# Dynamics Models of Synchronized Piecewise Linear Discrete Chaotic Systems of High Order

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Phase Portraits of the Onset of Instability of Fixed Points of Piecewise Linear Expressions of the Third Order

## 3. SPS Model with Saw-Tooth Nonlinearity

## 4. Algorithm for Determining the Acquisition Bandwidth

- With increasing ${\alpha}_{1},{\text{}\alpha}_{2}$, the stability domains with respect to the amplification $D$ expand. The most significant increase is observed for large ${m}_{1}$. For example, for ${m}_{1}=0.8$ with increasing ${\alpha}_{1},{\text{}\alpha}_{2}$ from values 0.5–0.8 (Figure 3) to values 2–4 (Figure 3), the stability domains in parameter $D$ increase 2–4 times.
- The boundary of the areas with global stability on the initial mismatch $\beta $ also expands significantly with increasing ${\alpha}_{1},{\text{}\alpha}_{2}$. However, dependence on ${m}_{1}$ is more complex. A decrease in the upper bound $\beta $ with increasing ${m}_{1}$ is observed near the limits of the local stability. On the contrary, in the farther zone from the boundary of local stability (medium $D$), there is a significant increase in the upper boundary $\beta $ with increasing ${m}_{1}$.
- Limiting the stability of the bottom of the frequency detuning (limiting with the cycles of the first kind) is most expressed with small ${m}_{1}$ and, as stated above, is non-monotonous. The most significant restriction is observed for large $D$ (Figure 3) and can reach values of 0.3–0.4.

## 5. SPS Model with a Triangular Nonlinearity

## 6. Results

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Cross-sections of the body with the local stability synchronization systems of the third order ((

**a**) $\eta =-1.2$; (

**b**) $\eta =-0.6$; (

**c**) $\eta =0$; (

**d**) $\eta =0.5$).

**Figure 3.**Acquisition band SFS of the third ((

**a**) ${\alpha}_{1}=0.5$; (

**b**) ${\alpha}_{1}=1$) order with ${F}_{1}\left(\phi \right)$.

**Figure 4.**Shows the dependences of ${c}_{\mathrm{max}}$ on the time constant of one of the links of the filter.

**Figure 5.**The acquisition band of the pulsed SPS ((

**a**) ${\alpha}_{1}=1$; (

**b**) ${\alpha}_{1}=10$) with ${F}_{c}\left(\phi \right)$.

**Figure 6.**Dependencies of the capture band and the settling time in the dual-ring SPS ((

**a**) $\mu =-0.1$; (

**b**) $\mu =0.1$; (

**c**) ${m}_{1}=0$; (

**d**) ${m}_{1}=0.5$).

The Eigenvalues of the Matrix A | Type of a Stable Point |
---|---|

1) 0 < p < 1, 0 < p_{2} < 1, 0 < p_{3} < 1, p_{1}, p_{2}, p_{3} are real-valued | stable node of the 1^{st} type |

2) −1 < p_{3} < 0, p_{1}, p_{2}, p_{3} are real-valued. | stable node of the 2^{nd} type |

3) 0 < p < 1, are real-valued | stable node of the 3^{rd} type |

4) 0 < p < 1, 0 < p_{2} < 1, −1 < p_{b} < 0, p_{l}, p_{2}, p_{3} are real-valued | stable node of the 4^{th} type |

5) 0 < Re < l are real-valued, p_{2}, p_{3} | stable focus of the 1^{st} type |

6) −l < Re < 0 | stable focus of the 2^{nd} type |

7) 0 < R < l; p_{x} are real-valued, p_{2}, p_{3} | stable focus of the 3^{rd} type |

8) −1 < p < 0, are real-valued, p_{2}, p_{3} | stable focus of the 4^{th} type |

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**MDPI and ACS Style**

Sokolov, S.; Zhilenkov, A.; Chernyi, S.; Nyrkov, A.; Mamunts, D.
Dynamics Models of Synchronized Piecewise Linear Discrete Chaotic Systems of High Order. *Symmetry* **2019**, *11*, 236.
https://doi.org/10.3390/sym11020236

**AMA Style**

Sokolov S, Zhilenkov A, Chernyi S, Nyrkov A, Mamunts D.
Dynamics Models of Synchronized Piecewise Linear Discrete Chaotic Systems of High Order. *Symmetry*. 2019; 11(2):236.
https://doi.org/10.3390/sym11020236

**Chicago/Turabian Style**

Sokolov, Sergei, Anton Zhilenkov, Sergei Chernyi, Anatoliy Nyrkov, and David Mamunts.
2019. "Dynamics Models of Synchronized Piecewise Linear Discrete Chaotic Systems of High Order" *Symmetry* 11, no. 2: 236.
https://doi.org/10.3390/sym11020236