# Adaptive Control Synchronization of a Novel Memristive Chaotic System for Secure Communication Applications

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Memristive Model

_{1}and x

_{2}are the input and output states of the memristor scheme, respectively, in the memristor scheme, h(x, x

_{1}) describes the memristor output, where k is a positive parameter. Figure 1 depicts the hysteresis loop for the proposed new memristive element. This result is achieved by choosing k = 1 and the memristive deriving signal is a sinusoidal signal.

## 3. New Memristive Jerk System

_{1}, y, and z are the states, and a, b, and c are positive constant parameters for the system. The proposed system verifies the dynamics including the fixed (equilibrium) points, Lyapunov exponent, bifurcation, and chaoticity phenomena. For determining the equilibrium points of the proposed jerk system in Equation (4), the system is rearranged as in Equation (5).

_{1}= 0, y = 0, and z = 0. At the equilibrium point E(0, 0, 0,0), and when the parameters a = 3.846 × 10

^{−4}, b = 0.7, c = 0.1, and k = 1, the Jacobian matrix can be obtained as in Equation (6).

^{−4}, c = 0.1, k = 1, and the initial conditions are (x(t

_{0}), x

_{1}(t

_{0}), y(t

_{0}), z(t

_{0})) = (0, 0, 0.1, 0).

^{−4}, b = 0.7, c = 0.1, and k = 1 with the initial conditions (x(t

_{0}), x

_{1}(t

_{0}), y(t

_{0}), z(t

_{0})) = (0, 0.1, 0, 0).

## 4. Adaptive Synchronization

#### 4.1. Controller Design

_{i}

_{(1,2,3)}represent the feedback controllers to be designed. The synchronization errors can be assumed and given by Equation (10).

_{b}(t) of the master-slave parameters can be given by Equation (14), and b

_{2}(t) estimates parameter b

_{1}.

#### 4.2. Numerical Simulation

_{2}(t) is the single uncertain parameter on the slave side. Figure 6 illustrates the convergence of the estimated slave parameter b

_{2}(t) to the master parameter value b

_{1}= 0.7.

## 5. Application in Secure Communication

_{m}on the master side as in Equation (21). Here, the data signal was a stream of binary numbers generated by a Bernoulli binary generator in Matlab Simulink as shown in Figure 8.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**The memristive hysteresis loop: (

**a**) x

_{1}(0) = 0, amplitude (A) of driven sinusoidal signal = 1, and different frequencies (f); (

**b**) x

_{1}(0) = 0, frequency = 1, and different amplitudes. The memristive hysteresis loop (continued): (

**c**) frequency = 1, amplitude = 1, and different initial conditions of x

_{1}(t).

Master | Slave | ||
---|---|---|---|

Parameter | Value | Parameter | Value |

a_{1} | 3.846 × 10^{−4} | a_{2} | 3.846 × 10^{−4} |

b_{1} | 0.7 | b_{2}(t) | uncertain |

x_{m}(t_{0}) | 0 | x_{s}(t_{0}) | 0.5 |

y_{m}(t_{0}) | 0.1 | y_{s}(t_{0}) | 0.4 |

z_{m}(t_{0}) | 0 | z_{s}(t_{0}) | 1 |

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**MDPI and ACS Style**

A. Rahman, Z.-A.S.; Al-Kashoash, H.A.A.; Ramadhan, S.M.; Al-Yasir, Y.I.A.
Adaptive Control Synchronization of a Novel Memristive Chaotic System for Secure Communication Applications. *Inventions* **2019**, *4*, 30.
https://doi.org/10.3390/inventions4020030

**AMA Style**

A. Rahman Z-AS, Al-Kashoash HAA, Ramadhan SM, Al-Yasir YIA.
Adaptive Control Synchronization of a Novel Memristive Chaotic System for Secure Communication Applications. *Inventions*. 2019; 4(2):30.
https://doi.org/10.3390/inventions4020030

**Chicago/Turabian Style**

A. Rahman, Zain-Aldeen S., Hayder A. A. Al-Kashoash, Saif Muneam Ramadhan, and Yasir I. A. Al-Yasir.
2019. "Adaptive Control Synchronization of a Novel Memristive Chaotic System for Secure Communication Applications" *Inventions* 4, no. 2: 30.
https://doi.org/10.3390/inventions4020030