Viscosity solutions methods for converse KAM theory

Diogo A. Gomes, Adam Oberman

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


The main objective of this paper is to prove new necessary conditions to the existence of KAM tori. To do so, we develop a set of explicit a-priori estimates for smooth solutions of Hamilton-Jacobi equations, using a combination of methods from viscosity solutions, KAM and Aubry-Mather theories. These estimates are valid in any space dimension, and can be checked numerically to detect gaps between KAM tori and Aubry-Mather sets. We apply these results to detect non-integrable regions in several examples such as a forced pendulum, two coupled penduli, and the double pendulum.

Original languageEnglish (US)
Pages (from-to)1047-1064
Number of pages18
JournalMathematical Modelling and Numerical Analysis
Issue number6
StatePublished - Nov 2008
Externally publishedYes


  • Aubry-Mather theory
  • Hamilton-Jacobi integrability
  • Viscosity solutions

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Modeling and Simulation
  • Computational Mathematics
  • Applied Mathematics


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