Geometric Structure of Mutually Coupled Phase-Locked Loops

Hisa-Aki Tanaka, Shin'ichi Oishi, Kazuo Horiuchi
IEEE Transactions on Circuit and Systems - I: Fundamental Theory and Applications, vol. 43, No.6, pp.438-443, June 1996.


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 anafyzed 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.

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