Synchronization limit and chaos onset in mutually coupled phase-locked loops

Dynamical property such as lock-in or out-of-lock condition of mutually coupled phase-locked loops (PLLs) is a problem 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 system is analysed in the context of nonlinear dynamical system theory. The symmetry of the mutually coupled PLLs system reduces the original 4th order ordinary differential equation (ODE) that governs the phase dynamics of the voltage-controlled oscillators (VCO) outputs to the 3rd order ODE, for which the geometric structure of the invariant manifolds provides an understanding as to how and when lock-in can be obtained or out-of-lock behavior persists. In addition, two-parameter diagrams of the one-homoclinic orbit are obtained by solving a set of nonlinear (finite dimensional) equations. This graphical results are confirmed to be useful in determining whether the system undergoes lock-in or continues out-of-lock behavior by numerical simulations. Presented theoretical results make it possible to understand experimental results of mutually coupled PLLs on the onset of chaos using the geometry of the invariant manifolds, where the resultant dynamical chaotic phenomena is postulated to represent an unfolding of the orbit-flip homoclinic point. Motivated by the numerical study of the system generated invariant manifolds, the topological horseshoe is proven to be generated even in the unfolding of a degenerated orbit-flip homoclinic point for the piecewise linear system under consideration