta u_6^2-(a_1+\alpha )-3\beta u_6^2-1-1-a_4\\ =-2(a_1+\alpha +1)-6\beta u_6^2-a_4\end{array}$$

(7)

where $a_1,a_4,\alpha ,\beta$ are positive parameters as mentioned, which can guarantee $\nabla V<0$, so the system (6) is dissipative.

2.3.3. Equilibria and Stability

By solving ${\dot{u}}_1={\dot{u}}_2={\dot{u}}_3={\dot{u}}_4={\dot{u}}_5={\dot{u}}_6=0$, we can obtain the equilibria of system (6):

$$\{\begin{array}{l}{E}_{01}=\{(u_1,u_2,u_3,u_4,u_5,u_6)|u_1=u_2=u_3=u_4=u_5=0,u_6=c\}\\ E_{02}=\{(u_1,u_2,u_3,u_4,u_5,u_6)|u_1=0,u_2=m,u_3=0,u_4=n,u_5=p,u_6=c\}\\ E_{03}=\{(u_1,u_2,u_3,u_4,u_5,u_6)|u_1=0,u_2=-m,u_3=0,u_4=-n,u_5=-p,u_6=c\}\end{array}$$

(8)

where $m,n,p,c\in \text{R}$, and

$$\begin{array}{l}m=\sqrt{-a_4(a_1+\alpha +3\beta c^2)(a_1+\alpha -a_2{a}_3+3\beta c^2)}/(a_1+\alpha +3\beta c^2)\\ n=\sqrt{-a_4(a_1+\alpha +3\beta c^2)(a_1+\alpha -a_2{a}_3+3\beta c^2)}/a_2\\ p=\sqrt{-(a_1+\alpha -a_2{a}_3+3\beta c^2)}/a_2\end{array}$$

The system has infinite equilibrium points, in which, $E_{01}$ represents the equilibrium points on $u_6$ axes, obviously. For convenient description, we study the stability of $E_{01}$, and the Jacobian matrix of system (6) at $E_{01}$ is calculated as:

$$J_{E_{01}}=\left[\begin{array}{llllll}-(a_1+\alpha +3\beta c^2)\hfill & 0\hfill & a_2\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & -(a_1+\alpha +3\beta c^2)\hfill & 0\hfill & a_2\hfill & 0\hfill & 0\hfill \\ a_3\hfill & 0\hfill & -1\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & a_3\hfill & 0\hfill & -1\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & -a_4\hfill & 0\hfill \\ -1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right]$$

(9)

The characteristic polynomial of Equation (9) is:

$$\lambda (\lambda +a_4){[{\lambda }^2+(a_1+\alpha +3\beta c^2+1)\lambda +(a_1+\alpha +3\beta c^2-a_2{a}_3)]}^2=0$$

(10)

The eigenvalues of Equation (10) are:

$$\begin{array}{l}{\lambda }_1=0\\ {\lambda }_2=-a_4\\ {\lambda }_3={\lambda }_4=\frac{-(1+a_1+\alpha +3\beta c^2)}{2}-\frac{\sqrt{{(1-a_1-\alpha -3\beta c^2)}^2+4a_2{a}_3}}{2}\\ {\lambda }_5={\lambda }_6=\frac{-(1+a_1+\alpha +3\beta c^2)}{2}+\frac{\sqrt{{(1-a_1-\alpha -3\beta c^2)}^2+4a_2{a}_3}}{2}\end{array}$$

(11)

Since $a_1,a_2,a_3,a_4,\alpha ,\beta$ are positive, the system of (6) has a zero eigenvalue ${\lambda }_1$) and three negative eigenvalues ( ${\lambda }_2,{\lambda }_3,{\lambda }_4$). According to the Routh–Hurwitz theorem, the necessary and sufficient condition for the other roots to have negative real parts is if and only if $a_1+\alpha +3\beta c^2-a_2{a}_3>0$, i.e., if the inequality constraint is satisfied and the equilibrium points are stable, and vice versa. Therefore, the system has infinite stable and unstable equilibrium points with different parameters.

3. Dynamical Behaviors of the Memristor-Based Complex Lorenz System

In this section, we explore dynamical behaviors of the memristor-based complex Lorenz system with different parameters and initial conditions based on conventional numerical methods.

