Two-parameter diagrams obtained through the 0–۱ 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 0–۱ 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 0–۱ test for chaos. A comparison between the two-parameter diagrams for the 0–۱ 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.

Introduction

The 0–۱ test for chaos is a computational tool to analyze nonlinear dynamical systems based on their time series responses [1]–[۵]. Mathematical model of the system is not needed in the analysis. If such a model is known, then one can generate a time series response and feed it into the 0–۱ test for chaos. The purpose of the test is to differentiate between periodic and chaotic responses. Typical situation is that a dynamical system is periodic for a particular interval of parameter values while chaotic for others. The well-known Lorenz, Rössler and Chua systems are examples of such continuous nonlinear systems or circuits. Time series periodic and chaotic signals can also be analyzed by other available tools, such as, for example, Lyapunov exponents, Fourier transforms, bifurcation diagrams, return maps, and others [6], [7], [14]. The 0–۱ test for chaos is a relatively new test that can be applied to both continuous and discrete nonlinear systems or their time series responses. However, because of the possibility for a time series to be oversampled, the continuous case seems to be more challenging than the discrete one [3]. The oversampling phenomenon has been analyzed in [13] and its relation to the Fourier spectrum of the time series has been established.