Iterations of perturbed tent maps with applications to chaos control

B. T. Polyak, E. N. Gryazina

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1 Citation (Scopus)


Iterations of 1D simple maps such as logistic, tent, cubic ones are very well studied. However perturbed versions of these maps (close in uniform norm but with strongly varying derivatives) can exhibit completely different behavior. We encounter such situation when dealing with chaos stabilization via small control. In this paper we present analytical investigation of this effect for one particular case - piecewise linear perturbation of the tent map. Surprisingly, iterations of this map converge to the unique fixed point very fast for all initial points. The result is in sharp contrast with iterations of the original tent map but explains fast stabilization of unstable periodic orbits by predictive control, proposed in Polyak & Maslov (2005); Polyak (2005).

Original languageEnglish
Title of host publication1st IFAC Conference on Analysis and Control of Chaotic Systems, CHAOS'06
PublisherIFAC Secretariat
Number of pages5
EditionPART 1
ISBN (Print)9783902661098
Publication statusPublished - 2006
Externally publishedYes

Publication series

NameIFAC Proceedings Volumes (IFAC-PapersOnline)
NumberPART 1
ISSN (Print)1474-6670


  • Chaos
  • Iterations
  • Markov chains
  • Nonlinear maps
  • Predictive control
  • Stabilization


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