# Two New Asymmetric Boolean Chaos Oscillators with No Dependence on Incommensurate Time-Delays and Their Circuit Implementation

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Preliminaries

#### 2.1. Boolean Differential Equations

#### 2.2. Boolean Chaos

**Theorem**

**1.**

**Lemma**

**1.**

**Theorem**

**2.**

**Lemma**

**2.**

#### 2.3. Lyapunov Exponents for ABNs

## 3. The Proposed Boolean Chaos Oscillators (BCOs) and Their Fixed Points

#### 3.1. BCO-1

**Theorem**

**3.**

**Proof.**

#### 3.2. BCO-2

**Theorem**

**4.**

**Proof.**

#### 3.3. Boolean Sensitivity Caused by Asymmetric Logic Functions

## 4. Boolean Chaos Robust to Different Incommensurate Time-Delays

**Lemma**

**3.**

**Corollary**

**1.**

**Proof.**

#### Boolean Chaos Robust to Distinct Discrete Physical Implementation

## 5. An Application Specific Integrated Circuit for the Proposed Boolean Chaos Oscillators

#### 5.1. Chip Design

#### 5.2. Experimental Results of the Integrated BCO-1 and BCO-2

#### 5.3. Comparison with Similar Implementations

## 6. Conclusions

^{2}and $0.000832$ mm

^{2}for BCO-1 and BCO-2, respectively, as well as high-speed chaotic oscillations with relevant amplitude content up to 200 $\mathrm{M}$$\mathrm{Hz}$. Several dynamical analyses such as time-series, chaotic attractors, Poincaré maps, and Lyapunov exponents validated the experimental results.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Autonomous Boolean networks (ABN) for the proposed Boolean chaos oscillator (BCO-1). (

**b**) An implementation of BCO-1 using electronic logic gates and its look-up table.

**Figure 2.**(

**a**) ABN for the second Boolean chaos oscillator (BCO-2). (

**b**) An implementation of BCO-2 using electronic logic gates and its look-up table.

**Figure 4.**Chaotic oscillations from the output voltage in the node C for BCO-1. The measurements exhibit a 100 ns of time and a 2 V of voltage grid per square. Case 1 in Table 1 with (

**a**) logic gates 74HCXXX, (

**b**) GAL22V10, and (

**c**) FPGA Spartan6. Case 3 in Table 1 with (

**d**) logic gates 74HCXXX, (

**e**) GAL22V10, and (

**f**) FPGA Spartan6. Case 7 in Table 1 with (

**g**) logic gates 74HCXXX, (

**h**) GAL22V10, and (

**i**) FPGA Spartan6.

**Figure 5.**Chaotic oscillations from the output voltage in the node B for BCO-2. The measurements exhibit a 100 ns of time and a 2 V of voltage grid per square. Case 1 in Table 2 with (

**a**) logic gates 74HCXXX, (

**b**) GAL22V10, and (

**c**) FPGA Spartan6. Case 5 in Table 2 with (

**d**) logic gates 74HCXXX, (

**e**) GAL22V10, and (

**f**) FPGA Spartan6.

**Figure 6.**The divergence $ln\langle d\left(s\right)\rangle $ to determine the largest Lyapunov exponent of the attractor for cases in Table 3 from each discrete physical implementation (Logic gates, GAL, FPGA).

**Figure 9.**Chaotic dynamics measured experimentally from the integrated circuit of 180 nm at distinct settings for both Boolean chaos oscillators. Top to bottom: Time-series, time-lag embedded attractor, frequency spectrum, Poincaré map, the divergence $ln\langle d\left(s\right)\rangle $ to determine the largest Lyapunov exponent ${\lambda}_{max}$ of the attractor. (

**a**) Experimental results for BCO-1 @ ${V}_{DD}=3.3$ V with ${\lambda}_{max}$ = 0.4496; (

**b**) Experimental results for BCO-1 @ ${V}_{DD}=2.8$ V with ${\lambda}_{max}$ = 0.4243; (

**c**) Experimental results for BCO-2 @ ${V}_{DD}=3.3$ V with ${\lambda}_{max}$ = 0.2492.

**Table 1.**Largest Lyapunov exponent $\left({\lambda}_{max}\right)$ of the BCO in Figure 1 for different time-delays in the feedback paths. The symbol “-” means no extra time-delay, while “√” refers to a time-delay composed of two logic NOT gates.

