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

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.INDEX TERMS Oscillatory chaotic and periodic circuits and systems, the 0-1 test for chaos, bifurcation diagrams, oversampling, reconstruction of chaotic attractors.

Two-Parameter 0-1 Test for Chaos and Sample Entropy Bifurcation Diagrams for Nonlinear Oscillating Systems

“…The test has been applied in engineering, physics, chemistry, biology, weather prediction, stock exchange analysis and other areas. Several problems of using the test have been reported [11]- [14]. Those problems are mainly due to the oversampling issue and difficulty in selecting parameters, such as the length of the time interval, t f , in 0 ≤ t ≤ t f , the output sampling step, dt, and another parameter T , an integer used in the downsampling process, see [10], [13], [14].…”

“…Several problems of using the test have been reported [11]- [14]. Those problems are mainly due to the oversampling issue and difficulty in selecting parameters, such as the length of the time interval, t f , in 0 ≤ t ≤ t f , the output sampling step, dt, and another parameter T , an integer used in the downsampling process, see [10], [13], [14]. On the other hand, the entropy concept, and the sample entropy in particular [15]- [17], is used mainly in physics and medicine (physiology, cardiology and enzymology) and to a lesser extend in engineering.…”

Computational Analysis of Ca2+ Oscillatory Bio-Signals: Two-Parameter Bifurcation Diagrams

“…It is certainly possible to compute the two-parameter diagrams for Lyapunov exponents [ 22 ], stability domains [ 23 ], Poincaré return maps [ 24 ], entropy [ 25 ] and the 0–1 test for chaos [ 26 ]. The later two-parameter diagrams for the electric circuits, Lorenz and Rössler chaotic systems, are presented in [ 20 , 27 , 28 ].…”

“…The diagrams in Figure 3 a and Figure 4 a are compared with the sample entropy (SE) [ 29 , 30 ] and 0–1 test (T01) [ 19 , 20 ] for chaos diagrams in Figure 13 in the same rectangular areas of the two varying parameters. The diagrams in Figure 13 are of size 1000 × 1000.…”

“…The frequency diagram may be essential in other kinds of analysis. For examples, it has been established recently that applying the 0–1 test for chaos may require a prior knowledge of the maximum frequency of the continuous spectrum of chaotic signals to properly choose certain parameters used in that test [ 19 , 20 ].…”