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 neighbourhood 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 infinitly 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-integrability 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 equation of PLL.

The singular point analysis, such as the Painlev test and Yoshida's test, is a computational method and has been implemented in a symbolic computational manner. But, in applying the singular point analysis to high dimensional and/or "complex" dynamical systems, we face with some computational difficulties. To cope with these difficulties, we propose a new numerical technique of the singular point analysis with the aid of the self-validating numerics. Using this technique, the singular point analysis can now be applicable to a wide class of high dimensional and/or "complex" dynamical systems, and in many cases dynamical properties such as the algebraic non-integrability can be proven for such systems.

Transient responses by a dielectric sphere have been analyzed here for a dipole source located at the center. The formulation has been constructed first in the frequency domain, then transformed into the time domain to obtain for an impulsive response by two analytical methods, namely the Singularity Expansion Method and the Wavefront Expansion Method. While the former method collects the contributions around the singularities in the complex frequency domain, the latter gives us a result which is a summation of each successive wavefront arrivals. A Gaussian pulse has been introduced to simulate an impulse response result. The Gaussian pulse response is analytically formulated by convolving Gaussian pulse with the corresponding impulse response. Numercal inversion results are also calculated by Fast Fourier Transform Algorithm. Numerical examples are shown here to compare the results obtained by these three methods and good agreement are obtained between them. Comments are often made in connection with the corresponding two dimensional cylindrical case.