(Differencialnie Uravnenia i Protsesy Upravlenia)

About
History
Editorial Page
Addresses
Scope
Editorial Staff
Submission Review
For Authors
Publication Ethics
Issues
Русская версия

**Mizin D. A**

St. Petersburg State Politechnical University

The aim of the paper is to develop constructive
methods for investigation of global structure of dynamical system
trajectories. The proposed method is a combination of theoretical results and
computer-oriented algorithms. The structure graph Γ describes the
global structure of system dynamics. Each vertex iof the structure graph
corresponds to the component Q_{i} of the chain recurrent set. The graph
Γ has a directed edge i→ jif there exists the connection
Q_{i}→ Q_{j}, i.e. there is a trajectory with α-limit set in
Q_{i} and ω-limit set in Q_{j}.Thus, we could infer on the global
dynamics by the structure graph. In particular, the structure graph contains
information on the attractors and their basins of attraction. The structure
matrix A=(a_{ij}) is defined as a transition matrix of the directed graph
Γ : a_{ij}=1if there exists the edge i→ j on Γ
otherwise a_{ij}=0. The paper presents a justification of the constructive
method to calculate the structure matrix of a dynamical system. The main
supposition is that the number of co mponents of the chain recurrent set is
finite. The symbolic image is constructed as a directed graph which
approximats the dynamics of a system. To obtain the structure graph a refined
sequence of symbolic images is constructed. We describe an algorithm for
computing the structure matrix. The results obtained in the paper are
illustrated by examples.