Temporally Normalizable Generic Unfoldings of Order-1 Weak Foci

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.…”

“…Proposition 2.2 (see [4]). There exists a germ of real analytic change of coordinates bringing (2.1) into a prepared forṁ…”

“…The isochronicity problem is closely related to a more general problem: the temporal normalizability of the family of analytic systems (see [4]). The analytical obstructions that prevent the system from being isochronous can be identified with the non-triviality of a functional component of the modulus of analytic classification under weak conjugacy plus an additional vanishing condition.…”

“…However, it proves analytic in the case of elliptic singularities (see [4]). However, it proves analytic in the case of elliptic singularities (see [4]).…”

“…In [4], we ask the question of whether there exists a weak conjugacy bringing the family (2.2), with the time part (2.8), into a family with the time part Ø ε = β(1 + C(ε)u) and the orbital part X ε ; see definition 2.3. Inasmuch as t ε X ε and Ø ε X ε would be weakly orbitally equivalent we can suppose, in particular, that the conjugacy fixes the orbital part and X ε = X ε .…”