Using chaos to shadow the quadratic map for all time

Nejib Smaoui, Eric Kostelich

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

We present a new way of proving that a computer-generated orbit for the chaotic attractor outside the periodic windows of the quadratic map fa = ax (1 - x) can be shadowed for all time (i-e., there exist true orbits {xy}kj-1i=0 which stay near a numerical orbit {pi}Ni=0 for all time). This is done by computing a numerical orbit for a particular value of a and show that {pi}Ni=0 ≈ ∪mj=1 {xij}kj-1i=0 where Σmj=1 kj = N. The true orbits are found using slightly different maps fa, = ajx(1 - x), where max1≤j≤m (aj - a) < √δp. This technique can therefore be applied to other chaotic differential equation and discrete systems.

Original languageEnglish (US)
Pages (from-to)117-129
Number of pages13
JournalInternational Journal of Computer Mathematics
Volume70
Issue number1
DOIs
StatePublished - Jan 1 1998

Keywords

  • Attractors
  • Chaotic trajectories
  • Quadratic map
  • Shadowing

ASJC Scopus subject areas

  • Computer Science Applications
  • Computational Theory and Mathematics
  • Applied Mathematics

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