Characterization of the generic unfolding of a weak focus

Abstract

In this paper we give a geometric description of the foliation of a generic real analytic family unfolding a real analytic vector field with a weak focus at the origin, and show that two such families are orbitally analytically equivalent if and only if the families of diffeomorphisms unfolding the complexified Poincaré map of the singularities are conjugate. Moreover, by shifting the leaves of the formal normal form in the blow-up (quasiconformal surgery) by means of a fibered transformation along a convenient complex cross-section, one constructs an abstract manifold of complex dimension 2 equipped with an elliptic holomorphic foliation whose monodromy map coincides with a given family of admissible diffeomorphisms.

The modulus of analytic classification for the unfolding of the codimension-one flip and Hopf bifurcations

“…The strong version of this result has been proven in [1], [2]. A simple analysis shows that the proof of Theorem 8.2 in the strong case applies verbatim to its weak version.…”

“…The proof in [16] was done for complex germs. In the case of real analytic germs a better proof of the other direction is given in [1], [2] for the case of families of real diffeomorphisms Q η . Indeed, given any two conjugate generic families of real analytic diffeomorphims of the form P η = Q •2 η , it is proved that they are embeddable as Poincaré monodromies of analytic vector fields unfolding a weak focus, which are analytically orbitally equivalent (a weak focus of a real vector field is a singular point with two pure imaginary eigenvalues and which is not a centre).…”

“…In [1], it is shown that two germs of analytic families of planar vector fields with a generic Hopf bifurcation of order 1 are orbitally analytically equivalent if and only if the germs of analytic families of their Poincaré monodromies are conjugate. The (unfolded) Poincaré monodromy is exactly a real diffeomorphism P ε : (R, 0) → (R, 0), which is the second iterate of a Poincaré semi-monodromy Q ε with a flip bifurcation.…”

Analytic obstructions to isochronicity in codimension 1*

“…(Algebraically, the divergence of the normalizing coordinate is identified with a cochain in the ring of summable power series.) Indeed, in [2,4,5] we have proved that the invariants of analytic classification under orbital equivalence of (1.1) coincide with the unfolding of the Ecalle-Voronin invariants of the monodromy; see, for example, [2,3]. In the case of a saddle point, these invariants coincide with the Ecalle-Voronin invariants of the holonomy of any separatrix.…”

Blow-up rates of large solutions for a ϕ-Laplacian problem with gradient term

“…Let g satisfy (H 3 ). If p > − 1 and γ +1 < p/(p +1), the Dirichlet problem (log θ (1 + µ + |u | ν )|u | −2 u ) + g(u ) = f (u) on (0, R), 1) and the Neumann problem…”