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.

A laser system which has a mirror outside of it to feedback a delayed output has been described by the Maxwell-Bloch equations with time delay. It is shown that a chaotic behavior in the equations can be controlled by using a OPF control algorithm. Our numerical simulation indicates that the chaotic behavior is stabilized on 1, 2 periodic unstable orbits.

This paper describes new methematical tools, taken from quantum field theory (QFT), which may make it possible to characterize localized excitations (including solitons, but also including chaotic modes) generated by PDE systems. The significance to computer hardware and neurocomputing is also discussed. This mathematics--IF further developed--may also have the potential to reorganize and simplify our understanding of QFT itself--a topic of very great intellectual and practical importance. The paper concludes by describing three new possibilities for research, which will be very important to achieving these goals.

One model of a laser is a set of differential equations called the Maxwell-Bloch equations. Actually, in a physical system, causing a chaotic behavior is very difficult. However the chaotic behavior can be observed easily in the system which has a mirror to feedback the delayed output.