Canonical forms of real homogeneous quadratic transformations

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The homogeneous quadratic transformations are encountered in a number of practical problems including two-body interaction processes such as second-order chemical reactions, biological interactions, asymptotic behavior of the errors of the Newton-Raphson process in numerical analysis, etc. Moreover, if, in a system of differential equations, the effect of the first-order terms vanishes, the theory of quadratic transformations plays substantial role in analyzing the stability of the critical points. Lawrence Markus first tried a classification of two-dimensional real homogeneous quadratic differential equation systems by means of nonassociative algebras in 1960 [l], and Rutherford Aris applied his results to the second-order chemical reactions [2]. However, Markus’ approach was not very systematic in the sense that the classification of algebras did not exactly correspond to that of differential systems and that there were a few duplications and omissions in the classifications. In this paper, we shall present a more systematic approach to the same problem on the basis of classical invariant theory, to obtain the “canonical forms” of two-dimensional real homogeneous quadratic transformations as well as an important class of “invariants” of the transformations. Then we shall apply the results to the stability analysis of the fixed points of the transformations. Applications to other problems, such as the analysis of a system of quadratic differential equations on the real plane and that of the asymptotic behavior of the errors of the Newton-Raphson process will be published elsewhere [7, 81. The key idea is to decompose the tensor (of contravariant valence 1 and covariant valence 2, symmetric in the covariant indices), which defines a two-dimensional quadratic transformation, into a binary cubic form (or a covariant symmetric tensor d-density of valence 3 and of weight 1) and a