1. MENCINGER, Matej. On the stability of Riccati differential equation X [dot] = TX + Q(x) in [R] [sup] n. Proc. Edinb. Math. Soc., 2002, vol. 45, isue 3, str. 601-615. [COBISS.SI-ID 7469334]
JCR IF: 0.386, SE (97/170), mathematics, x: 0.504
2. MENCINGER, Matej. On stability of the origin in quadratic systems of ODEs via Markus approach. Nonlinearity (Bristol), 2003, 16, str. 201-218. [COBISS.SI-ID 7691030]
JCR IF: 1.054, SE (29/153), mathematics, applied, x: 0.739, SE (17/31), physics, mathematical, x: 1.043
3. MENCINGER, Matej. Stability analysis of critical points in quadratic systems in [R][sup]3 which contain a plane of critical points. Prog. theor. phys., Suppl., 2003, no. 150, str. 388-392. [COBISS.SI-ID 8500758]
JCR IF: 0.368, SE (56/68), physics, multidisciplinary, x: 1.855
4. MENCINGER, Matej. On stability of the origin in homogenized quadratic systems of ODEs in [R][sup]3. Nonlinear phenom. complex syst., 2004, 7, 3, str. 263-272. [COBISS.SI-ID 9126422]
5. MENCINGER, Matej, ZALAR, Borut. A class of nonassociative algebras arising from quadratic ODEs. Commun. Algebra, 2005, vol. 33, no. 3, str. 807-828. [COBISS.SI-ID 13478489]
JCR IF: 0.303, SE (147/181), mathematics, x: 0.559
6. MENCINGER, Matej. On nonchaotic behaviour in quadratic systems. Nonlinear phenom. complex syst., 2006, vol. 9, no. 3, str. 283-287. [COBISS.SI-ID 10765846]
7. KUTNJAK, Milan, MENCINGER, Matej. A family of completely periodic quadratic discrete dynamical system. Int. j. bifurc. chaos appl. sci. eng., May 2008, vol. 18, iss. 5, str. 1425-1433. http://dx.doi.org/10.1142/S0218127408021087. [COBISS.SI-ID 12418582]
JCR IF (2007): 0.91, SE (30/74), mathematics, interdisciplinary applications, x: 0.979, SE (15/50), multidisciplinary sciences, x: 2.075
My research interests are on the filed of dynamical systems theory and in (nonassociative) algebra. Recently I consider a special class of nonlinear dynamical systems where the terms on the right hand side are polynomials of quadratic degree (i.e. quadratic systems). One of the possible approaches to the theory of quadratic systems was developed by Lawrence Markus in 1960. His idea was to associate an algebraic structure to the right hand side of the quadratic system. The one-to-one correspondence between quadratic systems and the corresponding algebra implies many interested connections between dynamical and algebraic properties. I work on two problems fitting into this algebraic frame. On one hand I consider the stability of the origin in quadratic systems of ODEs. And on the other hand I consider (together with my collaborator) the (non)chaotic behavior on the boundary of the basin of attraction of the origin in the discrete quadratic systems. In one of my papers I consider also the (non)chaotic behavior in systems of ODEs in 3D real space. Thus, my work is focused on dynamical systems and nonassociative algebra (some of the leading researchers on this field are S. Walcher, A. Sagle, M. Kinyon, etc.).
