Nonlinear circuit in complex time case of phase-locked loops

Hisa Aki Tanaka, Shinichi Oishi, Kazuo Horiuchi

    Research output: Contribution to journalArticle

    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 languageEnglish
    Pages (from-to)2055-2058
    Number of pages4
    JournalIEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
    VolumeE76-A
    Issue number12
    Publication statusPublished - 1993 Dec

    Fingerprint

    Nonlinear Circuits
    Phase-locked Loop
    Phase locked loops
    Singularity
    Networks (circuits)
    Chaos theory
    Fractals
    Poles
    Differential equations
    Non-integrability
    Multivalued Mapping
    Integrable Equation
    Chaotic Dynamics
    Series Expansion
    Small Parameter
    Nonlinear Dynamics
    Pole
    Fractal
    Chaos
    Clustering

    ASJC Scopus subject areas

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

    Cite this

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

    In: IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, Vol. E76-A, No. 12, 12.1993, p. 2055-2058.

    Research output: Contribution to journalArticle

    @article{5b497f6f9a1f401fa2957cb65f69eeb6,
    title = "Nonlinear circuit in complex time case of phase-locked loops",
    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.",
    author = "Tanaka, {Hisa Aki} and Shinichi Oishi and Kazuo Horiuchi",
    year = "1993",
    month = "12",
    language = "English",
    volume = "E76-A",
    pages = "2055--2058",
    journal = "IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences",
    issn = "0916-8508",
    publisher = "Maruzen Co., Ltd/Maruzen Kabushikikaisha",
    number = "12",

    }

    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 -