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

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 selfexcited 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.

Stochastic Chaos and Markov Blankets

“…The first section of this technical note describes one procedure using the Lorenz system [ 17 , 18 ] as the basis of a random dynamical system that exhibits stochastic chaos. This system possesses a pullback attractor that is itself a random variable [ 19 , 20 , 21 ].…”

“…Technically, nonequilibrium in this context refers to the time irreversibility of the dynamics (also known as loss of detailed balance) that inherits from solenoidal or conservative flow of systemic states [ 1 , 22 , 23 ]. It is this conservative component of flow that underwrites stochastic chaos as nicely described in [ 24 ] and quantified by positive Lyapunov exponents [ 25 ], i.e., exponential divergence of trajectories and associated sensitivity to initial conditions [ 17 ]. We emphasise that stochastic chaos is an attribute of the flow, as opposed to the noise, in a stochastic differential equation.…”

Identification of Chaos in Financial Time Series to Forecast Nonperforming Loan

“…, N − (m − 1)τ/Δt, m is named the embedding dimension (m ≥ d where d shows the dimension of the attractor), τ denotes the delay time, and Δt represents the sampling time. Computational errors caused by a finite precision arithmetic allow to consider one pseudotrajectory computed for a sufficiently large time interval [29][30][31].…”

“…e time delay is estimated using the AMI algorithm with a delay time belonging to [1,30]. e results for the two datasets are illustrated in Figure 3.…”

Dynamical transition and chaos for a five‐dimensional Lorenz model

“…By using the exhaustive computer search, they found the first hidden chaotic attractor in system (1.2) and pointed out that the new hidden chaotic attractor resembles the classic butterfly shape of the self-excited chaotic attractor. 21 Other results related to the system (1.2), one can refer to other works [22][23][24][25][26] and the references therein.…”