On the example of the famous Lorenz system, the difficulties and opportunities of reliable numerical analysis of chaotic dynamical systems are discussed in this article. For the Lorenz system, the boundaries of global stability are estimated and the difficulties of numerically studying the birth of self-excited and hidden attractors, caused by the loss of global stability, are discussed. The problem of reliable numerical computation of the finite-time Lyapunov dimension along the trajectories over large time intervals is discussed. Estimating the Lyapunov dimension of attractors via the Pyragas time-delayed feedback control technique and the Leonov method is demonstrated. Taking into account the problems of reliable numerical experiments in the context of the shadowing and hyperbolicity theories, experiments are carried out on small time intervals and for trajectories on a grid of initial points in the attractor’s basin of attraction.

Original languageEnglish
Pages (from-to)713-732
Number of pages20
JournalNonlinear Dynamics
Volume102
Issue number2
Early online date11 Aug 2020
DOIs
StatePublished - Oct 2020

    Scopus subject areas

  • Mechanical Engineering
  • Aerospace Engineering
  • Ocean Engineering
  • Applied Mathematics
  • Electrical and Electronic Engineering
  • Control and Systems Engineering

    Research areas

  • Chaos, Global stability, Hidden attractor, Lyapunov dimension, Lyapunov exponents, Time-delayed feedback control, Transient set, Unstable periodic orbit, HAUSDORFF DIMENSION, LOCALIZATION, LIMITATIONS, TRAJECTORIES, SIMULATION, VARIABILITY, PERIODIC-ORBITS, EQUATION, CHAOTIC ATTRACTOR, LONG

ID: 61326334