Quadratic dynamical systems and algebras
It was observed by \textit{L. Markus} [Ann. Math. Stud. 45, 185-213 (1960; Zbl 0119.298)] that one can assign to every differential equation \(\dot x = Q(x)\) in \(\mathbb{R}^n\) with homogeneous quadratic \(Q\) a commutative, nonassociative algebra whose product is defined by \(xy = {1\over 2} (Q(x + y) - Q(x) - Q(y))\). This opens up the possibility to use properties of and structure theorems about nonassociative algebras in the investigation of quadratic differential equations, just as methods of linear algebra can be employed in the study of linear differential equations. Subsequent work on this correspondence between quadratic differential equations and algebras was done by \textit{H. Röhrl} [Nagoya Math. J. 68, 59-122 (1977; Zbl 0419.34062)], the reviewer and others. As it turns out, this approach via nonassociative algebras is far from guaranteeing success in every instance (which one should not expect anyway for the class of all quadratic differential equations), but is useful for a number of interesting equations. The present paper contains an extensive survey of known results, and illustrates the correspondence with numerous examples. Among other results, it is indicated how the solution of every differential equation with polynomial right hand side can be found from the solution of a (homogeneous) quadratic system in a higher dimensional space. The importance of idempotents and index 2 nilpotents for the existence of special solutions of the differential equation is discussed. (Their existence is known from work of Kaplan/Yorke and Röhrl.) Stability of equilibria (in particular the equilibrium point 0 of a homogeneous quadratic system) is discussed; some of these results seem to carry over to differential equations with even right-hand sides. The influence of automorphisms resp. derivations of the algebra on the differential equation and its invariant (e.g., limit) sets is exhibited; much of this can be extended to a more general situation involving (local) Lie symmetry groups and the corresponding Lie algebras of vector fields. The last section deals with iterates of the map \(x \to x^2\) for a nonassociative algebra, and therefore, in a sense, with discrete dynamical systems. (Note that this map is never invertible on \(\mathbb{R}^n\), but its restriction to certain invariant subsets may be. It should also be mentioned that such iterates do occur in certain mathematical models in population genetics; see \textit{A. Wörz-Busekros} [Algebras in genetics. Springer Lecture Notes in Biomathematics 36 (1980; Zbl 0431.92017)].) Again, the role of idempotents, nilpotents and automorphisms is discussed and illustrated in examples.
scientific article; zbMATH DE number 740279