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

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.…”

“…In [5] we studied the obstructions that prevent the orbital part of the system (2.2) from being equivalent to (2.6), through identification of a complete modulus under orbital equivalence. On values of ε = 0, the invariant is constructed via comparison of the orbit space of the monodromy (2.4) and the orbit space of its formal normal form.…”

“…We refer the reader to [5] for further details on the orbital part of the invariant and symmetries. The orbital part of the modulus measures the obstruction to match together in a holomorphic chart the two (generically) different normalizing charts ϕ − and ϕ + .…”

“…where q ε is a iπ/ε-periodic multi-valued map with target space a 2-folded Riemann surface R ε deprived of a countable number of holes [2,5,17,19]; see figure 6. The maps Φ 0,∞ ±,ε : Q 0,∞ ± → C are germs of holomorphic sectorial trivializations conjugating the monodromy P ε = q ε • P ε • q −1 ε with the translation by one W → W +1, and known as real Fatou-Glutsyuk coordinates (see [5]).…”

Parametric rigidity of real families of conformal diffeomorphisms tangent to x→−x

“…(The term 'generic' means that (∂ 2 Q/∂x∂η)(0, 0) = 0, where Q(x, η) = Q η (x).) These families are tangent to the standard flip x → −x [3,9], and after removal of the quadratic term they take the form…”