NUMERICAL ANALYSIS OF DYNAMICAL SYSTEMS AND THEIR STRUCTURAL STABILITY

Abstract

     The method for investigating the dynamics of concrete systems of small dimension and obtaining strict results is demonstrated on the example of M. Henon’s system. The program realized as the C# – application and with usage of technology Open Maple is used here. The program allows to discover strange attractors for dynamical systems and to prove the hyperbolic dynamics on them, using outcomes of evaluations on the computer. However we get the strict a posteriori results here based on theorems of the article while the numerical evaluations are used only for checking the validity of assumptions of these statements.
     A structural stability of the model leads to a possibility of mathematically justified numerical analysis. It is the based concept of two traditional university courses: “Mathematical modeling and system analysis” and “Methods of calculations”. This article is an introduction to a solution of this problem proposed by the author. It became clear that for this purpose it suffices to consider the dynamics with an explicit account of unavoidable random fluctuations. More precisely, for a given classical system we construct its perturbation by a Markov process called a dynamic quantum model (DQM). The structurally stable realizations of DQM are dense everywhere, that allows one to investigate DQM by numerical evaluations. On the other hand, as the fluctuations tend to zero, the results obtained for DQM become statements about initial classical dynamics.