Lyapunov exponents and stability in interval maps

Hüseyin Koçak, Kenneth Palmer

Research output: Contribution to journalArticlepeer-review

Abstract

Determination of stability or instability of a given orbit of a scalar interval map is investigated in terms of the sign of the Lyapunov exponent of the orbit. It is proved that an orbit of such a C2 map with a negative Lyapunov exponent is stable. To prove instability, the classical notion of Lyapunov exponent is strengthened by introducing a new quantity called strong Lyapunov exponent. Then, it is proved that an orbit of a C1 interval map with a positive strong Lyapunov exponent is unstable, or equivalently, exhibits sensitive dependence on initial conditions. It is also shown that positive Lyapunov exponent suffices if an additional assumption is made about the critical points of the interval map.

Original languageEnglish (US)
Pages (from-to)79-82
Number of pages4
JournalSeMA Journal
Volume51
Issue number1
DOIs
StatePublished - Jun 2010
Externally publishedYes

Keywords

  • 37C75
  • 37D45
  • 37E05
  • Interval maps
  • Lyapunov exponent
  • sensitive dependence on initial conditions
  • stability
  • strong Lyapunov exponent

ASJC Scopus subject areas

  • Applied Mathematics
  • Modeling and Simulation
  • Numerical Analysis
  • Control and Optimization

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