On algebraic approach in quadratic systems

Matej Mencinger,Marko Robnik,Valery Romanovski

doi:10.1063/1.3046248

Abstract

We consider the one to one correspondence between homogeneous quadratic dynamical systems and algebra, which was introduced by Lawrence Markus (c.f. [1]) in order to classify the dynamics of all possible homogeneous quadratic continuous systems in the plane. Markus idea can be applied also for the homogeneous quadratic discrete systems. This one to one correspondence implies many interesting connections between systems and (algebraic) properties of the corresponding algebras. In this article we consider the influence of existing of some special algebraic elements and some special algebraic structure (power‐associativity) and algebraic homomorphisms and isomorphisms to the dynamics of the corresponding continuous and discrete dynamical system. The interplay between both areas is recently used in discrete systems in order to examine the (non)chaotic behavior of quadratic discrete dynamical systems in the plane.

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