3.1. Dynamical Behaviors with Different Parameters

In order to reveal dynamical behaviors of the proposed memristor-based complex Lorenz system, we fix $a_1=8,a_2=11,a_4=8/3,\alpha =0.67\times {10}^{-3},\beta =0.02\times {10}^{-3}$ [9], $u(0)=(2,0,1,4,0.1,0)$ and vary $a_3$ in range of $[0,200]$. Based on the Wolf algorithm and local maximum method, Lyapunov exponents and a bifurcation diagram are calculated and plotted with different computational time intervals 0–3000 s and 0–30000 s as shown in Figure 1 and Figure 2, respectively, which can provide an overall perspective view of the system. However, the two figures are not consistent to each other, apparently, and Figure 2 is different from Figure 1 in some intervals of $a_3$, e.g., $(18.2,45.3],[87.6,92.2],[161.4,200]$, which indicates that transient dynamical behaviors exist in such intervals due to a boundary crisis [16]. For the sake of exploring the dynamical behaviors in detail, time-domain waveforms, phase portraits, and Lyapunov exponents against time are presented under some fixed parameters as shown in Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8.

Figure 1. Dynamical behaviors with computational time interval 0–3000 s. (a) Lyapunov exponents; (b) bifurcation diagram.

Figure 1. Dynamical behaviors with computational time interval 0–3000 s. (a) Lyapunov exponents; (b) bifurcation diagram.

Figure 2. Dynamical behaviors with computational time interval 0–30000 s. (a) Lyapunov exponents; (b) bifurcation diagram.

Figure 2. Dynamical behaviors with computational time interval 0–30000 s. (a) Lyapunov exponents; (b) bifurcation diagram.

(1)

Fixed points exist for $a_3\in [0,18.2]$. When $a_3=5$, the system converges to a fixed point $E(0,0,0,0,0,-891.2)$, and the six Lyapunov exponents are non-positive as shown in Figure 3.

(2)

Transient chaos to fixed points exist for $a_3\in (18.2,45.3]$. When $a_3=25$, the system goes through transient chaos and converges to fixed point $E(0,0,0,0,0,-2120)$ as shown in Figure 4. One Lyapunov exponent (i.e., L1) is positive incipiently, and then tends to zero asymptotically.

(3)

A chaotic zone covers most region of $a_3\in (45.3,161.4)$. When $a_3=50$, the system operates chaotically with a positive Lyapunov exponent as shown in Figure 5.

(4)

Transient chaos to Period-5 orbits exist in a narrow interval $a_3\in (52.7479,52.7482)$. When $a_3=52.7481$, the system goes through transient chaos and Period-5 orbit intermittently, and enters into the steady state of Period-5 eventually, as shown in Figure 6.

(5)

Transient Period-3 to tours exist for $a_3\in [87.6,92.2]$. When $a_3=90$, the system operates in Period-3 orbit at first, and then enters into the state of tours as shown in Figure 7.

(6)

Transient Period-1 to chaos exist for $a_3\in [161.4,200]$. When $a_3=165$, as shown in Figure 8, in the beginning the system operates in Period-1 orbit, then enters into the state of tours, and slides into a chaotic state in the end. The Lyapunov exponent L1 changes from zero to a positive number.

Figure 3. Fixed point ($a_3=5$). (a1) Waveforms of $u_1$–$u_5$; (a2) waveform of $u_6$; (b1) phase portrait of transient state; (b2) phase portrait of steady state; (c1) Lyapunov exponents of $L_1$–$L_4$; (c2) Lyapunov exponents of $L_5$– $L_6$.

Figure 3. Fixed point ($a_3=5$). (a1) Waveforms of $u_1$–$u_5$; (a2) waveform of $u_6$; (b1) phase portrait of transient state; (b2) phase portrait of steady state; (c1) Lyapunov exponents of $L_1$–$L_4$; (c2) Lyapunov exponents of $L_5$– $L_6$.

