Galerkin-based analysis of bifurcation in a phase locked loop

A Galerkin-based algorithm is used to analyze the bifurcation responsible for breaking the false lock state in a second-order, Type I PLL. The algorithm has two major components. The first deals with approximating the limit cycle associated with the PLL´s false lock state by a number of harmonics. The amplitudes of these harmonics are formulated as unknowns in a system of algebraic equations. These equations depend on the PLL´s detuning ω_os, and the value of this parameter at which bifurcation occurs is an additional unknown. The second component of the algorithm deals formulating an additional equation so that the unknown harmonics and ω_os can be solved for simultaneously