### Abstract

The analytic structure of the governing equation for a 2nd order Phase-Locked Loops (PLL) is studied in the complex time plane. By a local reduction of the PLL equation to the Riccati equation, the PLL equation is analytically shown to have singularities which form a fractal structure in the complex time plane. Such a fractal structure of complex time singularities is known to be characteristic for nonintegrable, especially chaotic systems. On the other hand, a direct numerical detection of the complex time singularities is performed to verify the fractal structure. The numerical results show the reality of complex time singularities and the fractal structure of singularities on a curve.

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

Pages (from-to) | 1777-1782 |

Number of pages | 6 |

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

Volume | E77-A |

Issue number | 11 |

Publication status | Published - 1994 Nov |

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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*,

*E77-A*(11), 1777-1782.

**Analytic structure of phase-locked loops in complex time.** / Tanaka, Hisa Aki; Matsuda, Toshiya; Oishi, Shinichi; Horiuchi, Kazuo.

Research output: Contribution to journal › Article

*IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences*, vol. E77-A, no. 11, pp. 1777-1782.

}

TY - JOUR

T1 - Analytic structure of phase-locked loops in complex time

AU - Tanaka, Hisa Aki

AU - Matsuda, Toshiya

AU - Oishi, Shinichi

AU - Horiuchi, Kazuo

PY - 1994/11

Y1 - 1994/11

N2 - The analytic structure of the governing equation for a 2nd order Phase-Locked Loops (PLL) is studied in the complex time plane. By a local reduction of the PLL equation to the Riccati equation, the PLL equation is analytically shown to have singularities which form a fractal structure in the complex time plane. Such a fractal structure of complex time singularities is known to be characteristic for nonintegrable, especially chaotic systems. On the other hand, a direct numerical detection of the complex time singularities is performed to verify the fractal structure. The numerical results show the reality of complex time singularities and the fractal structure of singularities on a curve.

AB - The analytic structure of the governing equation for a 2nd order Phase-Locked Loops (PLL) is studied in the complex time plane. By a local reduction of the PLL equation to the Riccati equation, the PLL equation is analytically shown to have singularities which form a fractal structure in the complex time plane. Such a fractal structure of complex time singularities is known to be characteristic for nonintegrable, especially chaotic systems. On the other hand, a direct numerical detection of the complex time singularities is performed to verify the fractal structure. The numerical results show the reality of complex time singularities and the fractal structure of singularities on a curve.

UR - http://www.scopus.com/inward/record.url?scp=0028546652&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0028546652&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0028546652

VL - E77-A

SP - 1777

EP - 1782

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 - 11

ER -