Figure 4. Transient chaos to fixed point ($a_3=25$). (a1) Waveforms of $u_1$–$u_5$; (a2) waveform of $u_6$; (b1) phase portrait of transient state; (b2) phase portrait of steady state; (c1) Lyapunov exponents of $L_1$–$L_4$; (c2) Lyapunov exponents of $L_5$–$L_6$.

Figure 4. Transient chaos to fixed point ($a_3=25$). (a1) Waveforms of $u_1$–$u_5$; (a2) waveform of $u_6$; (b1) phase portrait of transient state; (b2) phase portrait of steady state; (c1) Lyapunov exponents of $L_1$–$L_4$; (c2) Lyapunov exponents of $L_5$–$L_6$.

Figure 5. Chaos ($a_3=50$). (a1) Waveform of$u_1$; (a2) subinterval waveform of $u_1$; (b1) phase portrait of transient state; (b2) phase portrait of steady state; (c1) Lyapunov exponents of $L_1$–$L_4$; (c2) Lyapunov exponents of $L_5$–$L_6$.

Figure 5. Chaos ($a_3=50$). (a1) Waveform of$u_1$; (a2) subinterval waveform of $u_1$; (b1) phase portrait of transient state; (b2) phase portrait of steady state; (c1) Lyapunov exponents of $L_1$–$L_4$; (c2) Lyapunov exponents of $L_5$–$L_6$.

Figure 6. Transient chaos to Period-5 ($a_3=50$). (a1) Waveform of $u_1$; (a2) subinterval waveform of $u_1$; (b1) phase portrait of transient state; (b2) phase portrait of steady state; (c1) Lyapunov exponents of $L_1$–$L_4$; (c2) Lyapunov exponents of $L_5$–$L_6$.

Figure 6. Transient chaos to Period-5 ($a_3=50$). (a1) Waveform of $u_1$; (a2) subinterval waveform of $u_1$; (b1) phase portrait of transient state; (b2) phase portrait of steady state; (c1) Lyapunov exponents of $L_1$–$L_4$; (c2) Lyapunov exponents of $L_5$–$L_6$.

Figure 7. Transient Period-3 to tours ($a_3=90$). (a1) Waveform of $u_1$; (a2) subinterval waveform of $u_1$; (b1) phase portrait of transient state; (b2) phase portrait of steady state; (c1) Lyapunov exponents of $L_1$–$L_4$; (c2) Lyapunov exponents of $L_5$–$L_6$.

Figure 7. Transient Period-3 to tours ($a_3=90$). (a1) Waveform of $u_1$; (a2) subinterval waveform of $u_1$; (b1) phase portrait of transient state; (b2) phase portrait of steady state; (c1) Lyapunov exponents of $L_1$–$L_4$; (c2) Lyapunov exponents of $L_5$–$L_6$.

Figure 8. Transient Period-1 to chaos ($a_3=165$). (a1) Waveform of $u_1$; (a2) subinterval waveform of $u_1$; (b1) phase portrait of transient state; (b2) phase portrait of steady state; (c1) Lyapunov exponents of $L_1$–$L_4$; (c2) Lyapunov exponents of $L_5$–$L_6$.

Figure 8. Transient Period-1 to chaos ($a_3=165$). (a1) Waveform of $u_1$; (a2) subinterval waveform of $u_1$; (b1) phase portrait of transient state; (b2) phase portrait of steady state; (c1) Lyapunov exponents of $L_1$–$L_4$; (c2) Lyapunov exponents of $L_5$–$L_6$.

Remark 1. Due to transient phenomena, the dynamical behaviors of nonlinear dynamical systems become more complicated and difficult to depict. The time-domain waveform, phase portrait, bifurcation diagram, and Lyapunov exponent should be considered conjunctively, and the computational time interval should be large enough to describe the system behaviors completely and accurately.

3.2. Dynamical Behaviors with Different Initial Conditions

By fixing a₁ = 8, a₂ = 11, a₃ = 163, a₄ = 8/3, α = 0.67 × 10⁻³, β =0.02 × 10⁻³, u(0) = (2, 0, 1, 4, 0.1, u₆(0)), and choosing different $u_6(0)$, we plot time-domain waveforms and phase portraits for system (6) as shown in Figure 9. The figure indicates that the system operates periodically or chaotically with different initial values of $u_6(0)$; for example, Period-1 limit cycle, Period-1 (red cycle in Figure 9b) to torus, and chaos are found as shown in Figure 9.

