Invariant curves, attractors, and phase diagram of a piecewise linear map with chaos

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We introduce equations describing the invariant curves associated with periodic points in a wide class of two-dimensional invertible maps, which in the special case of the map T(x, z)=(1-a|x|+bz,x) can be solved by analytical methods. In the dissipative case several branches of the separatrices of the fixed points, as well as, of the period-2 and -4 points, are constructed. The regions of the parameter space where a given type of strange attractor exists are located. We point out that the disappearance of homoclinic intersections between the separatrices of the fixed point and that of heteroclinic intersections between the unstable manifolds of the period-2 points and the stable manifold of the fixed point may occur separately, and the latter leads already to the appearance of a two-piece strange attractor. This phenomenon may happen at weak dissipation in other maps, too. In the conservative case b=1 separatrices and certain invariant tori are calculated.

Original languageEnglish
Pages (from-to)195-221
Number of pages27
JournalJournal of Statistical Physics
Issue number1
Publication statusPublished - Oct 1983



  • Two-dimensional map
  • evolution of strange attractors
  • homoclinic and heteroclinic points
  • invariant curves
  • invariant tori
  • phase diagram

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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