Pseudonormal form and finite-smooth equivalence of real autonomous systems with two purely imaginary eigenvalues

Abstract

This paper studies real autonomous infinitely smooth systems of ordinary differential equations of class C ∞ in a neighborhood of a nondegenerate singu� lar point. We consider systems whose matrix of linear part has two purely imaginary eigenvalues; all of the other eigenvalues are outside the imaginary axis. We are interested in the possibility of reducing such sys� tems to a form similar to the wellknown normal form. Since we allow transformations with singularities, we refer to the forms of transformed systems as pseudo� normal forms. The reduction of systems to such forms makes it possible to solve the problem of local finitely smooth equivalence of the systems of equations under consideration and more deeply comprehend the notion of resonance. The problem of finitely smooth equivalence has been well studied for systems with lin� ear part whose spectrum lies outside the imaginary axis (see the Sternberg-Chen theorem in (1, Chapter 9)), while even weakly degenerate systems have been stud� ied very little (see, e.g., (2-9)). Results of (5-7) and of this paper allow us to assert that real autonomous infi� nitely smooth systems of class C ∞ with one zero or two purely imaginary eigenvalues (except systems from a certain exceptional set of infinite codimension) can be reduced by nondegenerate finitely smooth transfor� mations to resonance polynomial normal forms. Consider the real autonomous system