TY - JOUR

T1 - The Lorenz system

T2 - hidden boundary of practical stability and the Lyapunov dimension

AU - Kuznetsov, N. V.

AU - Mokaev, T. N.

AU - Kuznetsova, O. A.

AU - Kudryashova, E. V.

PY - 2020/10

Y1 - 2020/10

N2 - 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.

AB - 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.

KW - Chaos

KW - Global stability

KW - Hidden attractor

KW - Lyapunov dimension

KW - Lyapunov exponents

KW - Time-delayed feedback control

KW - Transient set

KW - Unstable periodic orbit

KW - HAUSDORFF DIMENSION

KW - LOCALIZATION

KW - LIMITATIONS

KW - TRAJECTORIES

KW - SIMULATION

KW - VARIABILITY

KW - PERIODIC-ORBITS

KW - EQUATION

KW - CHAOTIC ATTRACTOR

KW - LONG

UR - http://www.scopus.com/inward/record.url?scp=85089289298&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/eaea7580-3832-31e4-93c1-b89fa9bf097b/

U2 - 10.1007/s11071-020-05856-4

DO - 10.1007/s11071-020-05856-4

M3 - Article

AN - SCOPUS:85089289298

VL - 102

SP - 713

EP - 732

JO - Nonlinear Dynamics

JF - Nonlinear Dynamics

SN - 0924-090X

IS - 2

ER -