# Dynamical Properties of Fractional-Order Memristor

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Fractional Derivative

- (1)
- ${}_{a}D_{t}^{\alpha}\left[uf(t)+vg(t)\right]={u}_{a}{D}_{t}^{\alpha}f(t)+{v}_{a}{D}_{t}^{\alpha}g(t)$
- (2)
- ${}_{a}D_{t}^{\alpha}{}_{a}D_{t}^{\beta}f(t)={}_{a}D_{t}^{\beta}{}_{a}D_{t}^{\alpha}f(t)={}_{a}D_{t}^{\alpha +\beta}f(t)$,
- (3)
- ${}_{a}{}^{L}D_{t}^{\alpha}K=\frac{K{t}^{-\alpha}}{\mathsf{\Gamma}(1-\alpha )}$, $\alpha >0$

## 3. The Model of Fractional-Order Memristor

^{−1}V

^{−1}, the length ${x}_{0}$ = 0.01, and the frequency $\omega =5\mathrm{rad}/\mathrm{s}$. The transient current curve and the voltage curve of the memristor are shown in Figure 2a,b respectively.

## 4. The Properties of Fractional-Order Memristor

#### 4.1. The Two Fractional-Order Memristors in Serial

^{−1}V

^{−1}), M2(${u}_{v}=2\times {10}^{-14}$ m.s

^{−1}V

^{−1}), and $\alpha =0.98$, one can obtain the curves of $v-i$ that are are shown in Figure 6.

#### 4.2. The Circuit of Fractional-Order MR in Parallel

^{−1}V

^{−1}, ${R}_{ON2}=120\mathsf{\Omega}$, ${R}_{OFF2}=18\mathrm{k}\mathsf{\Omega}$, and $\omega $ = 5. The simulation result when using these parameters is shown in Figure 8.

#### 4.3. The Circuit of Fractal-Order Memristor and Capacitor That Are Serially Connected

#### 4.4. The Circuit of Fractal-Order Memristor and Capacitor That Were Connected in Parallel

#### 4.5. The Circuit of Fractal-Order Memristor and Inductor That Are Serially Connected

#### 4.6. The Circuit of Fractal-Order Memristor and Inductor Connected in Parallel

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 4.**The $v-i$ curves of a fractional-order MR ($\alpha =0.98$). (

**a**) Varied with frequency $\omega $; (

**b**) varied with different switch resistance (R

_{OFF}/R

_{ON}); and (

**c**) varied with different average mobility values ${\mu}_{v}$.

**Figure 6.**Simulation results of two fractional-order, serially connected memristors $v(t)-i(t)$ curves.

**Figure 10.**${M}^{\alpha}C$ series circuit and its features. (

**a**) Effect of the order $\alpha $; (

**b**) effect of the exciting frequency $\omega $; and (

**c**) effect of the capacitance $C$.

**Figure 12.**The response of the circuit of a memristor and a capacitor that were connected in parallel. (

**a**) $\alpha $ = 0.98, 0.9, and 0.8; and (

**b**) $\omega =$ 5, 10, 15 rad/s.

**Figure 14.**${M}^{\alpha}L$ series circuit and its features: (

**a**) Effect of $\alpha $ on the hysteresis loop; (

**b**) effect of $\omega $ on the hysteresis loop; and (

**c**) effect of $L$ on the hysteresis loop.

**Figure 16.**The response of the circuit of the memristor and inductor connected in parallel. (

**a**) $\alpha $ = 0.98, 0.9, and 0.8; and (

**b**) $\omega =$ 5, 10, 15 rad/s.

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**MDPI and ACS Style**

Wang, S.F.; Ye, A.
Dynamical Properties of Fractional-Order Memristor. *Symmetry* **2020**, *12*, 437.
https://doi.org/10.3390/sym12030437

**AMA Style**

Wang SF, Ye A.
Dynamical Properties of Fractional-Order Memristor. *Symmetry*. 2020; 12(3):437.
https://doi.org/10.3390/sym12030437

**Chicago/Turabian Style**

Wang, Shao Fu, and Aiqin Ye.
2020. "Dynamical Properties of Fractional-Order Memristor" *Symmetry* 12, no. 3: 437.
https://doi.org/10.3390/sym12030437