Exploring Phase-Lock Equations as Nonlinear Ordinary Differential Equations

In this paper, we derive a simplified version of the phase-lock equations (PLE) that arises in the study of Ginzburg-Landau equations of superconductivity. Our goal is to diffuse the results to an audience who is not entirely familiar with theoretical dynamic behavior, existence, uniqueness, and continuation of solutions of partial differential equations (PDE) on abstract space, but might nonetheless be interested in the practical simulation and connections of dynamical systems versus the numerical issues of PDE in a simpler space. The significant features observed in PLE are their strong attracting sets and basin of attractions resident in the PLE system, which may be beneficial to physicists and engineers in the context of superconductivity. We also explore numerically and graphically the sensitivity of PLE solutions with a variety of forcing functions and parameters.