Stability of an Iterative Dynamical System Liming Wang and Dan Schonfeld Abstract— 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. I. I NTRODUCTION The stability of the dynamical system is a classical yet still active research area [1], [2], [3]. The Lyapunov stability theorem provides the sufficient condition for solution of differential equations [4]. In many areas such as signal pro- cessing, operators as de-noising filters, smoothing filters and certain algorithms could be applied iteratively to the data [5]. Naturally, the important question that whether the analysis result is robust and stable to the perturbations of the data rises in this situation. Moreover, due to the presence of noises and errors in acquisition of the data, the stability issue under the iterations of operators should not be neglected. However, because of the different assumptions and modelings, many classical results of stability can not be applied easily in these cases. The dynamical system in many of these situations could be modelled as the iteration of a complex operator Φ : C N → C N , where N ∈ N. The stability in this case can be stated as that whether a small perturbation of the initial point will end up in a small changes for the outcome under Φ ◦n , i.e. n-th iteration of Φ. The stability of this model is closely related to the complex dynamics [6], [7], [8]. We will see later, the concepts of Julia and Fatou set play an important role in complex dynamics. We also point out that the stability for this kind of dynamical system coincides with the mapping equivalence theory proposed by the authors [9]. L. Wang is with the Department of Electrical Engineering, Columbia University, 500 W. 120th Street, New York, NY 10027, USA. Email: liming@ee.columbia.edu D. Schonfeld is with the Department of Electrical and Computer Engi- neering, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607, USA. Email: dans@uic.edu More specifically, signals are often represented in a discrete way by their natures. In order to employ the signal processing techniques, one may map the symbolic signal into numerical domain. It is conceivable that different such mappings could lead to contradictory conclusions. Interestingly, the stability in this case is related to the mapping consistency problem [9]. The stability can be seen as an equivalence to the mapping equivalence problem under the iterations of operator. In this paper, we propose a framework for analyzing stabil- ity for a dynamical system whose dynamics are characterized by iterations of certain operator. In Section II, we provide the preliminaries for the paper. We first define the stability and introduce the concepts of Fatou and Julia Set. We introduce a few important properties of Fatou and Julia set. We establish the connection between stability between Fatou and Julia set. We also show the necessary and sufficient condition for stability under iteration of affine operator. In Section III, we introduce concepts of quasi-stability as a generalization of stability and present several theoretical results on quasi- stability. In particular, In Section IV, we present experimental results which illustrate the theoretical results in genomic signal processing. Finally, we provide a brief summary and discussion of our results in Section V. II. DYNAMICAL MODEL AND STABILITY We model the operator as a complex operator (function). Let Φ : C N → C N be a holomorphic (analytic) operator. The input signal can be embedded as one point z in C N . In the case of the symbolic sequence, for example, a symbolic data sequence {a i } n−1 i=0 , where a i ∈A. The set A is the alphabet. In order to apply the signal processing techniques, we may use a map f from A n to C N . For example, if we have a mapping method ˜ f : A → C k , then it naturally induces the map f : {a i } n−1 i=0 → z, z ∈ C nk , where ([z] jk+1 , [z] jk+2 , ..., [z] jk+k ) T = g(a j ), j =0, 1, ..., n − 1. Therefore, for a given symbolic sequence and a mapping method f , the corresponding numerical sequence is a point in C N . We denote this point as z f . We will call the collection of iterations under function compositions {Φ ◦n } n=1 as the dynamical system. In this paper we will assume Φ is a polynomial, i.e. (Φ(z)) i = P i (z 1 ,z 2 , ..., z N ),i =1, ..., N , where P i is a polynomial. Note that by Taylor’s theorem, we know that any holomorphic map can be approximated by polynomials. The stability issue is equivalent to the question that whether small changes of the given input sequence will cause a small changes for the outcomem which could be defined in many different ways. In this paper, we give 2012 American Control Conference Fairmont Queen Elizabeth, Montréal, Canada June 27-June 29, 2012 978-1-4577-1094-0/12/$26.00 ©2012 AACC 3328