The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension

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9 Scopus citations


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
JournalNonlinear Dynamics
Early online date11 Aug 2020
StateE-pub ahead of print - 11 Aug 2020

Scopus subject areas

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


  • Chaos
  • Global stability
  • Hidden attractor
  • Lyapunov dimension
  • Lyapunov exponents
  • Time-delayed feedback control
  • Transient set
  • Unstable periodic orbit

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