On Algebraic Approach in Quadratic Systems

Matej Mencinger

doi:10.1155/2011/230939

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.

Highlights

The stability of hyperbolic critical points in nonlinear systems of ODEs is well-known
The critical or equilibrium or stationary or fixed point of x f x or xk 1 f xk is defined to be the solution of the following algebraic system of equation s, f x0 0 or f x0 x0, respectively
For the systems of ODEs, x f x, the critical point x0 is said to be hyperbolic if no eigenvalue of the corresponding Jacobian matrix, Jf x0, of the nonlinear vector function f has it is eigenvalue equal to zero i.e., Re λi / 0

Summary

Introduction

The stability of hyperbolic critical points in nonlinear systems of ODEs is well-known. Note that just one eigenvalue of the corresponding linear approximation of x f x or xk 1 f xk for which Re λi 0 or |λi| 1, respectively, International Journal of Mathematics and Mathematical Sciences implies that the stability must be investigated separately in each particular case because of the significance of the higher order terms Such articles where for the non-hyperbolic critical points the classes of stable and unstable systems are considered are published constantly. In the sequel we consider the existence of some special algebraic elements i.e., nilpotents of rank 2 and idempotents , as well as the reflection of algebra isomorphisms in the corresponding homogeneous quadratic systems, which represents the basis for the linear equivalence classification of homogeneous quadratic systems It was already used by the author in order to analyze the stability of the origin in the continuous case in IR2 and in IR3 the origin is namely a total degenerated critical point for x Q x in any dimension n 3. We obtain the following quadratic systems xk 1 xk2 − yk2 − z2k, yk 1 2xkyk, zk 1 2xkzk, 1.7 x x2 − y2 − z2, y 2xy, z 2xz

Some Connections between Systems and Their Corresponding Algebras

Algebraic Isomorphism and Linear Equivalence

Conclusions