Figure 9. Dynamical behaviors with different initial values of $u_6(0)$. (a) Period-1 ($u_6(0)=0$); (b) Transient Period-1 to tours ($u_6(0)=3$); (c) Chaos ($u_6(0)=300$).

Figure 9. Dynamical behaviors with different initial values of $u_6(0)$. (a) Period-1 ($u_6(0)=0$); (b) Transient Period-1 to tours ($u_6(0)=3$); (c) Chaos ($u_6(0)=300$).

4. MPS of the Memristor-Based Complex Lorenz System

In this section, we study the MPS of two identical memristor-based complex Lorenz systems, and provide numerical simulation results to verify the feasibility and effectiveness of the proposed schemes.

4.1. MPS Design

System (5) is our synchronization object, and the drive system and response system can be described as:

$$\{\begin{array}{l}{\dot{x}}_{1d}=-(a_1+a)x_{1d}+a_2{x}_{2d}-3\beta x_{4d}^2{x}_{1d}\hfill \\ {\dot{x}}_{2d}=a_3{x}_{1d}-x_{2d}-x_{1d}{x}_{3d}\hfill \\ {\dot{x}}_{3d}=(1/2)(x_{1d}{\overline{x}}_{2d}+{\overline{x}}_{1d}{x}_{2d})-a_4{x}_{3d}\hfill \\ {\dot{x}}_{4d}=(1/2)({\overline{x}}_{1d}+x_{1d})\hfill \end{array}$$

(12)

$$\{\begin{array}{l}{\dot{x}}_{1r}=-(a_1+a)x_{1r}+a_2{x}_{2r}-3\beta x_{4r}^2{x}_{1r}+L_1\hfill \\ {\dot{x}}_{2r}=a_3{x}_{1r}-x_{2r}-x_{1r}{x}_{3r}+L_2\hfill \\ {\dot{x}}_{3r}=(1/2)(x_{1r}{\overline{x}}_{2r}+{\overline{x}}_{1r}{x}_{2r})-a_4{x}_{3r}+L_3\hfill \\ {\dot{x}}_{4r}=(1/2)({\overline{x}}_{1r}+x_{1r})+L_4\hfill \end{array}$$

(13)

where $x_{1d}=u_{1d}+i{u}_{2d},x_{2d}=u_{3d}+i{u}_{4d}$, $x_{1r}=u_{1r}+i{u}_{2r},x_{2r}=u_{3r}+i{u}_{4r}$ are complex state variables, $x_{3d}=u_{5d},x_{4d}=u_{6d},x_{3r}=u_{5r},x_{4r}=u_{6r}$ are real state variables, $L_1=v_1+i{v}_2,L_2=v_3+i{v}_4$, $L_3=v_5$, $L_4=v_6$ are control functions, and then the complex systems (12) and (13) can be described by real Equation (14) and Equation (15) respectively:

$$\{\begin{array}{l}\begin{array}{l}{\dot{u}}_{1d}=-(a_1+\alpha )u_{1d}+a_2{u}_{3d}-3\beta u_{6d}^2{u}_{1d}\\ {\dot{u}}_{2d}=-(a_1+\alpha )u_{2d}+a_2{u}_{4d}-3\beta u_{6d}^2{u}_{2d}\end{array}\hfill \\ \begin{array}{l}{\dot{u}}_{3d}=a_3{u}_{1d}-u_{3d}-u_{1d}{u}_{5d}\\ {\dot{u}}_{4d}=a_3{u}_{2d}-u_{4d}-u_{2d}{u}_{5d}\end{array}\hfill \\ {\dot{u}}_{5d}=u_{1d}{u}_{3d}+u_{2d}{u}_{4d}-a_4{u}_{5d}\hfill \\ {\dot{u}}_{6d}=-u_{1d}\hfill \end{array}$$

