# Building Fixed Point-Free Maps with Memristor

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## Abstract

**:**

## 1. Introduction

## 2. General Model of Fixed Point-Free Maps

## 3. FPFM${}_{\mathbf{1}}$ Map

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Representation of recent trends, from chaotic maps to fixed point-free maps based on memristor.

**Figure 2.**The map built with a cosine function $cos(.)$ and a memristor with discrete memristance $M\left(y\left(n\right)\right)$.

**Figure 4.**By varying ${a}_{2}$ of the FPFM${}_{1}$ map from 1.35 to 1.85 (

**a**) bifurcation diagram, (

**b**) Lyapunov exponents are obtained.

**Figure 5.**By changing ${a}_{1}$ of the FPFM${}_{1}$ map from 0 to 0.25 we get (

**a**) bifurcation diagram, (

**b**) Lyapunov exponents, where the red and blue curves present the first and second Lyapunov exponents, respectively.

**Figure 7.**Iterative plots obtained from: (

**a**) FPFM${}_{2}$ map, (

**b**) FPFM${}_{3}$ map, (

**c**) FPFM${}_{4}$ map.

**Figure 9.**Captured signals of the FPFM${}_{1}$ map realized with a hardware board (

**a**) signal x, (

**b**) signal y.

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

FPFM${}_{1}$ | $x\left(n+1\right)={a}_{1}cos\left(x\left(n\right)\right)+{a}_{2}\left({\left(y\left(n\right)\right)}^{2}-1\right)x\left(n\right)$ | ${a}_{1}=0.1$ | $x\left(0\right)=1$ |

$y\left(n+1\right)=y\left(n\right)+x\left(n\right)$ | ${a}_{2}=1.7$ | $y\left(0\right)=-0.5$ | |

FPFM${}_{2}$ | $x\left(n+1\right)={a}_{1}cos\left(x\left(n\right)\right)+{a}_{2}\left(\left|y\left(n\right)\right|-1\right)x\left(n\right)$ | ${a}_{1}=0.05$ | $x\left(0\right)=1$ |

$y\left(n+1\right)=y\left(n\right)+x\left(n\right)$ | ${a}_{2}=2.3$ | $y\left(0\right)=-0.5$ | |

FPFM${}_{3}$ | $x\left(n+1\right)={a}_{1}cos\left(x\left(n\right)\right)+{a}_{2}sin\left(\pi y\left(n\right)\right)x\left(n\right)$ | ${a}_{1}=0.05$ | $x\left(0\right)=0.5$ |

$y\left(n+1\right)=y\left(n\right)+x\left(n\right)$ | ${a}_{2}=1.82$ | $y\left(0\right)=-0.8$ | |

FPFM${}_{4}$ | $x\left(n+1\right)={a}_{1}cos\left(x\left(n\right)\right)+{a}_{2}\left({e}^{-cos\left(\pi y\left(n\right)\right)}-1\right)x\left(n\right)$ | ${a}_{1}=0.05$ | $x\left(0\right)=-0.5$ |

$y\left(n+1\right)=y\left(n\right)+x\left(n\right)$ | ${a}_{2}=2.6$ | $y\left(0\right)=0.4$ |

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Almatroud, O.A.; Pham, V.-T. Building Fixed Point-Free Maps with Memristor. *Mathematics* **2023**, *11*, 1319.
https://doi.org/10.3390/math11061319

**AMA Style**

Almatroud OA, Pham V-T. Building Fixed Point-Free Maps with Memristor. *Mathematics*. 2023; 11(6):1319.
https://doi.org/10.3390/math11061319

**Chicago/Turabian Style**

Almatroud, Othman Abdullah, and Viet-Thanh Pham. 2023. "Building Fixed Point-Free Maps with Memristor" *Mathematics* 11, no. 6: 1319.
https://doi.org/10.3390/math11061319