Finite smooth normal forms and integrability of local families of vector fields
2010
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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