Global dynamics of the lorenz system with invariant algebraic surfaces

Jaume Llibre, Marcelo Messias, Paulo Ricardo Da Silva

Research output: Contribution to journalArticleResearchpeer-review

42 Citations (Scopus)


In this paper by using the Poincaré compactification of ℝ3 we describe the global dynamics of the Lorenz system x = s(-x + y), y = rx - y - xz, z = -bz + xy, having some invariant algebraic surfaces. Of course (x, y, z) ∈ ℝ3 are the state variables and (s, r, b) ∈ ℝ3 are the parameters. For six sets of the parameter values, the Lorenz system has invariant algebraic surfaces. For these six sets, we provide the global phase portrait of the system in the Poincaré ball (i.e. in the compactification of ℝ3 with the sphere S 2 of the infinity). © 2010 World Scientific Publishing Company.
Original languageEnglish
Pages (from-to)3137-3155
JournalInternational Journal of Bifurcation and Chaos
Issue number10
Publication statusPublished - 1 Jan 2010


  • dynamics at infinity
  • Integrability
  • invariant algebraic surface
  • Lorenz system
  • Poincaré compactification


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