Testing Deterministic Chaos: Incorrect Results of the 0–1 Test and How to Avoid Them

Wieslaw Marszalek,Jan Sadecki,Maciej Walczak

doi:10.1109/access.2019.2960378

Abstract

The `false-negative' and `false-positive' outcomes of the 0-1 test for chaos in continuous dynamical systems are described and analyzed in this paper. First, typical false outcomes of the 0-1 test for chaos are illustrated through several numerical examples of the solutions of chaotic continuous systems. Those examples are based on computation of the K values in the 0-1 test (0 ≤ K ≤ 1) for a selection of two parameters, namely the dt, output step in the numerical solver, and the T value (integer denoting the step of the output sample selection). The central role in the `false-negative' outcome is played by the oversampling phenomenon in the 0-1 test, while the `false-positive' results are possible for a complicated periodic signal having a spectrum with multiple frequencies. Analyzing the spectra of the signals is the key method to avoid the false outcomes and also an important tool in the process of reconstructing of chaotic attractors from the time series signals. The correct computing process for continuous dynamical systems and selection of the parameters dt and T depend on the analyzed system (dynamical model) and should always be preceded (or combined with) the frequency analysis of the examined signals. The computation of special multi-parameter (n-parameter; n ≥ 2) bifurcation diagrams for the 0-1 test should, in most cases, be done by parallel computing, since, obtaining one such multi-parameter bifurcation diagram in practice requires solving of the underlying mathematical model (system of ODEs) millions of times.

Highlights

THE 0-1 TEST FOR CHAOS The relatively new 0-1 test for chaos continues to interest more and more researchers dealing with various continuous and discrete oscillatory dynamical systems
Typical continuous systems being tested with the 0-1 test are the well-known Lorenz, Rössler, Lü and Chua systems or circuits, oscillatory memristor and arc circuits, mechanical systems, stock market, chaotic plasma, epilepsy and traffic models, laser systems, interactions in industrial production models, experimental data and others [1]–[14]
For each pair of discrete values of the two parameters the ordinary differential equations (ODEs) system was solved in the interval 0 ≤ t ≤ 10000 and the solutions were output with the constant time step of dt = 0.001 to form various sequences being analyzed with the 0-1 test

Summary

Introduction

THE 0-1 TEST FOR CHAOS The relatively new 0-1 test for chaos continues to interest more and more researchers dealing with various continuous and discrete oscillatory dynamical systems. The numbers are the K values obtained by using the 0-1 test for chaos for the chaotic Lorenz system with parameters σ = 10, ρ = 28 and β = 8/3.

Results

Conclusion