Finite smooth normal forms and integrability of local families of  vector fields

Vincent Naudot; Jiazhong Yang

doi:10.3934/dcdss.2010.3.667

Article Contents

2010, Volume 3, Issue 4: 667-682. Doi: 10.3934/dcdss.2010.3.667

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Finite smooth normal forms and integrability of local families of vector fields

Vincent Naudot^1, and
Jiazhong Yang^2,

1.
Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Road, 33431 Boca Raton
2.
School of Mathematical Sciences, Peking University, Beijing, 100871

Received: February 28, 2009

Revised: April 30, 2010

Published: November 2010

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Abstract

In this paper we study a class of smooth vector fields which depend on small parameters and their eigenvalues may admit certain resonances. We shall derive the polynomial normal forms for such systems under $C^k$ conjugacy, where $k$ can be arbitrarily large. When the smoothness of normalization is less required, we can even reduce these systems to their quasi-linearizable normal forms under $C^{k_0}$ conjugacy, where $k_0$ is good enough to preserve certain qualitative properties of the original systems while the normal forms are as convenient as the linearized ones in applications. Concerning the normalization procedure, we prove that the transformation can be expressed in terms of Logarithmic Mourtada Type (LMT) functions, which makes both qualitative and quantitative analysis possible.

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Mathematics Subject Classification: Primary: 37E30; Secondary: 37G07.

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