Matej Mencinger

A CLASS OF NONASSOCIATIVE ALGEBRAS ARISING FROM QUADRATIC ODEs

Abstract We present an algebraic part of our research on quadratic differential equations in 3-dimensional space, which possess a plane of critical points. Our goal is to provide a classification, up to isomorphism, of those algebras that arise from the above-mentioned systems via Markus construction.

Investigation of center manifolds of three-dimensional systems using computer algebra

For a family of three dimensional systems with center manifolds filled with closed trajectories (corresponding to periodic solutions of the system) we give criteria on the coefficients of the system to distinguish between the cases of isochronous and non-isochronous oscillations. Bifurcations of critical periods of the system are studied as well. The study is performed using computer algebra systems MATHEMATICA and SINGULAR.

Cyclicity of some analytic maps

Abstract In this paper, we describe an approach to estimate the cyclicity of centers in maps given by f ( x ) = − x − ∑ k = 1 ∞ a k x k + 1 . The main motivation for this problem originates from the study of cyclicity of planar systems of ODEs. We also consider the bifurcation of limit cycles from each component of the center variety of some particular cases of maps f ( x ) = − x − ∑ k = 1 ∞ a k x k + 1 arising from algebraic equations of the form x + y + h . o . t . = 0 where higher order terms up to degree four are present.

Isochronicity of centers at a center manifold

For a three dimensional system with a center manifold filled with closed trajectories (corresponding to periodic solutions of the system) we give criteria on the coefficients of the system to distinguish between the cases of isochronous and non-isochronous oscillations. Bifurcations of critical periods of the system are studied as well.

Stability analysis of critical points in quadratic systems in R3 which contain a plane of critical points

Markus idea [L. Markus, Quadratic Differential Equations and Nonassociative Algebras, Ann. Math. Studies 45 (Princeton Univ. Press, 1960), p. 185] of treating the quadratic systems of ODEs via commutative algebras which was introduced in 1960 is used in this paper to consider the stability of the origin. In [M. Mencinger, submitted to Communications in Algebra all three-dimensional commutative algebras which contain a two-dimensional nil subalgebra were classified up to the algebraic isomorphism. This classification is used here to study the stability of the origin in the systems in R 3 with a plane of critical points.

The center and cyclicity problems for some analytic maps

Abstract The center variety and bifurcations of limit cycles from the center for maps f ( x ) = − ∑ k = 0 ∞ a k x k + 1 arising from x + y + ∑ j = 0 n α n − j , j x n − j y j = 0 are considered. Motivated by a general result for n = 2 l + 1 we investigate the center and cyclicity problem for n being even. We review results for n = 2 and n = 4 and perform the analysis for n = 6 , 8 , 10 . Finally, we state some conjectures for general n = 2 l .

On algebraic approach in quadratic systems

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.