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

The Structure Matrix of Dynamical System

Author(s):

Salih Aytar

Suleyman Demirel University, Turkey

aytar@fef.sdu.edu.tr

Sergey Kobyakov

St. Petersburg State Technical University, Russia

aytar@fef.sdu.edu.tr

George Osipenko

St. Petersburg State Technical University, Russia

math@math.hop.stu.neva.ru

Abstract:

Let {Q₁, Q₂, Q₃,...} be chain recurrent set components of a dynamical system. A connection Q_i→Q_j is said to exist if there is a point x such that α - limit set of x is in Q_i and ω - limit set of x is in Q_j. Let the matrix S=(s_ij) be such that s_ij=1 if there is the connection Q_i→Q_j, s_ii=1 and s_ij=0 in other case. The matrix S is named the structure matrix of dynamical system f. By the definition, the structure matrix is a topological invariant. The main result: If the dynamical system has a finite number of chain recurrent components with the stable connections then there exist a finite algorithm for construction of the structure matrix.

Full text (pdf)