A Lemma in the Theory of Structural Stability of Differential Equations1

The object of this note is to answer this question in the affirmative when F(x) is of class C2. (It can be mentioned that, even if F(x) is analytic, there need not exist such a map R of class C1 with nonvanishing Jacobian; [2].) The mapping (1.3) to be obtained below maps solution paths of (1.1) into those of (1.4) preserving parametrizations. The first part of the paper concerns the linearization of a local homeomorphism of a Euclidean space into another of the same dimension. The second part concerns (1.1) or rather the "group" of homeomorphisms associated with (1.4). The reduction of the problem to mappings is suggested by Sternberg [3].