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

被引：73
作者：
Kuznetsov, N., V [1 ,2 ,3 ]
Mokaev, T. N. [1 ]
Kuznetsova, O. A. [1 ]
Kudryashova, E., V [1 ]
机构：
[1] St Petersburg State Univ, Math & Mech Fac, St Petersburg, Russia
[2] Univ Jyvaskyla, Fac Informat Technol, Jyvaskyla, Finland
[3] RAS, Inst Problems Mech Engn, St Petersburg, Russia
来源：
NONLINEAR DYNAMICS | 2020年 / 102卷 / 02期
基金：
俄罗斯科学基金会;
关键词：
Global stability; Chaos; Hidden attractor; Transient set; Lyapunov exponents; Lyapunov dimension; Unstable periodic orbit; Time-delayed feedback control; HAUSDORFF DIMENSION; CHAOTIC ATTRACTOR; PERIODIC-ORBITS; SIMULATION; LONG; LOCALIZATION; TRAJECTORIES; VARIABILITY; LIMITATIONS; EQUATION;
D O I：
10.1007/s11071-020-05856-4
中图分类号：
TH [机械、仪表工业];
学科分类号：
0802 ;
摘要：
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.
引用
收藏
页码：713 / 732
页数：20
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