A Julia Set Methodology For The Detection OfTransient Chaotic Oscillations In ContinuousControl Systems

Abstract

The paper defines and demonstrates the occurrence of chaos in the iteration of mathematical formulae and shows that such systems can be exposed by their dependence on the initial values of variables and parametric coefficients. Filled Julia sets, although computationally intensive, give visual explanation of the fate of an iteration and indicate regions of crossover between stable and unstable operation. The paper then demonstrates that even the simplest process controllers may be subject to chaos in their component parts while still maintaining control of the primary outputs of the system. A variation on the Julia set is then described and offered as a possible “chaotic function analyzer” for such systems. The paper closes with some illustrations of phase plots in the stable and chaotic regions for a nonlinear plant that is examined by the proposed method.