The modulus of analytic classification for the unfolding of the codimension-one flip and Hopf bifurcations (Q650982)

The authors study equivalence classes of generic 1-parameter germs of real analytic families \(\mathcal{Q}_{\varepsilon}\) unfolding codimension 1 germs of diffeomorphisms \(\mathcal{Q}_{0}: (\mathbb{R},0)\to (\mathbb{R},0)\) with a fixed point at the origin and multiplier \(-1\) under weak analytic conjugacy. It is shown that these germs present generic unfoldings of the flip bifurcation. Necessary and sufficient condition for analytic conjugacy of two such germs is given. The Glutsyuk modulus is described on two sectors which do not cover a full neighborhood of the origin. From this modulus the Lavaurs can be recovered. Since the modulus depends analytically on \(\varepsilon\), the Glutsyuk modulus, defined only on a union of two sectors in the parameter space, determines the Lavaurs modulus for parameter values in a full neighborhood of the origin. As an application, the germs of generic analytic families of vector fields undergoing a Hopf bifurcation of order 1 are completely classified under orbital equivalence.