Quadratic dynamical systems and algebras
type: Item
instance of: scholarly article
title: Quadratic dynamical systems and algebras
zbMATH Open document ID: 0830.17001
author: Michael K. Kinyon
author: Arthur A. Sagle
published in: Journal of Differential Equations
publication date: 1995-04-04T00:00:00Z
review text: 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 (loca…
reviewed by: S Walcher
Mathematics Subject Classification ID: 17A01
Mathematics Subject Classification ID: 17-02
Mathematics Subject Classification ID: 34-02
Mathematics Subject Classification ID: 34A05
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C20
Mathematics Subject Classification ID: 37-XX
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 740279
zbMATH Keywords: stability of equilibria
zbMATH Keywords: automorphisms
zbMATH Keywords: nonassociative algebras
zbMATH Keywords: quadratic differential equations
zbMATH Keywords: extensive survey
zbMATH Keywords: idempotents
zbMATH Keywords: index 2 nilpotents
zbMATH Keywords: derivations
zbMATH Keywords: iterates
zbMATH Keywords: discrete dynamical systems
MaRDI profile type: Publication
full work available at URL: jdeq.1995.1049
OpenAlex ID: W2078808548
DOI: 10.1006/JDEQ.1995.1049
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description: scientific article; zbMATH DE number 740279