Quasi-conformal mapping theorem and bifurcations

Abstract

LetH be a germ of holomorphic diffeomorphism at 0 ∈ ℂ. Using the existence theorem for quasi-conformal mappings, it is possible to prove that there exists a multivalued germS at 0, such thatS(ze ^2πi)=H○S(z) (1). IfH _λ is an unfolding of diffeomorphisms depending on λ ∈ (ℂ,0), withH ₀=Id, one introduces its ideal$\mathcal{I}_H$. It is the ideal generated by the germs of coefficients (a _i (λ), 0) at 0 ∈ ℂ^k, whereH _λ(z)−z=Σa _i (λ)z ⁱ. Then one can find a parameter solutionS _λ (z) of (1) which has at each pointz ₀ belonging to the domain of definition ofS ₀, an expansion in seriesS _λ(z)=z+Σb _i (λ)(z−z ₀)ⁱ with$(b_i ,0) \in \mathcal{I}_H$, for alli.

This result may be applied to the bifurcation theory of vector fields of the plane. LetX _λ be an unfolding of analytic vector fields at 0 ∈ ℝ² such that this point is a hyperbolic saddle point for each λ. LetH _λ(z) be the holonomy map ofX _λ at the saddle point and$\mathcal{I}_H$ its associated ideal of coefficients. A consequence of the above result is that one can find analytic intervals σ, τ, transversal to the separatrices of the saddle point, such that the difference between the transition mapD _λ(z) and the identity is divisible in the ideal$\mathcal{I}_H$. Finally, suppose thatX _λ is an unfolding of a saddle connection for a vector fieldX ₀, with a return map equal to identity. It follows from the above result that the Bautin ideal of the unfolding, defined as the ideal of coefficients of the difference between the return map and the identity at any regular pointz∈σ, can also be computed at the singular pointz=0. From this last observation it follows easily that the cyclicity of the unfoldingX _λ, is finite and can be computed explicity in terms of the Bautin ideal.