Displaying similar documents to “The return map for a planar vector field with nilpotent linear part: a direct and explicit derivation.”

Integrable analytic vector fields with a nilpotent linear part

Xianghong Gong (1995)

Annales de l'institut Fourier

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We study the normalization of analytic vector fields with a nilpotent linear part. We prove that such an analytic vector field can be transformed into a certain form by convergent transformations when it has a non-singular formal integral. We then prove that there are smoothly linearizable parabolic analytic transformations which cannot be embedded into the flows of any analytic vector fields with a nilpotent linear part.

Smoothness property for bifurcation diagrams.

Robert Roussarie (1997)

Publicacions Matemàtiques

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Strata of bifurcation sets related to the nature of the singular points or to connections between hyperbolic saddles in smooth families of planar vector fields, are smoothly equivalent to subanalytic sets. But it is no longer true when the bifurcation is related to transition near singular points, for instance for a line of double limit cycles in a generic 2-parameter family at its end point which is a codimension 2 saddle connection bifurcation point. This line has a flat contact with...

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

Waldo Arriagada-Silva, Christiane Rousseau (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

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In this paper we study equivalence classes of generic 1 -parameter germs of real analytic families 𝒬 ε unfolding codimension 1 germs of diffeomorphisms 𝒬 0 : ( ℝ , 0 ) → ( ℝ , 0 ) with a fixed point at the origin and multiplier - 1 , under (weak) analytic conjugacy. These germs are generic unfoldings of the flip bifurcation. Two such germs are analytically conjugate if and only if their second iterates, 𝒫 ε = 𝒬 ε ∘ 2 , are analytically conjugate. We give a complete modulus of analytic classification: this modulus is an unfolding...

Convergence of power series along vector fields and their commutators; a Cartan-Kähler type theorem

B. Jakubczyk (2000)

Annales Polonici Mathematici

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We study convergence of formal power series along families of formal or analytic vector fields. One of our results says that if a formal power series converges along a family of vector fields, then it also converges along their commutators. Using this theorem and a result of T. Morimoto, we prove analyticity of formal solutions for a class of nonlinear singular PDEs. In the proofs we use results from control theory.