Metric properties of ε-trajectories of dynamical systems with stochastic behaviour

A highly developed branch of the modern theory of dynamical systems is the study of deterministic ones with statistical properties in behaviour. During the last decade such systems were discovered in various domains of physics, chemistry, biology and technology. It is due to their complexity, that only the simplest of such systems have been analytically investigated (Lorenz system, Rikitake dynamo, billiard systems), and that is why numerical methods are widely used, especially in applied investigations. In numerical modelling we have no true trajectory of a dynamical system f , but an approximation = ( x 1 , x 2 ,…) such that the sequence of distances is small in some sense. For the case of round off errors in computer modelling, such a sequence is uniformly small, i.e. there exists some ε > 0, such that sup n ρ( x n +1 , fx n )<ε. The sequence in this case is called an ε-trajectory of the dynamical system f 1 . In a series of investigations 1–14 a study was made of the properties and applications of ε-trajectories.