A rigorous lower bound for the stability regions of the quadratic map

Warwick Tucker, Daniel Wilczak

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Abstract

We establish a lower bound on the measure of the set of stable parameters a for the quadratic map Qa (x) = a x (1 - x). For these parameters, we prove that Qa either has a single stable periodic orbit or a period-doubling bifurcation. From this result, we also obtain a non-trivial upper bound on the set of stochastic parameters for Qa.

Original languageEnglish
Pages (from-to)1923-1936
Number of pages14
JournalPhysica D: Nonlinear Phenomena
Volume238
Issue number18
DOIs
Publication statusPublished - Sep 2009
Externally publishedYes

Keywords

  • Interval analysis
  • Period-doubling bifurcation
  • Periodic orbit
  • Quadratic map

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