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New sufficient conditions for a center and global phase portraits for polynomial systems.

Hector Giacomini, Malick Ndiaye (1996)

Publicacions Matemàtiques

In this paper we consider cubic polynomial systems of the form: x' = y + P(x, y), y' = −x + Q(x, y), where P and Q are polynomials of degree 3 without linear part. If M(x, y) is an integrating factor of the system, we propose its reciprocal V (x, y) = 1 / M(x,y) as a linear function of certain coefficients of the system. We find in this way several new sets of sufficient conditions for a center. The resulting integrating factors are of Darboux type and the first integrals are in the Liouville form.By...

Non-smooth variational bifurcation

Marco Degiovanni, Antonio Marino (1987)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We consider bifurcation problems associated with some lower semicontinuous functionals that do not satisfy the usual regularity assumptions. For such functionals it is possible to define a generalized "Hessian form" and to show that certain eigenvalues of this one are bifurcation values. The results are applied to a bifurcation problem for elliptic variational inequalities.

Non-uniformly hyperbolic horseshoes arising from bifurcations of Poincaré heteroclinic cycles

Jacob Palis, Jean-Christophe Yoccoz (2009)

Publications Mathématiques de l'IHÉS

In the present paper, we advance considerably the current knowledge on the topic of bifurcations of heteroclinic cycles for smooth, meaning C ∞, parametrized families {g t ∣t∈ℝ} of surface diffeomorphisms. We assume that a quadratic tangency q is formed at t=0 between the stable and unstable lines of two periodic points, not belonging to the same orbit, of a (uniformly hyperbolic) horseshoe K (see an example at the Introduction) and that such lines cross each other with positive relative speed as...

Normal forms of analytic perturbations of quasihomogeneous vector fields: Rigidity, invariant analytic sets and exponentially small approximation

Eric Lombardi, Laurent Stolovitch (2010)

Annales scientifiques de l'École Normale Supérieure

In this article, we study germs of holomorphic vector fields which are “higher order” perturbations of a quasihomogeneous vector field in a neighborhood of the origin of ℂ n , fixed point of the vector fields. We define a “Diophantine condition” on the quasihomogeneous initial part S which ensures that if such a perturbation of S is formally conjugate to S then it is also holomorphically conjugate to it. We study the normal form problem relatively to S . We give a condition on S that ensures that there...

Normal forms of invariant vector fields under a finite group action.

Federico Sánchez-Bringas (1993)

Publicacions Matemàtiques

Let Γ be a finite subgroup of GL(n, C). This subgroup acts on the space of germs of holomorphic vector fields vanishing at the origin in Cn and on the group of germs of holomorphic diffeomorphisms of (Cn, 0). We prove a theorem of invariant conjugacy to a normal form and linearization for the subspace of invariant germs of holomorphic vector fields and we give a description of this type of normal forms in dimension n = 2.

Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part

Patrick Bonckaert, Freek Verstringe (2012)

Annales de l’institut Fourier

We explore the convergence/divergence of the normal form for a singularity of a vector field on ℂ n with nilpotent linear part. We show that a Gevrey- α vector field X with a nilpotent linear part can be reduced to a normal form of Gevrey- 1 + α type with the use of a Gevrey- 1 + α transformation. We also give a proof of the existence of an optimal order to stop the normal form procedure. If one stops the normal form procedure at this order, the remainder becomes exponentially small.

Normalization of bundle holomorphic contractions and applications to dynamics

François Berteloot, Christophe Dupont, Laura Molino (2008)

Annales de l’institut Fourier

We establish a Poincaré-Dulac theorem for sequences ( G n ) n ∈ ℤ of holomorphic contractions whose differentials d 0 G n split regularly. The resonant relations determining the normal forms hold on the moduli of the exponential rates of contraction. Our results are actually stated in the framework of bundle maps.Such sequences of holomorphic contractions appear naturally as iterated inverse branches of endomorphisms of ℂ ℙ k . In this context, our normalization result allows to estimate precisely the distortions of ellipsoids...

Normalization of Poincaré singularities via variation of constants.

Timoteo Carletti, Alessandro Margheri, Massimo Villarin (2005)

Publicacions Matemàtiques

We present a geometric proof of the Poincaré-Dulac Normalization Theorem for analytic vector fields with singularities of Poincaré type. Our approach allows us to relate the size of the convergence domain of the linearizing transformation to the geometry of the complex foliation associated to the vector field.

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