Comparative time grainyness and asymptotic stability of dynamical systems

Variability in the underlying time structure of dynamical systems and its consequences are encountered in a variety of mathematical contexts, though are not always recognized as such. An obvious example is the numerical time-discretization of ordinary differential equations, where the relationship between the behaviour of an approximate discrete-time dynamical system and that of the original continuous-time system is usually assumed or needs to be verified. Until recently different mathematical proofs have often been required to establish analogous results, for example the Hartman-Grobman theorem, for continuoustime and discrete-time systems. The measure chain calculus introduced by Hilger now provides a unifying framework to investigate not only continuous-time and discrete-time systems, but also those with a mixture of continuousand discrete-time regimes, as well as desynchronized systems. In this paper it provides a framework for establishing with the aid of Lyapunov functions sufficient conditions for asymptotic stability of nonlinear systems with a mixture of continuousand discrete-time regimes. A Russian language version of this paper will appear in Automatika i Telemekhanika. This work was partially supported by the ARC grant A89132609