Note on the Markus-Yamabe conjecture for gradient dynamical systems

F. Mañosas, D. Peralta-Salas

Research output: Contribution to journalArticleResearchpeer-review

3 Citations (Scopus)

Abstract

Let v : Rn → Rn be a C1 vector field which has a singular point O and its linearization is asymptotically stable at every point of Rn. We say that the vector field v satisfies the Markus-Yamabe conjecture if the critical point O is a global attractor of the dynamical system over(x, ̇) = v (x). In this note we prove that if v is a gradient vector field, i.e. v = ∇ f (f ∈ C2), then the basin of attraction of the critical point O is the whole Rn, thus implying the Markus-Yamabe conjecture for this class of vector fields. An analogous result for discrete dynamical systems of the form xm + 1 = ∇ f (xm) is proved. © 2005 Elsevier Inc. All rights reserved.
Original languageEnglish
Pages (from-to)580-586
JournalJournal of Mathematical Analysis and Applications
Volume322
DOIs
Publication statusPublished - 15 Oct 2006

Keywords

  • Global attractor
  • Gradient dynamical system
  • Markus-Yamabe conjecture

Fingerprint Dive into the research topics of 'Note on the Markus-Yamabe conjecture for gradient dynamical systems'. Together they form a unique fingerprint.

  • Cite this