The Completely Integrable Differential Systems are Essentially Linear Differential Systems

Jaume Llibre, Claudia Valls, Xiang Zhang

Research output: Contribution to journalArticleResearchpeer-review

6 Citations (Scopus)

Abstract

© 2015, Springer Science+Business Media New York. Let x˙=f(x) be a Ck autonomous differential system with k∈N∪{∞,ω} defined in an open subset Ω of Rn. Assume that the system x˙=f(x) is Cr completely integrable, i.e., there exist n-1 functionally independent first integrals of class Cr with 2≤r≤k. As we shall see, we can assume without loss of generality that the divergence of the system x˙=f(x) is not zero in a full Lebesgue measure subset of Ω. Then, any Jacobian multiplier is functionally independent of the n-1 first integrals. Moreover, the system x˙=f(x) is Cr-1 orbitally equivalent to the linear differential system y˙=y in a full Lebesgue measure subset of Ω. Additionally, for integrable polynomial differential systems, we characterize their type of Jacobian multipliers.
Original languageEnglish
Pages (from-to)815-826
JournalJournal of Nonlinear Science
Volume25
Issue number4
DOIs
Publication statusPublished - 27 Aug 2015

Keywords

  • Completely integrability
  • Differential systems
  • Jacobian multiplier
  • Normal form
  • Orbital equivalence
  • Polynomial differential systems

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