Two-parameter diagrams obtained through the 0-1 test of chaos for nonlinear oscillatory continuous systems are presented in this paper. The diagrams are the results of a parallel approach to tackle enormous memory and computational time requirements due to the known oversampling problem associated with the use of the 0-1 test for chaos in continuous systems. Our rectangular diagrams with black-and-white shades of gray levels correspond to the numbers between 0 and 1 obtained as the result of the 0-1 test for chaos. A comparison between the two-parameter diagrams for the 0-1 test with the color bifurcation diagrams for oscillatory systems obtained from another method (period-n identification) is also considered. Illustrative examples are based on both the well-known Lorenz model and a model describing two equivalent electric arc circuits. INDEX TERMS The 0-1 test of chaos, two-parameter bifurcation diagrams, oscillatory continuous dynamics, parallel computation, electric arc circuits.

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.

This paper presents and compares two-parameter bifurcation diagrams obtained from the 0-1 test for chaos and sample entropy methods for nonlinear oscillating systems. Based on the computed diagrams it is often possible to establish a unique correspondence between the two types of diagrams. Comparison of the diagrams are presented for typical chaotic systems, including those of Lorenz, Rössler, electric arc circuits and chemical oscillating systems. The two types of bifurcation diagrams may also be related to the period−n and frequency distribution diagrams. The main goal is to address the issue of obtaining similar results from the 0-1 test for chaos and sample entropy methods and discuss difficulty in properly selecting parameters needed in the two methods. A possible link between the 0-1 test for chaos and surrogate data methods is also mentioned.INDEX TERMS Time-series identification, sample entropy, the 0-1 test for chaos, two-parameter bifurcation diagrams, oscillatory nonlinear systems

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