Chaotic phenomena in desynhcronized systems and stability analysis

Abstract

(Received February 1992) 1. INTRODUCTION Complex systems tend to be desynchronized, as part and parcel of their internal organisation and internal connections. One way that this may arise is from quite small mismatching of operating times of system components. On the other hand, lack of synchronization can be built into the system. For example, iterations and asynchronous algorithms are exploited in parallel processing. Too, a system may be susceptible of change by numerous factors at different, asynchronous times. In all cases, it is important to understand the effect that desynchronization can have on the stability of the system and the convergence properties of processes. Robert [1] has considered convergence of chaotic iterations in both linear and nonlinear cases, using notions of vectorial norm and contraction mappings. Baudet [2] has extended applications to more general asynchronous iterations. Quite general convergence theorems have been derived by Bertsekas [3] and Tsitsiklis [4]. A cogent treatment and many references are in Bertsekas and Tsitsiklis [5]. Some classes of discrete, linear desychronized systems have been extensively studied [6-9]. This work was largely involved with showing sufficient conditions for stability in terms of the spectral radius of combinations of coefficient matrices. Kosyakin [10] additionally showed that the general problem of absolute stability for such systems could not be solved in finite arithmetic terms, and extended this study to the nonlinear case. This paper treats asynchronous systems from a somewhat different point of view. We consider the symbolic dynamics of desynchronized switching times and extract a numerical quantity whose values determine the stability characteristics of the system. An important role in the proof is played by Hilbert's projective metric. In the next section we introduce various preliminary definitions and results, and define the model of asynchronous systems that is studied. For completeness, the Hilbert metric is introduced in Section 3 and some known results are sketched. The last section is devoted to the proof of the main theorem. 2. DEFINITIONS AND STATEMENT OF RESULTS Consider a system G the state of which can be updated by v managers (such as processors) at various discrete time instants which we call