Computational Modeling and Structural Stability

Abstract

The structural stability of a mathematical model with respect to small changes is a necessary condition for its correctness. The same condition is also necessary for the applicability of numerical methods, a computational experiment. But after S. Smale’s works it became clear that in smooth dynamics the system of a general form is not structurally stable, therefore there is no strict mathematical basis for modeling and computational analysis of systems. The contradiction appeared in science: according to physicists dynamics is simple and universal. The paper proposes a solution to this problem based on the construction of dynamic quantum models (DQM). DQM is a perturbation of a smooth dynamical system by a Markov cascade (time is discrete). The dynamics obtained in this way are simpler than smooth dynamics: the structurally stable DQM realizations are everywhere dense and open on the set of all DQM realizations. This dynamics in contrast to the classical one has a clear structural theory, which makes it possible to construct effective algorithms for study of concrete systems. For example this paper shows the use of computer simulation for rigorous proof of hyperbolicity of the Henon system attractor. On the other hand, when fluctuations tend to zero, i.e. in the semiclassical limit, the dynamics of the DQM goes into the initial smooth dynamics. In this paper the equivalence of structural stability and hyperbolicity for smooth discrete dynamical systems is established along this path.