The academician Sibirskys area of research was dynamic systems of differential equations, i.e. namely those systems, which describe the various processes or phenomena in the universe, nature, economy, society, etc. For some of them, the analytical solutions can be found, but the most important of them are very complex and cannot be solved through integration. Even for the simplest polynomial ordinary differential systems, the solutions for some problems, formulated more than 100 years ago, are still unknown. Thats why the interest in these equations is very present, while their research is increasingly promoted in numerous and prestigious scientific centers of the world.

The French mathematician Poincaré has founded the qualitative theory of these systems, in which he proposed to conduct the geometrical and qualitative study of the relations between the solutions instead of the quantitative study of the solutions. The classical theory of the invariants, and namely the theory of polynomial invariants, under the action of polynomial linear group GL(n, R), over the forms in n variables of degree m, has been a very active area of research in the latter part of the 19th century.

The original idea of Sibirsky was that of developing a similar theory, in which the forms in n variables are replaced by differential systems of degree n. Having approached this area, academician Sibirsky has developed the “Method of algebraic invariants in the qualitative theory of differential equations”. He discovered the algebraic invariants of the dynamic systems for various classical groups of transformations. Subsequently it was shown, that the multitudes of these invariants for every system in part, form an algebra, which currently carries the name of Sibirskys algebra. 

So, in what consists this above mentioned method? We will try to demostrate this thing by anology. Let us imagine that the analized differential system is identical to a planet in the space. In such a situation, Sibirskys algebra can be considered a sattelite of this system. Then, the influence of Sibirskys algebras invariants over the geometry of differential systems is analogical to the mutual influence of objects from nature, named sattelites and planets. And, as the phenomena of the Earths sea and ocean tides could be explained through the influence of the Moon, so the behavior of some complex dynamic systems that we cannot explicitly solve, could be explained through Sibirskys algebras. 

Nowaday it is recognised that, thanks to this method, some quite complicated problems from the theory of differential equations, could have definitively been solved. The method of algebraic invariants was taken up and developed in various scientific centers of the world (in Canada, United States, Spain, France, Slovenia, etc.), the influence of Sibirskys school being in a continuous ascent.

Dr. hab., Svetlana Cojocaru
Academician, Gheorghe Duca