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

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.