Theory and implementation of the multiple valued iterative dynamics algorithms for a class of singularly disturbed nonlinear control dynamical systems

Abstract

We discuss the visualization algorithms and their justification theorems of the steady state sets of a class of nonlinear disturbed control dynamical systems. We focus upon the systems that include sudden large disturbance that is significant enough to cause a bifurcation (sudden change of the qualitative behavior) of the dynamics. We call such a disturbance, the singular disturbance. Under the singular disturbance, the traditional models of control theory generally fail to produce the desired point-convergence. Instead, the steady state sets often exhibit a fractal structure. The multiple valued iterative dynamics modeling proved to be a useful tool establishing the theoretical framework for this case. The main objective of this paper is to establish the theoretical foundation of this framework. First, we present our model and state the basic theorems that justify the algorithmic aspect of the model. Next, we prove the justification theorems. We use the iteration of the predecessor operators and the successor operators as the main tool. Afterwards, we discuss the usage of the justification theorems. We review the known visualization algorithms, with a due emphases on the usage of our main theorems.