Geometric structure of mutually coupled phase-locked loops

Dynamical properties such as lock-in or out-of-lock condition of mutually coupled phase-locked loops (PLLs) are problems of practical interest. The present paper describes a study of such dynamical properties for mutually coupled PLLs incorporating lag filters and triangular phase detectors. The fourth-order ordinary differential equation (ODE) governing the mutually coupled PLLs 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 third-order 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 lock-in or continues out-of-lock behavior, are verified by numerical simulations.