ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

On the Global Structure of a Dynamical System


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 Qi of the chain recurrent set. The graph Γ has a directed edge i→ jif there exists the connection Qi→ Qj, i.e. there is a trajectory with α-limit set in Qi and ω-limit set in Qj.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=(aij) is defined as a transition matrix of the directed graph Γ : aij=1if there exists the edge i→ j on Γ otherwise aij=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.

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