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Nonlinear differential algebraic equations.

Shcheglova, A.A. (2007)

Sibirskij Matematicheskij Zhurnal

Normal forms for singularities of one dimensional holomorphic vector fields.

Garijo, Antonio, Gasull, Armengol, Jarque, Xavier (2004)

Electronic Journal of Differential Equations (EJDE) [electronic only]

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 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 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.

Note on canonical forms for functional-differential equations.

Čermák, Jan (2000)

Mathematica Pannonica

Note on Kummer's transformation

František Neuman (1970)

Archivum Mathematicum