GEOMETRIC STRUCTURE OF MUTUALLY COUPLED PHASE-LOCKED LOOPS

Dynamical properties such as lock-in or out-of-lock condition of mutua lly coupled phase-locked loops (PLL's) are problems of practical inter est, The present paper describes a study of such dynamical properties for mutually coupled PLL's incorporating lag filters and triangular ph ase detectors, The fourth-order ordinary differential equation (ODE) g overning the mutually coupled PLL's is reduced to the equivalent third -order ODE due to the symmetry, where the system is analyzed in the co ntext 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 genera ted in the vector field from the third-order ODE. In addition, a conne ction to the recently developed theory on chaos and bifurcations from degenerated homoclinic points is also found to exist. The two-paramete r diagrams of the one-homoclinic orbit are obtained by graphical solut ion of a set of nonlinear (finite dimensional) equations. Their graphi cal results useful in determining whether the system undergoes lock-in or continues out-of-lock behavior, are verified by numerical simulati ons.