Advertisement

The Convergence Problem in Finite Dimensions

  • Alain HarauxEmail author
  • Mohamed Ali Jendoubi
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

This core chapter concerns the convergence problem in finite dimensions. We recall that even gradient systems in two dimensions may be divergent, and we show that this phenomenon can happen even for potentials which are almost as smooth as analytic functions. Analyticity, through the Łojasiewicz gradient produces convergence of bounded trajectories, for gradient systems and some classes of gradient-like systems as well. We illustrate all the results by carefully selected examples.

References

  1. 1.
    J. Palis, W. de Melo, Geometric theory of dynamical systems (An introduction. Translated from the Portuguese by A. K. Manning. Springer, New York, 1982)Google Scholar
  2. 2.
    S. Łojasiewicz, Ensembles semi-analytiques. Preprint, I.H.E.S. Bures-sur-Yvette (1965), http://perso.univ-rennes1.fr/michel.coste/Lojasiewicz.pdf
  3. 3.
    S. Lojasiewicz, Une propriété topologique des sous-ensembles analytiques réels. In Les Équations aux Dérivées Partielles (Paris, 1962), pp. 87–89. Éditions du Centre National de la Recherche Scientifique, Paris, 1963Google Scholar
  4. 4.
    L. Véron, Un exemple concernant le comportement asymptotique de la solution bornée de l’équation \(d^2u/dt^2\in \partial \varphi (u)\). Monatsh. Math. 89(1), 57–67 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    H. Attouch, X. Goudou, P. Redont, The heavy ball with friction method. I. The continuous dynamical system: global exploration of the local minima of a real-valued function by asymptotic analysis of a dissipative dynamical system. Commun. Contemp. Math. 2, 1–34 (2000)MathSciNetzbMATHGoogle Scholar
  6. 6.
    M.A. Jendoubi, P. Poláčik, Non-stabilizing solutions of semilinear hyperbolic and elliptic equations with damping. Proc. Roy. Soc. Edinburgh Sect. A 133(5), 1137–1153 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    A. Haraux, M.A. Jendoubi, Convergence of solutions of second-order gradient-like systems with analytic nonlinearities. J. Diff. Equat. 144(2), 313–320 (1998)Google Scholar
  8. 8.
    L. Chergui, Convergence of global and bounded solutions of a second order gradient like system with nonlinear dissipation and analytic nonlinearity. J. Dynam. Diff. Equat. 20, 643–652 (2008)Google Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Laboratoire Jacques-Louis LionsSorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7598ParisFrance
  2. 2.Institut Préparatoire aux Etudes Scientifiques et TechniquesUniversité de CarthageLa MarsaTunisia
  3. 3.Faculté des sciences de Tunis, Laboratoire EDP-LR03ES04Université de Tunis El ManarTunisTunisia

Personalised recommendations