Analytic Structure of Phase–Locked Loops in Complex Time

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

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E77-A   No.11   pp.1777-1782
Publication Date: 1994/11/25
Online ISSN: 
DOI: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Nonlinear Theory and Its Applications)
Category: Analysis of Phase Locked Loops
Keyword: 
singular point analysis,  chaotic dynamics,  fractal,  phase–locked loops,  

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Summary: 
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.