Local Orbitals of Normal Forms of Vector Fields on a Plane

Real-Formal Orbital Rigidity for Germs of Real Analytic Vector Fields on the Real Plane

“…As a consequence ρ(x) := x(u 0 , x) is an invertible formal transformation. In what follows, ρ will be called the shift from the formal cross-section to the line {u = u 0 } respect to the differential equation (3). To emphasize this shift, we write ρ : −→ {u = u 0 } as is usually done for functions.…”

“…For example, nonresonant complex saddles (eigenvalues have irrational negative ratio) may appear to be formally linearizable but analytically non-linearizable [10,14,26,36,37]. On the other hand, the formal (orbital) classification of resonant complex saddles (eigenvalues have rational negative ratio) depends on scalar parameters [3,4,12,13], while their analytic orbital classification gives rise to functional moduli (Martinet-Ramis moduli, [9,13,19]) in the same way as their analytic classification (Ecalle-Voronin moduli, [7,13,18,20,32,34]). Similarly, the formal orbital classification of saddle-nodes (one eigenvalue is equal to zero) depends on a finite number of scalar parameters, while their analytic orbital classification is given by Martinet-Ramis functional moduli [8,13,20,33].…”

Normal Form in Filtered Lie Algebra Representations

“…Maybe they are, but they certainly do the bookkeeping right! The first application of this idea to normal form theory of ordinary differential equations at equilibrium under orbital equivalence is given by Bogdanov in his treatment of the planar nilpotent unique normal form (Bogdanov, 1979). This paper was never translated into English and did not have the impact it should have had.…”

Singularities of implicit differential systems and their integrability

“…In this case we gave Example 5.1 which is related to Bogdanov-Takens singularity (cf. [3]) with the unique separatrix and solution which cannot be analytic.…”