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Applications of the group analysis of differential equations to some systems of noncommuting C 1-smooth vector fields

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Abstract

Given a canonical basis of C 1-smooth vector fields \( \{ \tilde X_i \} \) satisfying certain restrictions on commutators, we prove an existence theorem for their local nilpotent homogeneous approximation at the origin using the methods of the group analysis of differential equations. We study the properties of the quasimetrics induced by some systems of vector fields related to \( \{ \tilde X_i \} \).

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Correspondence to A. V. Greshnov.

Additional information

Original Russian Text Copyright © 2009 Greshnov A. V.

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 1, pp. 47–62, January–February, 2009.

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Greshnov, A.V. Applications of the group analysis of differential equations to some systems of noncommuting C 1-smooth vector fields. Sib Math J 50, 37–48 (2009). https://doi.org/10.1007/s11202-009-0005-8

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Keywords

  • vector field
  • Arzelà-Ascoli theorem
  • theorem on the existence and uniqueness for ODEs
  • commutator
  • quasimetric