On new resonances and normal forms of autonomous systems with one zero eigenvalue

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

“…Theorem 1. There exists a weakly degenerate trans formation (5) reducing system (3) to the pseudonormal form (6) Here, are real numbers; are constant diagonal matrices; A p is a constant Jordan matrix; and . The pseudonormal form (6) makes it possible to better comprehend the notion of resonance.…”

“…The most impor tant problem is transforming system (3) so that the matrix A(x) becomes Jordan. This cannot be achieved by applying a smooth change variables; therefore, we use so called shearing transformations ( [5][6][7]9] where , the are rationals, , and the are integers. Definition 2.…”

“…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][3][4][5][6][7][8][9]). Results of [5][6][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.…”