Stability of an iterative dynamical system

Stability is a classical yet active research topic for dynamical systems. Certain operators such as de-noising filters, smoothing filters and many algorithms may be applied iteratively. In many cases, they can be modelled as a complex dynamical system. Due to the errors and noises in acquisition of data, the stability of analysis results is vital to the validity of the analysis. However, little is known about the stability of analysis results in these situations. In this paper, we propose a method for analyzing the stability under iterations of operator. First we give the definition of stability under iterations of operator. We model the dynamics as an complex dynamical system. We introduce the concepts of Fatou and Julia set. We establish the connection of stability to Fatou and Julia set. We define different concepts of quasi-stability including asymptotical, bounded quasi-stability, which generalize the notion of stability. We provide the necessary and sufficient condition for quasi-stability under iteration of affine operator. We present a few results for the quasi-stability based on the concept of Fatou and Julia Set. Finally, we provide the numerical example to illustrate the theory.