Analytic Structure of Phase–Locked Loops in Complex Time

Hisa–Aki TANAKA; Toshiya MATSUDA; Shin'ichi OISHI; Kazuo HORIUCHI

Analytic Structure of Phase–Locked Loops in Complex Time

Hisa–Aki TANAKA, Toshiya MATSUDA, Shin'ichi OISHI, Kazuo HORIUCHI

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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 Ricatti 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.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E77-A No.11 pp.1777-1782

Publication Date: 1994/11/25

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Type of Manuscript: Special Section PAPER (Special Section on Nonlinear Theory and Its Applications)

Category: Analysis of Phase Locked Loops

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Hisa–Aki TANAKA, Toshiya MATSUDA, Shin'ichi OISHI, Kazuo HORIUCHI, "Analytic Structure of Phase–Locked Loops in Complex Time" in IEICE TRANSACTIONS on Fundamentals, vol. E77-A, no. 11, pp. 1777-1782, November 1994, doi: .
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 Ricatti 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.
URL: https://globals.ieice.org/en_transactions/fundamentals/10.1587/e77-a_11_1777/_p

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@ARTICLE{e77-a_11_1777,
author={Hisa–Aki TANAKA, Toshiya MATSUDA, Shin'ichi OISHI, Kazuo HORIUCHI, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Analytic Structure of Phase–Locked Loops in Complex Time},
year={1994},
volume={E77-A},
number={11},
pages={1777-1782},
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 Ricatti 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.},
keywords={},
doi={},
ISSN={},
month={November},}

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TY - JOUR
TI - Analytic Structure of Phase–Locked Loops in Complex Time
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1777
EP - 1782
AU - Hisa–Aki TANAKA
AU - Toshiya MATSUDA
AU - Shin'ichi OISHI
AU - Kazuo HORIUCHI
PY - 1994
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E77-A
IS - 11
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - November 1994
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 Ricatti 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.
ER -