# Adaptive Neural Network Synchronization Control for Uncertain Fractional-Order Time-Delay Chaotic Systems

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- On the basis of Barbarat’s lemma, an adaptive RBF neural network controller was designed that realizes the synchronization control of FOTDCSs with nonlinear uncertainty and external disturbance.
- (2)
- When the driving system and the response system have different time delays, they could also achieve synchronization under the action of the controller.
- (3)
- A numerical simulation realized the synchronous control of the uncertain fractional time-delay Liu chaotic system and the uncertain fractional time-delay financial chaotic system. The theoretical proof and simulation results show the effectiveness of the controller.

## 2. Preliminaries

#### 2.1. Introduction to Fractional Calculus

**Proposition 1**

**.**If $x\left(t\right)\in {C}^{1}[0,T](T>0)$, then ${D}_{t}^{\alpha}{D}_{t}^{\alpha -1}x\left(t\right)=x\left(t\right)$.

**Proposition**

**2**

**.**If $x\left(t\right)\in R$, then there is ${D}_{t}^{\alpha -1}\left|x\left(t\right)\right|\ge 0$ and ${D}_{t}^{\alpha -1}\left|x\left(t\right)\right|\ge {D}_{t}^{\alpha -1}x\left(t\right)$.

**Proposition 3**

**.**Set $\underset{t\to +\infty}{lim}{x}^{2}\left(\tau \right)d\tau <\infty $, then $x\left(t\right)\in {L}_{2}$; when $x\left(t\right)\in {L}_{\infty}$, if $\stackrel{\xb7}{x\left(t\right)}$ exists and is bounded, then $\underset{t\to +\infty}{lim}x\left(t\right)=0$.

**Proof.**

#### 2.2. Introduction to Radial Basis Neural Network

## 3. Design and Stability Analysis of the Adaptive Controller Based on the RBF Neural Network

#### 3.1. Synchronization of Uncertain FOTDCSs

#### 3.2. Adaptive Controller Based on the RBF Neural Network

#### 3.3. Stability Analysis

**Proposition 2**Given the initial conditions, the adaptive RBF neural network controller proposed in this paper could realize the synchronization.

**Proof.**

## 4. Numerical Example

#### 4.1. Fractional-Order Time-Delay Chaotic System

#### 4.2. Simulation Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Phase–space trajectory distribution of fractional delay financial system: (

**a**) X–Y plane; (

**b**) X–Z plane; (

**c**) Y–Z plane; (

**d**) X–Y–Z plane.

**Figure 2.**Phase–space trajectory distribution of fractional delay Liu system: (

**a**) X–Y plane; (

**b**) X–Z plane; (

**c**) Y–Z plane; (

**d**) X–Y–Z plane.

**Figure 3.**State trajectories of synchronization error vectors (

**a**) ${e}_{1},{e}_{2}$,and ${e}_{3}$, (

**b**) ${x}_{1}$ and ${y}_{1}$, (

**c**) ${x}_{2}$ and ${y}_{2}$, and (

**d**) ${x}_{3}$ and ${y}_{3}$.

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**MDPI and ACS Style**

Yan, W.; Jiang, Z.; Huang, X.; Ding, Q. Adaptive Neural Network Synchronization Control for Uncertain Fractional-Order Time-Delay Chaotic Systems. *Fractal Fract.* **2023**, *7*, 288.
https://doi.org/10.3390/fractalfract7040288

**AMA Style**

Yan W, Jiang Z, Huang X, Ding Q. Adaptive Neural Network Synchronization Control for Uncertain Fractional-Order Time-Delay Chaotic Systems. *Fractal and Fractional*. 2023; 7(4):288.
https://doi.org/10.3390/fractalfract7040288

**Chicago/Turabian Style**

Yan, Wenhao, Zijing Jiang, Xin Huang, and Qun Ding. 2023. "Adaptive Neural Network Synchronization Control for Uncertain Fractional-Order Time-Delay Chaotic Systems" *Fractal and Fractional* 7, no. 4: 288.
https://doi.org/10.3390/fractalfract7040288