Case | Time-Delay | Lyapunov Exponent | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

${\mathit{\tau}}_{\mathit{a}\mathit{a}}$ | ${\mathit{\tau}}_{\mathit{a}\mathit{b}}$ | ${\mathit{\tau}}_{\mathit{a}\mathit{c}}$ | ${\mathit{\tau}}_{\mathit{b}\mathit{a}}$ | ${\mathit{\tau}}_{\mathit{b}\mathit{b}}$ | ${\mathit{\tau}}_{\mathit{b}\mathit{c}}$ | ${\mathit{\tau}}_{\mathit{c}\mathit{a}}$ | ${\mathit{\tau}}_{\mathit{c}\mathit{b}}$ | ${\mathit{\tau}}_{\mathit{c}\mathit{c}}$ | ${\mathbf{\lambda}}_{\mathit{m}\mathit{a}\mathit{x}}$ | |

1 | - | - | - | - | - | - | - | - | - | 0.2306 |

2 | - | - | - | - | - | - | - | - | √ | 0.2079 |

3 | √ | - | - | - | - | - | - | - | - | 0.2275 |

4 | √ | - | - | - | - | - | - | - | √ | 0.2057 |

5 | √ | - | - | - | √ | - | - | - | √ | 0.2076 |

6 | √ | √ | √ | - | - | - | - | - | - | 0.2101 |

7 | - | - | - | - | - | - | √ | √ | √ | 0.2121 |

8 | √ | √ | √ | - | - | - | √ | √ | √ | 0.1774 |

9 | √ | √ | √ | √ | √ | √ | - | - | - | 0.1808 |

10 | √ | √ | √ | √ | √ | √ | √ | √ | √ | 0.1896 |

11 | - | √ | √ | √ | √ | √ | √ | √ | √ | 0.1862 |

12 | - | √ | √ | √ | √ | √ | √ | √ | - | 0.1707 |

**Table 2.**Lyapunov exponent of BCO in Figure 2 for different time-delays in the feedback paths.

Case | Time-Delay | Lyapunov Exponent | |||||
---|---|---|---|---|---|---|---|

${\mathit{\tau}}_{\mathit{a}\mathit{a}}$ | ${\mathit{\tau}}_{\mathit{a}\mathit{b}}$ | ${\mathit{\tau}}_{\mathit{a}\tilde{\mathit{a}}}$ | ${\mathit{\tau}}_{\mathit{b}\mathit{b}}$ | ${\mathit{\tau}}_{\mathit{b}\mathit{a}}$ | ${\mathit{\tau}}_{\mathit{b}\tilde{\mathit{b}}}$ | ${\mathbf{\lambda}}_{\mathit{m}\mathit{a}\mathit{x}}$ | |

1 | - | - | - | - | - | - | 0.1644 |

2 | - | - | √ | - | - | - | 0.0960 |

3 | - | - | - | - | - | √ | 0.1495 |

4 | √ | - | - | √ | - | - | 0.1442 |

5 | - | √ | - | - | - | - | 0.1525 |

6 | - | - | - | - | √ | - | 0.1448 |

Logic Gates | GAL | FPGA | |
---|---|---|---|

BCO-1 | |||

${\lambda}_{max}$ (case 1, Table 1) | 0.230 | 0.224 | 0.209 |

${\lambda}_{max}$ (case 3, Table 1) | 0.227 | 0.221 | 0.194 |

${\lambda}_{max}$ (case 7, Table 1) | 0.212 | 0.211 | 0.185 |

BCO-2 | |||

${\lambda}_{max}$ (case 1, Table 2) | 0.164 | 0.160 | 0.157 |

${\lambda}_{max}$ (case 5, Table 2) | 0.152 | 0.150 | 0.148 |

This Work BCO-1 | This Work BCO-2 | [20] | [42] | |
---|---|---|---|---|

Chaos source | Boolean chaos | Boolean chaos | Chaotic oscillation | Multiattractor |

Integrated | Fully | Fully | Partially | Fully |

Technology | 180 nm | 180 nm | 180 nm | 180 nm |

Size ${\left(\mathsf{\mu}\mathrm{m}\right)}^{2}$ | 4500 | 832 | 28,000 | (315,000 × 383,000) |

Static power $\left(\mathsf{\mu}\mathrm{w}\right)$ | 0.2 | 0.09 | 25 | 3660 |

Speed limit $\left(\mathrm{M}\mathrm{H}\mathrm{z}\right)$ | 200 | 160 | 10 | NA |

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**MDPI and ACS Style**

Munoz-Pacheco, J.M.; García-Chávez, T.; Gonzalez-Diaz, V.R.; de La Fuente-Cortes, G.; del Carmen Gómez-Pavón, L.
Two New Asymmetric Boolean Chaos Oscillators with No Dependence on Incommensurate Time-Delays and Their Circuit Implementation. *Symmetry* **2020**, *12*, 506.
https://doi.org/10.3390/sym12040506

**AMA Style**

Munoz-Pacheco JM, García-Chávez T, Gonzalez-Diaz VR, de La Fuente-Cortes G, del Carmen Gómez-Pavón L.
Two New Asymmetric Boolean Chaos Oscillators with No Dependence on Incommensurate Time-Delays and Their Circuit Implementation. *Symmetry*. 2020; 12(4):506.
https://doi.org/10.3390/sym12040506

**Chicago/Turabian Style**

Munoz-Pacheco, Jesus M., Tonatiuh García-Chávez, Victor R. Gonzalez-Diaz, Gisela de La Fuente-Cortes, and Luz del Carmen Gómez-Pavón.
2020. "Two New Asymmetric Boolean Chaos Oscillators with No Dependence on Incommensurate Time-Delays and Their Circuit Implementation" *Symmetry* 12, no. 4: 506.
https://doi.org/10.3390/sym12040506