It is well known that one can completely determine the (in)stability of a nonlinear flow near a hyperbolic critical point by considering the corresponding Jacobian matrix of the nonlinear flow. This is the statement of the stable manifold theorem and the Hartman's theorem. If the critical point is nonhyperbolic (degenerate) with some pair(s) of purely imaginary eigenvalues or with some simple or double-zero eigenvalue(s) one can apply the centre manifold theorem in order to determine the (in)stability of the critical point. However, the origin in homogeneous quadratic systems is totally degenerated (the corresponding Jacobian equals the zero matrix) and the centre manifold is actually the whole space. Therefore some completely new tools are needed to consider the stability in such cases. I used the algebraic approach in 2D real space and (partially) in 3D real space in order to obtain the algebraic results which distinguish between stable and unstable cases. I considered completely a large class of systems which corresponds to a class of commutative algebras with two dimensional nill subalgebra. Together with my thesis advisor professor Borut Zalar (University of Maribor) we made the algebraic classification of these algebras (published in "Communications in Algebra"). This was the basis for a case-by-case analysis (published in "Progress in Theoretical Physics Supplements", Kyoto) which results in a theorem which holds for 2D systems and this 9-parameter class of 3D systems. The form of the obtained theorem is simple enough to allow the generalization in an arbitrary dimension, which was published as a conjecture in the well-known journal "Nonlinearity" (Bristol). Roughly speaking, the existence of a pair of complex idempotents in the complexification of the corresponding algebra seems to imply the stability of the origin in a homogeneous quadratic system of ODEs. In the journal "Proceedings of Edinburgh Mathematical Society" I published the stability analysis of nonhomogeneous quadratic systems where the corresponding Jacobian admits two purely imaginary eigenvalues and the corresponding eigenvectors are forming a two-dimensional subalgebra of the corresponding algebra. In the journal "Nonlinear Phenomena in Complex Systems" I considered a special class of homogeneous systems in 3D space which are obtained by a process called homogenization. The results published in this paper also confirm the conjecture given in "Nonlinearity".
In another article published in "Nonlinear Phenomena in Complex Systems" I proved that the systems of ODEs corresponding to 3D algebras with a 2D nill subalgebra are not chaotic. The algebraic approach played here the crucial role, since it would be extremely hard to deduce that there is no chaos in these systems directly from the general 9-parameter form.
Next, let me explain briefly my work on the second topic (chaos in discrete systems). The existence of some special algebraic elements called idempotents and nilpotents of rank two in the corresponding algebra plays an important role in both continuous and in the discrete systems. Concerning the chaotic behavior on the boundary of the attraction of the origin in 2D the existence of idempotents seems to be important. The most simple and well-known example where chaotic dynamics is found is the system of the complex squaring map (which corresponds to the algebra of complex numbers). We have a result published in "International Journal of Bifurcation and Chaos" where on the boundary of the attraction of the origin in systems corresponding to 2D algebras containing some nilpotents of rank two a very special completely periodic behavior was discovered. We want to continue the work began in 2006 and consider the (non)chaotic dynamics in the plane in all other homogenous systems. We want to discover some algebraic properties related with chaos in homogeneous systems.
I want to continue my work on the field of nonassociative algebras in general, dynamical systems in general and topology in general. We want to consider the algebraic properties in some special class of algebras, the relations between some special class of algebras and the corresponding systems, the relations between the rank of algebra and the dynamics in the corresponding system. Since the existence of a subalgebra in the algebra under consideration implies an invariant subflow in the corresponding dynamical system, the rank of the corresponding algebra might result somehow on the dynamics of the quadratic system (hopefully on the stability and chaotic behavior).
On the other hand I want to consider (independently from the algebraic approach) the (chaotic) dynamics in homogeneous continuous systems in 3D space on one hand and in (non)homogeneous discrete systems in 2D space on the other hand. The effect of the so called homogenization and quadratization process on nonlinear systems might also be considered.
I joined many international conferences. Let me list some important ones: I joined the 4th, 5th, 6th and 7th International Summer School/Conference at the University of Maribor, Slovenia, "Let's face chaos through nonlinear dynamics" organized by professor Marko Robnik (CAMTP, University of Maribor), I joined the "International conference on nonlinear dynamics and evolution equations", University of Newfoundland, St. John's, Canada, the international conference "Dynamics Days Europe", University of Loughborough, England, while joining the international conference "Systèmes dynamiques, equations différentielles et applications" at the University of Poitiers, France, I was running a special section on ODEs.
-2003- Dokotor matematičnih znanosti, Univerza v Mariboru
-1999- Magisterij iz matematike, Univerza v Mariboru
-1996- Univerzitetna izobrazba, Univerza v Mariboru
-2008-...IZREDNI PROFESOR ZA MATEMATIKO, FG UM
-2003 - 2008 DOCENT ZA MATEMATIKO, FG UM
-1996 - 2003 ASISTENT ZA MATEMATIKO, FG UM
Recenzent in kritik pri MR Mathematical Reviews; pri American Mathematical Societ. Vodja tutorjev na FG.
Član projekta Mladi za napredek Maribora (ocenjevalec za matematiko)