(14)

$$\{\begin{array}{l}\begin{array}{l}{\dot{u}}_{1r}=-(a_1+\alpha )u_{1r}+a_2{u}_{3r}-3\beta u_{6r}^2{u}_{1r}+v_1\\ {\dot{u}}_{2r}=-(a_1+\alpha )u_{2r}+a_2{u}_{4r}-3\beta u_{6r}^2{u}_{2r}+v_2\end{array}\hfill \\ \begin{array}{l}{\dot{u}}_{3r}=a_3{u}_{1r}-u_{3r}-u_{1r}{u}_{5r}+v_3\\ {\dot{u}}_{4r}=a_3{u}_{2r}-u_{4r}-u_{2r}{u}_{5r}+v_4\end{array}\hfill \\ {\dot{u}}_{5r}=u_1{u}_{3r}+u_{2r}{u}_{4r}-a_4{u}_{5r}+v_5\hfill \\ {\dot{u}}_{6r}=-u_{1r}+v_6\hfill \end{array}$$

(15)

Errors between the drive system (12) and the response system (13) are defined as:

$$\{\begin{array}{l}{e}_1=e_{u1}+i{e}_{u2}=(u_{1d}-k_1{u}_{1r})+i(u_{2d}-k_2{u}_{2r})\\ e_2=e_{u3}+i{e}_{u4}=(u_{3d}-k_3{u}_{3r})+i(u_{4d}-k_4{u}_{4r})\\ e_3=e_{u5}=u_{5d}-k_5{u}_{5r}\\ e_4=e_{u6}=u_{6d}-k_6{u}_{6r}\end{array}$$

(16)

where $e_{u1}=u_{1d}-k_1{u}_{1r}$, $e_{u2}=u_{2d}-k_2{u}_{2r}$, $e_{u3}=u_{3d}-k_3{u}_{3r}$, $e_{u4}=u_{4d}-k_4{u}_{4r}$, $e_{u5}=u_{5d}-k_5{u}_{5r}$, $e_{u6}=u_{6d}-k_6{u}_{6r}$, which are errors between systems (14) and (15), $k_j(j=1-6)$ denotes projective factor. By substituting Equations (14) and (15), we can obtain the derivative of $e_{ui}(i=1-6)$ as:

$$\{\begin{array}{l}\begin{array}{l}{\dot{e}}_{u1}=-(a_1+\alpha )e_{u1}+a_2{u}_{3d}-3\beta u_{6d}^2{u}_{1d}-k_1(a_2{u}_{3r}-3\beta u_{6r}^2{u}_{1r})-k_1{v}_1\\ {\dot{e}}_{u2}=-(a_1+\alpha )e_{u2}+a_2{u}_{4d}-3\beta u_{6d}^2{u}_{2d}-k_2(a_2{u}_{4r}-3\beta u_{6r}^2{u}_{2r})-k_2{v}_2\end{array}\hfill \\ \begin{array}{l}{\dot{e}}_{u3}=-e_{u3}+a_3{u}_{1d}-u_{1d}{u}_{5d}-k_3(a_3{u}_{1r}-u_{1r}{u}_{5r})-k_3{v}_3\\ {\dot{e}}_{u4}=-e_{u4}+a_3{u}_{2d}-u_{2d}{u}_{5d}-k_4(a_3{u}_{2r}-u_{2r}{u}_{5r})-k_4{v}_4\end{array}\hfill \\ {\dot{e}}_{u5}=-a_4{e}_{u5}+u_{1d}{u}_{3d}+u_{2d}{u}_{4d}-k_5(u_{1r}{u}_{3r}+u_{2r}{u}_{4r})-k_5{v}_5\hfill \\ {\dot{e}}_{u6}=-e_{u6}+u_{6d}-u_{1d}-k_6(u_{6r}-u_{1r})-k_6{v}_6\hfill \end{array}$$

(17)

