Eigenvalue method for approximating cycle slip rates in PLLs

The small eigenvalues of a PLL´s Fokker-Planck operator are useful for approximating the rates at which noise-induced cycle slips occur. A method is presented for numerically approximating these small eigenvalues. An a finite dimensional subspace of 2πm-periodic functions (m is a user-supplied problem-dependent integer), the eigenvalue problem is reformulated (in an approximate sense) by a finite dimensional system of differential equations with 2π-periodic coefficients. The unknown eigenvalue appears as a parameter of this system; it must be selected so that the system has a particular periodic solution. Next, a method is discussed for numerically approximating the periodic solutions and small eigenvalues. The method utilizes as a constraint the known scale factor change that the eigenfunctions must experience from one 2π interval to the next. This scale factor is shown to be a multiplier of the system of differential equations. Finally, the method is applied to a simple example