### Abstract

We analyze the nonlinear dynamics of PLL from the complex' singularity structure by introducing the complex time. The most important results which we have obtained in this work are as follow: (1) From the psi-series expansion of the solution, the local behavior in the neighborhood of a movable singularity is mapped onto an integrable differential equation: the Ricatti equation. (2) From the movable pole of the Ricatti equation, a set of infinitely clustered singularities about a movable singularity is shown to exist for the equation of PLL by the multivalued mapping. The above results are interesting because the clustering and/or the fractal distribution of singularities is known to be a characteristic feature of the non-integrabilities or chaos. By using the method in this letter, we can present a circumstantial evidence for chaotic dynamics without assuming any small parameters in the in the equation of PLL.

Original language | English |
---|---|

Pages (from-to) | 2055-2058 |

Number of pages | 4 |

Journal | IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences |

Volume | E76-A |

Issue number | 12 |

Publication status | Published - 1993 Dec |

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### ASJC Scopus subject areas

- Hardware and Architecture
- Information Systems
- Electrical and Electronic Engineering

### Cite this

*IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences*,

*E76-A*(12), 2055-2058.

**Nonlinear circuit in complex time case of phase-locked loops.** / Tanaka, Hisa Aki; Oishi, Shinichi; Horiuchi, Kazuo.

Research output: Contribution to journal › Article

*IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences*, vol. E76-A, no. 12, pp. 2055-2058.

}

TY - JOUR

T1 - Nonlinear circuit in complex time case of phase-locked loops

AU - Tanaka, Hisa Aki

AU - Oishi, Shinichi

AU - Horiuchi, Kazuo

PY - 1993/12

Y1 - 1993/12

N2 - We analyze the nonlinear dynamics of PLL from the complex' singularity structure by introducing the complex time. The most important results which we have obtained in this work are as follow: (1) From the psi-series expansion of the solution, the local behavior in the neighborhood of a movable singularity is mapped onto an integrable differential equation: the Ricatti equation. (2) From the movable pole of the Ricatti equation, a set of infinitely clustered singularities about a movable singularity is shown to exist for the equation of PLL by the multivalued mapping. The above results are interesting because the clustering and/or the fractal distribution of singularities is known to be a characteristic feature of the non-integrabilities or chaos. By using the method in this letter, we can present a circumstantial evidence for chaotic dynamics without assuming any small parameters in the in the equation of PLL.

AB - We analyze the nonlinear dynamics of PLL from the complex' singularity structure by introducing the complex time. The most important results which we have obtained in this work are as follow: (1) From the psi-series expansion of the solution, the local behavior in the neighborhood of a movable singularity is mapped onto an integrable differential equation: the Ricatti equation. (2) From the movable pole of the Ricatti equation, a set of infinitely clustered singularities about a movable singularity is shown to exist for the equation of PLL by the multivalued mapping. The above results are interesting because the clustering and/or the fractal distribution of singularities is known to be a characteristic feature of the non-integrabilities or chaos. By using the method in this letter, we can present a circumstantial evidence for chaotic dynamics without assuming any small parameters in the in the equation of PLL.

UR - http://www.scopus.com/inward/record.url?scp=0027816579&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0027816579&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0027816579

VL - E76-A

SP - 2055

EP - 2058

JO - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

JF - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

SN - 0916-8508

IS - 12

ER -