# On Fractional Lyapunov Functions of Nonlinear Dynamic Systems and Mittag-Leffler Stability Thereof

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## Abstract

**:**

## 1. Introduction

## 2. Basic Preliminary

**Definition**

**1**

**Definition**

**2**

**Remark**

**1.**

**Definition**

**3**

**Definition**

**4**

**Definition**

**5**

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

## 3. Model Description

## 4. Mittag-Leffler Stability

**Definition**

**6**

**Definition**

**7**

**Remark**

**5.**

**Remark**

**6.**

- (i)
- $m\left(y\right)$is Lipschitz in regard to y.
- (ii)
- ∃${m}_{0}$ satisfying $\parallel m\left(I\right)-m\left(R\right)\parallel \le {m}_{0}\parallel I-R\parallel ,$ and if $R=0,$ then $\parallel m\left(I\right)\parallel \le {m}_{0}\parallel I\parallel .$

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 5. Numerical Simulations

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SIR | Susceptible–infected–recovered |

SIRS | Susceptible–infected–recovered–susceptible |

RL | Riemann–Liouville |

C | Caputo |

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**MDPI and ACS Style**

Rehman, A.u.; Singh, R.; Agarwal, P.
On Fractional Lyapunov Functions of Nonlinear Dynamic Systems and Mittag-Leffler Stability Thereof. *Foundations* **2022**, *2*, 209-217.
https://doi.org/10.3390/foundations2010013

**AMA Style**

Rehman Au, Singh R, Agarwal P.
On Fractional Lyapunov Functions of Nonlinear Dynamic Systems and Mittag-Leffler Stability Thereof. *Foundations*. 2022; 2(1):209-217.
https://doi.org/10.3390/foundations2010013

**Chicago/Turabian Style**

Rehman, Attiq ul, Ram Singh, and Praveen Agarwal.
2022. "On Fractional Lyapunov Functions of Nonlinear Dynamic Systems and Mittag-Leffler Stability Thereof" *Foundations* 2, no. 1: 209-217.
https://doi.org/10.3390/foundations2010013