Theorem 1. The MPS between the drive system (12) and the response system (13) can be achieved, if the controllers are designed as

$$\{\begin{array}{l}\begin{array}{l}{v}_1=[a_2{u}_{3d}-3\beta u_{6d}^2{u}_{1d}-k_1(a_2{u}_{3r}-3\beta u_{6r}^2{u}_{1r})]/k_1\\ v_2=[a_2{u}_{4d}-3\beta u_{6d}^2{u}_{2d}-k_2(a_2{u}_{4r}-3\beta u_{6r}^2{u}_{2r})]/k_2\end{array}\hfill \\ \begin{array}{l}{v}_3=[a_3{u}_{1d}-u_{1d}{u}_{5d}-k_3(a_3{u}_{1r}-u_{1r}{u}_{5r})]/k_3\\ v_4=[a_3{u}_{2d}-u_{2d}{u}_{5d}-k_4(a_3{u}_{2r}-u_{2r}{u}_{5r})]/k_4\end{array}\hfill \\ v_5=[u_{1d}{u}_{3d}+u_{2d}{u}_{4d}-k_5(u_{1r}{u}_{3r}+u_{2r}{u}_{4r})]/k_5\hfill \\ v_6=[u_{6d}-u_{1d}-k_6(u_{6r}-u_{1r})]/k_6\hfill \end{array}$$

(18)

Proof. Choose the Lyapunov function as:

$$V(t)=\frac{1}{2}{\displaystyle \sum _{j=1}^6{e}_{uj}^2}=\frac{1}{2}(e_{u1}^2+e_{u2}^2+e_{u3}^2+e_{u4}^2+e_{u5}^2+e_{u6}^2)$$

(19)

Solve the derivative of $V(t)$ along the trajectory of error system (16):

$$\begin{array}{c}\dot{V}(t)={\displaystyle \sum _{i=1}^6{e}_{uj}}{\dot{e}}_{uj}\\ =e_{u1}[-(a_1+\alpha )e_{u1}+a_2{u}_{3d}-3\beta u_{6d}^2{u}_{1d}-k_1(a_2{u}_{3r}-3\beta u_{6r}^2{u}_{1r})-k_1{v}_1]\\ +e_{u2}[-(a_1+\alpha )e_{u2}+a_2{u}_{4d}-3\beta u_{6d}^2{u}_{2d}-k_2(a_2{u}_{4r}-3\beta u_{6r}^2{u}_{2r})-k_2{v}_2]\\ +e_{u3}[-e_{u3}+a_3{u}_{1d}-u_{1d}{u}_{5d}-k_3(a_3{u}_{1r}-u_{1r}{u}_{5r})-k_3{v}_3]\\ +e_{u4}[-e_{u4}+a_3{u}_{2d}-u_{2d}{u}_{5d}-k_4(a_3{u}_{2r}-u_{2r}{u}_{5r})-k_4{v}_4]\\ +e_{u5}[-a_4{e}_{u5}+u_{1d}{u}_{3d}+u_{2d}{u}_{4d}-k_5(u_{1r}{u}_{3r}+u_{2r}{u}_{4r})-k_5{v}_5]\\ +e_{u6}[-e_{u6}+u_{6d}-u_{1d}-k_6(u_{6r}-u_{1r})-k_6{v}_6]\end{array}$$

(20)

Substitute Equation (17) into Equation (20), then

$$\dot{V}(t)=-(a_1+\alpha )e_{u1}^2-(a_1+\alpha )e_{u2}^2-e_{u3}^2-e_{u4}^2-a_4{e}_{u5}^2-e_{u6}^2<0$$

(21)

According to the Lyapunov stability theory, since $V(t)$ is a positive function and its derivative $\dot{V}(t)$ is negative, the errors $e_{uj}(j=1-6)$ and $e_j(j=1-4)$ tend to zero as $t\to \infty$. Therefore, the response system (13) is synchronized with the drive system (12) asymptotically. □

