# Generalized Attracting Horseshoe in the Rössler Attractor

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## Abstract

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## 1. Introduction

## 2. Poincaré Map Algorithm

## 3. A Constructed GAH System

#### 3.1. The GAH Map

- (★)
- f maps the keystone region K (containing a portion of the arch of the horseshoe) to the left of the fixed point p and the portion of its corresponding stable manifold ${W}^{s}\left(p\right)$ containing p and contained in $f\left(Q\right)$.

#### 3.2. A GAH Producing System

## 4. Poincaré Maps and Circuit Realization of the Rössler Attractor

#### 4.1. The Poincaré Map

#### 4.2. Circuit Realization of the Rössler System

## 5. Potential Applications

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**The Rössler attractor with parameters $a=0.2$, $b=0.1$, and $c=10$, and a rotation (represented by $\widehat{x}$ and $\widehat{y}$) of $\theta =2\pi /5$ in spherical coordinates.

**Figure 4.**Poincaré section ($r=5,\theta =2\pi /5$) of the Rössler attractor containing a horseshoe-like structure. Plot is shown in the rotated frame.

**Figure 5.**The first return (blue markers) of the quadrilateral trapping region (red markers) with vertices located at $(\widehat{x},\widehat{y})=(-3.55,-27),\phantom{\rule{0.166667em}{0ex}}(11.91,-6.6),\phantom{\rule{0.166667em}{0ex}}(12,0),\phantom{\rule{0.166667em}{0ex}}(-8.5,3.5)$. While the quadrilateral edges look “continuous”, it should be noted that it is in fact discretized using four thousand points, which are then mapped back to the Poincaré section ($r=5,\theta =2\pi /5$). Plot is shown in the rotated frame with $\widehat{x}$ and $\widehat{y}$ denoting rotated axes.

**Figure 6.**First five iterations of the Poincaré map (blue markers) of the quadrilateral trapping region (red markers) with vertices located at $(\widehat{x},\widehat{y})=(-3.55,-27),\phantom{\rule{0.166667em}{0ex}}(11.91,-6.6),\phantom{\rule{0.166667em}{0ex}}(12,0),\phantom{\rule{0.166667em}{0ex}}(-8.5,3.5)$. While the quadrilateral edges look “continuous”, it should be noted that it is in fact discretized using four thousand points, which are then mapped back to the Poincaré section ($r=5,\theta =2\pi /5$). Plot is shown in the rotated frame with $\widehat{x}$ and $\widehat{y}$ denoting rotated axes.

**Figure 8.**Multisim outputs of the Rössler attractor showing a period doubling Hopf bifurcation leading to chaos.

Type | Quantity | Code |
---|---|---|

10 k$\Omega $ Resistor | 11 | |

100 k$\Omega $ Resistor | 3 | |

390 k$\Omega $ Resistor | 1 | |

56 k$\Omega $ Resistor | 1 | |

560 k$\Omega $ Resistor | 1 | |

5.1 k$\Omega $ Resistor | 1 | |

100 k$\Omega $ Potentiometer | 1 | |

100 nF Capacitor | 6 | |

$2.2$ nF Capacitor | 3 | |

Op-Amp | 2 | AD633JN |

Multiplier | 1 | TL074CN |

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**MDPI and ACS Style**

Murthy, K.; Jordan, I.; Sojitra, P.; Rahman, A.; Blackmore, D.
Generalized Attracting Horseshoe in the Rössler Attractor. *Symmetry* **2021**, *13*, 30.
https://doi.org/10.3390/sym13010030

**AMA Style**

Murthy K, Jordan I, Sojitra P, Rahman A, Blackmore D.
Generalized Attracting Horseshoe in the Rössler Attractor. *Symmetry*. 2021; 13(1):30.
https://doi.org/10.3390/sym13010030

**Chicago/Turabian Style**

Murthy, Karthik, Ian Jordan, Parth Sojitra, Aminur Rahman, and Denis Blackmore.
2021. "Generalized Attracting Horseshoe in the Rössler Attractor" *Symmetry* 13, no. 1: 30.
https://doi.org/10.3390/sym13010030