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

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## Abstract

**:**

## 1. Introduction

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

#### 2.1. Memristor Model

#### 2.2. Memristor-Based Complex Lorenz System

#### 2.3. Properties of the Memristor-Based Complex Lorenz System

#### 2.3.1. Symmetry and Invariance

#### 2.3.2. Dissipation

#### 2.3.3. Equilibria and Stability

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

#### 3.1. Dynamical Behaviors with Different Parameters

**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.

- (1)
- Fixed points exist for ${a}_{3}\in [0,\text{}18.2]$. 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 ${a}_{3}\in (18.2,\text{}45.3]$. 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 ${a}_{3}\in (45.3,\text{}161.4)$. 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 ${a}_{3}\in (52.7479,\text{}52.7482)$. 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 ${a}_{3}\in [87.6,\text{}92.2]$. 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 ${a}_{3}\in [161.4,\text{}200]$. 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 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 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 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

_{1}= 8, a

_{2}= 11, a

_{3}= 163, a

_{4}= 8/3, α = 0.67 × 10

^{−3}, β =0.02 × 10

^{−3}, u(0) = (2, 0, 1, 4, 0.1, u

_{6}(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 (${u}_{6}(0)=300$).

## 4. MPS of the Memristor-Based Complex Lorenz System

#### 4.1. MPS Design

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

**Proof.**Choose the Lyapunov function as:

#### 4.2. Numerical Simulations

_{1}= 8, a

_{2}= 11, a

_{3}= 50, a

_{4}= 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 ${k}_{1}=1,\text{}{k}_{2}=-1,\text{}{k}_{3}=2,\text{}{k}_{4}=-2,\text{}{k}_{5}=0.5,\text{}{k}_{6}=-0.5$, 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 ${k}_{1}={k}_{2}=-1,\text{}{k}_{3}={k}_{4}=2,\text{}{k}_{5}={k}_{6}=0.5$, 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**) ${u}_{3d}=2{u}_{3r}$; (

**d**) ${u}_{4d}=-2{u}_{4r}$; (

**e**) ${u}_{5d}=0.5{u}_{5r}$; (

**f**) ${u}_{6d}=-0.5{u}_{6r}$.

**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**) ${e}_{4}={e}_{u6}$.

**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,\text{}{k}_{2}=-1,\text{}{k}_{3}=2,$${k}_{4}=-2,\text{}{k}_{5}=0.5,\text{}{k}_{6}=-0.5$; (

**b**) ${k}_{1}={k}_{2}=-1,\text{}{k}_{3}={k}_{4}=2,\text{}{k}_{5}={k}_{6}=0.5$; (

**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

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Wang, S.; Wang, X.; Zhou, Y.
A Memristor-Based Complex Lorenz System and Its Modified Projective Synchronization. *Entropy* **2015**, *17*, 7628-7644.
https://doi.org/10.3390/e17117628

**AMA Style**

Wang S, Wang X, Zhou Y.
A Memristor-Based Complex Lorenz System and Its Modified Projective Synchronization. *Entropy*. 2015; 17(11):7628-7644.
https://doi.org/10.3390/e17117628

**Chicago/Turabian Style**

Wang, Shibing, Xingyuan Wang, and Yufei Zhou.
2015. "A Memristor-Based Complex Lorenz System and Its Modified Projective Synchronization" *Entropy* 17, no. 11: 7628-7644.
https://doi.org/10.3390/e17117628