Analytical study of the Lorenz system: Existence of infinitely many periodic orbits and their topological characterization

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Abstract

We consider the Lorenz equations, a system of three-dimensional ordinary differential equations modeling atmospheric convection. These equations are chaotic and hard to study even numerically, and so a simpler “geometric model” has been introduced in the seventies. One of the classical problems in dynamical systems is to relate the original equations to the geometric model. This has been achieved numerically by Tucker for the classical parameter values and remains open for general values. In this paper, we establish analytically a relation to the geometric model for a different set of parameter values that we prove must exist. This is facilitated by finding a way to apply topological tools developed for the study of surface dynamics to the more intricate case of three-dimensional flows.

Original languageEnglish
Article numbere2205552120
JournalProceedings of the National Academy of Sciences of the United States of America
Volume120
Issue number31
DOIs
StatePublished - 24 Jul 2023

Keywords

  • Lorenz system
  • chaos
  • knot theory

ASJC Scopus subject areas

  • General

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