# A New No Equilibrium Fractional Order Chaotic System, Dynamical Investigation, Synchronization, and Its Digital Implementation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fundamentals of Fractional Order Systems

^{q}) established by Riemann–Liouville for order (q ≥ 0) of function (f(t)) [41].

## 3. A New Fractional Order Chaotic Model

## 4. The System Dynamical Analyses

#### 4.1. Bifurcation Diagrams

#### 4.2. Lyapunov Exponents

## 5. Adaptive Synchronization of Two New Fractional Order Chaotic Systems

#### 5.1. Adaptive Controller Design Process

_{1}, u

_{2}, and u

_{3}are the adaptive synchronization controllers that we want to design; a

_{m}, b

_{m}, and c

_{m}are the known master parameters; and a

_{s}(t), b

_{s}(t), and c

_{s}(t) present the uncertain slave parameters that must be estimated. The master-slave synchronization errors can be determined by Equation (9).

_{a}, e

_{b}, and e

_{c}are the errors of master-slave parameters and can be determined as in Equation (11).

_{x}, k

_{y}, and k

_{z}present positive constants, and the uncertain slave system parameters included (a

_{s}, b

_{s}, and c

_{s}) are estimated by updated laws as in the following Equation (17).

#### 5.2. Simulation Results

_{x}, e

_{y}, and e

_{z}) are illustrated. It can be seen that the synchronization errors rapidly (in less than 1.5 s) decrease to zero values exponentially. As illustrated in Figure 9, these uncertain slave parameters were appropriately estimated to the corresponding master parameters using the updated laws in Equation (17).

_{s}and b

_{s}, are rapidly estimated corresponding to the master parameter a

_{m}and b

_{m}, respectively (in less than 1.5 s), which is an exact match to the time duration of the synchronization errors to reach zero. On the other hand, the time duration was about 3.5 s for estimating the third uncertain slave parameter c

_{s}corresponding to the master parameter c

_{m}, where it does not affect the synchronization process. This is because the dynamical behavior of the nonlinear systems is not affected by the same degree of sensitivity for all its parameters as mentioned in [49].

## 6. Digital Implementation of New Fractional Order Chaotic System

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 2.**System (Equation (5)) phase portraits: (

**a**) x-y chaotic attractor; (

**b**) y-z chaotic attractor; (

**c**) x-z chaotic attractor; (

**d**) three-dimensional chaotic attractor.

**Figure 7.**Tracking the slave trajectories corresponding to master trajectories; (

**a**) x

_{s}track x

_{m}; (

**b**) y

_{s}track y

_{m}; (

**c**) z

_{s}track z

_{m}.

**Figure 11.**The experimental results of the proposed system phase portraits: (

**a**) x-y chaotic attractors; (

**b**) y-z chaotic attractors; (

**c**) x-z chaotic attractors.

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

Parameter | Value | Parameter | Value |

a_{m} | 0.5 | a_{s}(t) | Estimated |

b_{m} | 1.8 | b_{s}(t) | Estimated |

c_{m} | 8 | c_{m}(t) | Estimated |

fractional order (q) | 0.98 | fractional order (q) | 0.98 |

X_{m}(0) | 1 | X_{s}(0) | 2 |

y_{m}(0) | 1 | y_{s}(0) | 0 |

z_{m}(0) | 1 | z_{s}(0) | −1 |

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**MDPI and ACS Style**

Rahman, Z.-A.S.A.; Jasim, B.H.; Al-Yasir, Y.I.A.; Abd-Alhameed, R.A.; Alhasnawi, B.N.
A New No Equilibrium Fractional Order Chaotic System, Dynamical Investigation, Synchronization, and Its Digital Implementation. *Inventions* **2021**, *6*, 49.
https://doi.org/10.3390/inventions6030049

**AMA Style**

Rahman Z-ASA, Jasim BH, Al-Yasir YIA, Abd-Alhameed RA, Alhasnawi BN.
A New No Equilibrium Fractional Order Chaotic System, Dynamical Investigation, Synchronization, and Its Digital Implementation. *Inventions*. 2021; 6(3):49.
https://doi.org/10.3390/inventions6030049

**Chicago/Turabian Style**

Rahman, Zain-Aldeen S. A., Basil H. Jasim, Yasir I. A. Al-Yasir, Raed A. Abd-Alhameed, and Bilal Naji Alhasnawi.
2021. "A New No Equilibrium Fractional Order Chaotic System, Dynamical Investigation, Synchronization, and Its Digital Implementation" *Inventions* 6, no. 3: 49.
https://doi.org/10.3390/inventions6030049