Geometric structure of mutually coupled phase-locked loops

Hisa Aki Tanaka*, Shi N.Ichi Oishi, Kazuo Horiuchi

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)


Dynamical properties such as lock-in or out-of-lock condition of mutually coupled phase-locked loops (PLL's) are problems of practical interest. The present paper describes a study of such dynamical properties for mutually coupled PLL's incorporating lag filters and triangular phase detectors. The fourth-order ordinary differential equation (ODE) governing the mutually coupled PLL's is reduced to the equivalent third-order ODE due to the symmetry, where the system is analyzed in the context of nonlinear dynamical system theory. An understanding as to how and when lock-in can be obtained or out-of-lock behavior persists, is provided by the geometric structure of the invariant manifolds generated in the vector field from the thirdorder ODE. In addition, a connection to the recently developed theory on chaos and bifurcations from degenerated homoclinic points is also found to exist. The two-parameter diagrams of the one-homoclinic orbit are obtained by graphical solution of a set of nonlinear (finite dimensional) equations. Their graphical results useful in determining whether the system undergoes lockin or continues out-of-lock behavior, are verified by numerical simulations.

Original languageEnglish
Pages (from-to)438-443
Number of pages6
JournalIEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications
Issue number6
Publication statusPublished - 1996

ASJC Scopus subject areas

  • Electrical and Electronic Engineering


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