International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 230939, 12 pages
doi:10.1155/2011/230939
Review Article

On Algebraic Approach in Quadratic Systems

1Department of Basic Science, Faculty of Civil Engineering, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia
2Department of Mathematics, Institute of Mathematics, Physics and Mechanics Ljubljana, 1000 Ljubljana, Jadranska 19, Slovenia

Received 14 December 2010; Accepted 9 February 2011

Academic Editor: Ivan Chajda

Copyright © 2011 Matej Mencinger. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

When considering friction or resistance, many physical processes are mathematically simulated by quadratic systems of ODEs or discrete quadratic dynamical systems. Probably the most important problem when such systems are applied in engineering is the stability of critical points and (non)chaotic dynamics. In this paper we consider homogeneous quadratic systems via the so-called Markus approach. We use the one-to-one correspondence between homogeneous quadratic dynamical systems and algebra which was originally introduced by Markus in (1960). We resume some connections between the dynamics of the quadratic systems and (algebraic) properties of the corresponding algebras. We consider some general connections and the influence of power-associativity in the corresponding quadratic system.