IEICE TRANSACTIONS on Fundamentals
Nonlinear Circuit in Complex Time --Case of Phase-Locked Loops--
Summary :
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 neighbourhood 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 infinitly 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-integrability 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 equation of PLL.
- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E76-A No.12 pp.2055-2058
- Publication Date
- 1993/12/25
- Publicized
- Online ISSN
- DOI
- Type of Manuscript
- Special Section LETTER (Special Section of Letters Selected from the 1993 IEICE Fall Conference)
- Category
Authors
Keyword
Latest Issue
Copyrights notice of machine-translated contents
The copyright of the original papers published on this site belongs to IEICE. Unauthorized use of the original or translated papers is prohibited. See IEICE Provisions on Copyright for details.
Cite this
Copy
Hisa-Aki TANAKA, Shin'ichi OISHI, Kazuo HORIUCHI, "Nonlinear Circuit in Complex Time --Case of Phase-Locked Loops--" in IEICE TRANSACTIONS on Fundamentals,
vol. E76-A, no. 12, pp. 2055-2058, December 1993, doi: .
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 neighbourhood 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 infinitly 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-integrability 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 equation of PLL.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/e76-a_12_2055/_p
Copy
@ARTICLE{e76-a_12_2055,
author={Hisa-Aki TANAKA, Shin'ichi OISHI, Kazuo HORIUCHI, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Nonlinear Circuit in Complex Time --Case of Phase-Locked Loops--},
year={1993},
volume={E76-A},
number={12},
pages={2055-2058},
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 neighbourhood 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 infinitly 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-integrability 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 equation of PLL.},
keywords={},
doi={},
ISSN={},
month={December},}
Copy
TY - JOUR
TI - Nonlinear Circuit in Complex Time --Case of Phase-Locked Loops--
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 2055
EP - 2058
AU - Hisa-Aki TANAKA
AU - Shin'ichi OISHI
AU - Kazuo HORIUCHI
PY - 1993
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E76-A
IS - 12
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - December 1993
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 neighbourhood 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 infinitly 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-integrability 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 equation of PLL.
ER -
English