4.2. Numerical Simulations

To demonstrate the validity of the above-mentioned MPS scheme, we provide numerical simulation results between two identical memristor-based complex Lorenz systems. The parameters and initial conditions are selected as a₁ = 8, a₂ = 11, a₃ = 50, a₄ = 8/3, α = 0.67 × 10⁻³, β =0.02 × 10⁻³, u_d(0) = (2, −3, 3, −2, 3, 2), u_r(0) = (−1, −1, 0, 0, 0, −1), which can guarantee the drive system operating chaotically. When the projective factors $k_1=1,k_2=-1,k_3=2,k_4=-2,k_5=0.5,k_6=-0.5$, the results are illustrated in Figure 10, Figure 11 and Figure 12, which consistently indicate that MPS is achieved within a short time. In detail, Figure 10 presents time-domain waveforms of the drive system (12) and the response system (13), Figure 11 shows that synchronization errors eventually converge to zero within five seconds, and Figure 12a displays phase portrait of $u_1-u_3-u_5$ phase space. Furthermore, we investigate the MPS under different scaling factors. When $k_1=k_2=-1,k_3=k_4=2,k_5=k_6=0.5$, MPS is achieved as shown in Figure 12b. When $k_j=-1.5(j=1-6)$ and $k_j=0.5(j=1-6)$, projective synchronization (PS) is achieved as shown in Figure 12c,d. When $k_j=-1(j=1-6)$, anti-synchronization (AS) is achieved as shown in Figure 12e. When $k_j=1(j=1-6)$, complete synchronization (CS) is achieved as shown in Figure 12f.

Figure 10. Time-domain waveforms of the drive system (12) and response system (13). (a) $u_{1d}=u_{1r}$; (b) $u_{2d}=-u_{2r}$; (c) $u_{3d}=2u_{3r}$; (d) $u_{4d}=-2u_{4r}$; (e) $u_{5d}=0.5u_{5r}$; (f) $u_{6d}=-0.5u_{6r}$.

Figure 10. Time-domain waveforms of the drive system (12) and response system (13). (a) $u_{1d}=u_{1r}$; (b) $u_{2d}=-u_{2r}$; (c) $u_{3d}=2u_{3r}$; (d) $u_{4d}=-2u_{4r}$; (e) $u_{5d}=0.5u_{5r}$; (f) $u_{6d}=-0.5u_{6r}$.

Figure 11. Synchronization errors between the drive system (12) and response system (13). (a) $e_1=e_{u1}+i{e}_{u2}$; (b) $e_2=e_{u3}+i{e}_{u4}$; (c) $e_3=e_{u5}$; (d) $e_4=e_{u6}$.

Figure 11. Synchronization errors between the drive system (12) and response system (13). (a) $e_1=e_{u1}+i{e}_{u2}$; (b) $e_2=e_{u3}+i{e}_{u4}$; (c) $e_3=e_{u5}$; (d) $e_4=e_{u6}$.

Remark 2. It is worth noting that the above MPS of memristor-based complex Lorenz systems can be applied to digital cryptography and secure communication. The sender modulates the original signals into the chaotic sequences generated from memristor-based complex Lorenz system and sends the combined signals to the receiver through communication channels, and the receiver obtains and decodes the combined signals through the MPS. Compared to the conventional synchronization of chaotic real systems, the complex system can provide multiple state variables to improve the efficiency of information transmission, and MPS can achieve high unpredictability to improve the security of communication.

Figure 12. Synchronization with different projective factors. (a) $k_1=1,k_2=-1,k_3=2,$$k_4=-2,k_5=0.5,k_6=-0.5$; (b) $k_1=k_2=-1,k_3=k_4=2,k_5=k_6=0.5$; (c) $k_j=-1.5$$(j=1-6)$; (d) $k_j=0.5(j=1-6)$; (e) $k_j=-1(j=1-6)$;(f) $k_j=1(j=1-6)$.

Figure 12. Synchronization with different projective factors. (a) $k_1=1,k_2=-1,k_3=2,$$k_4=-2,k_5=0.5,k_6=-0.5$; (b) $k_1=k_2=-1,k_3=k_4=2,k_5=k_6=0.5$; (c) $k_j=-1.5$$(j=1-6)$; (d) $k_j=0.5(j=1-6)$; (e) $k_j=-1(j=1-6)$;(f) $k_j=1(j=1-6)$.

5. Conclusions

In this paper, a novel complex Lorenz system with a flux-controlled memristor has been constructed and investigated. We found that the proposed system has infinite stable and unstable equilibrium points and exhibits abundant dynamical behaviors including fixed points, limit cycles, tori, chaos, and complex transient behaviors. Furthermore, MPS controller has been designed to synchronize the memristor-based Lorenz systems up to different scaling factors quickly and efficiently, which has been demonstrated by numerical simulations. However, the relevant research has just begun, and the memristor-based complex systems and their synchronization will have to be investigated from the viewpoint of applications in the future.