Abstract
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.
Highlights
In 1963, meteorologist Edward Lorenz suggested an approximate mathematical model for the Rayleigh–Bénard convection and discovered numerically a chaotic attractor in this model [76]
The aim of this work is to discuss the estimates of the global stability and the difficulties of numerically studying the birth of self-excited and hidden attractors, caused by the loss of global stability
This problem relates to a generalization [44,63] of the second part of the celebrated Hilbert’s 16th problem [34] on the number and mutual disposition of attractors and repellers in the chaotic multidimensional dynamical systems, and, in particular, their dependence on the degree of polynomials in the model
Summary
In 1963, meteorologist Edward Lorenz suggested an approximate mathematical model (the Lorenz system) for the Rayleigh–Bénard convection and discovered numerically a chaotic attractor in this model [76]. In this definition, the stationary set can contain both stable (trivial oscillations) and unstable equilibrium states, i.e., the local stability of all equilibria is not required. This is due to the loss of stability of the stationary set Within this framework, it is naturally to classify oscillations in systems as self-excited or hidden [39,62,67,68]. Model has a global bounded convex absorbing set, over time, the state of the system, observed experimentally, will be attracted to the local attractor contained in the absorbing set This problem can be considered as a practical interpretation of the problem of determining the boundary of global stability
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