Nonlinear circuit in complex time case of phase-locked loops

We analyze the nonlinear dynamics of PLL from the complex' singularity structure by introducing the complex time. The most important results which we have obtained in this work are as follow: (1) From the psi-series expansion of the solution, the local behavior in the neighborhood of a movable singularity is mapped onto an integrable differential equation: the Ricatti equation. (2) From the movable pole of the Ricatti equation, a set of infinitely clustered singularities about a movable singularity is shown to exist for the equation of PLL by the multivalued mapping. The above results are interesting because the clustering and/or the fractal distribution of singularities is known to be a characteristic feature of the non-integrabilities or chaos. By using the method in this letter, we can present a circumstantial evidence for chaotic dynamics without assuming any small parameters in the in the equation of PLL.