Analytic structure of phase-locked loops in complex time

Hisa Aki Tanaka*, Toshiya Matsuda, Shin'ichi Oishi, Kazuo Horiuchi

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
Pages (from-to)1777-1782
Number of pages6
JournalIEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
Issue number11
Publication statusPublished - 1994 Nov 1

ASJC Scopus subject areas

  • Signal Processing
  • Computer Graphics and Computer-Aided Design
  • Electrical and Electronic Engineering
  • Applied Mathematics


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