# Discrete Memristance and Nonlinear Term for Designing Memristive Maps

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## Abstract

**:**

## 1. Introduction

## 2. General Model

## 3. MM${}_{1}$ Map

#### 3.1. Case 1 When $c\ne 0$

#### 3.2. Case 2 When $c=0$

#### 3.3. Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Relationships of three research topics. The overlapped area of three circles shows the memristive maps with hidden attractor.

**Figure 3.**Iterative plots obtained from: (

**a**) MM${}_{1}$ map, (

**b**) MM${}_{2}$ map, (

**c**) MM${}_{3}$ map, and (

**d**) MM${}_{4}$ map.

**Figure 4.**(

**a**) Bifurcation diagram, (

**b**) Lyapunov exponents of MM${}_{1}$ map when changing a from 2.3 to 2.7. The red and blue colors present the first and second Lyapunov exponents, respectively.

**Figure 5.**Coexisting iterative plots are observed when changing $\left(x\right(0),y(0\left)\right)$: $(0.1,0.2+6\pi )$ (yellow), $(0.1,0.2+4\pi )$ (magenta), $(0.1,0.2+2\pi )$ (red), $(0.1,0.2)$ (black), $(0.1,0.2-2\pi )$ (blue), $(0.1,0.2-4\pi )$ (green), $(0.1,0.2-6\pi )$ (cyan).

**Figure 6.**The presence of two iterative plots for $\left(x\right(0),y(0\left)\right)=(1,2)$ (black) and $\left(x\right(0),y(0\left)\right)=(-1,-2)$ (red).

**Figure 7.**Coexistence of seven iterative plots for different values $\left(x\right(0),y(0\left)\right)$: $(0.1,0.2+6\pi )$ (yellow), $(0.1,0.2+4\pi )$ (magenta), $(0.1,0.2+2\pi )$ (red), $(0.1,0.2)$ (black), $(0.1,0.2-2\pi )$ (blue), $(0.1,0.2-4\pi )$ (green), $(0.1,0.2-6\pi )$ (cyan).

Map | Equations | Parameters | $\left(\mathit{x}\right(0),\mathit{y}(0\left)\right)$ |
---|---|---|---|

MM${}_{1}$ | $x\left(n+1\right)=asin\left(bcos\left(y\left(n\right)\right)x\left(n\right)\right)+c$ | $a=2.6,b=1.1$ | $x\left(0\right)=1$ |

$y\left(n+1\right)=y\left(n\right)+x\left(n\right)$ | $c=0.001$ | $y\left(0\right)=2$ | |

MM${}_{2}$ | $x\left(n+1\right)=asin\left(bsin\left(y\left(n\right)\right)x\left(n\right)\right)+c$ | $a=2.6,b=1$ | $x\left(0\right)=1$ |

$y\left(n+1\right)=y\left(n\right)+x\left(n\right)$ | $c=0.001$ | $y\left(0\right)=2$ | |

MM${}_{3}$ | $x\left(n+1\right)=atanh\left(bcos\left(y\left(n\right)\right)x\left(n\right)\right)+c$ | $a=2.6,b=1.3$ | $x\left(0\right)=1$ |

$y\left(n+1\right)=y\left(n\right)+x\left(n\right)$ | $c=0.001$ | $y\left(0\right)=2$ | |

MM${}_{4}$ | $x\left(n+1\right)=atanh\left(bsin\left(y\left(n\right)\right)x\left(n\right)\right)+c$ | $a=2.7,b=1.1$ | $x\left(0\right)=1$ |

$y\left(n+1\right)=y\left(n\right)+x\left(n\right)$ | $c=0.001$ | $y\left(0\right)=2$ |

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**MDPI and ACS Style**

Ramadoss, J.; Almatroud, O.A.; Momani, S.; Pham, V.-T.; Thoai, V.P. Discrete Memristance and Nonlinear Term for Designing Memristive Maps. *Symmetry* **2022**, *14*, 2110.
https://doi.org/10.3390/sym14102110

**AMA Style**

Ramadoss J, Almatroud OA, Momani S, Pham V-T, Thoai VP. Discrete Memristance and Nonlinear Term for Designing Memristive Maps. *Symmetry*. 2022; 14(10):2110.
https://doi.org/10.3390/sym14102110

**Chicago/Turabian Style**

Ramadoss, Janarthanan, Othman Abdullah Almatroud, Shaher Momani, Viet-Thanh Pham, and Vo Phu Thoai. 2022. "Discrete Memristance and Nonlinear Term for Designing Memristive Maps" *Symmetry* 14, no. 10: 2110.
https://doi.org/10.3390/sym14102110