A tutorial and diagnostic tool for chaotic oscillators and time series

A tutorial and diagnostic visual tool is used to compute a set of strange attractors for a double oscillator. The tool provides a visual environment to detect chaotic behaviour from a time series. Here, computation and visualization are coupled together in a single environment. Common techniques to detect chaos involve the visualization of phase portraits and Poincare maps, apart from the computation of fractal dimension, Lyapunov exponents and Fast Fourier Transforms (FFT). Results from these computations must be viewed graphically before a decision about the chaotic behaviour of a system can be made. In many cases, the use of a single technique may not always guarantee conclusive evidence that a system's behaviour is chaotic. A few numerical integration schemes are built into the environment to generate a time series. The environment is demonstrated first with the use of the Duffing oscillator and the chaotic behaviour of a double oscillator is then presented, which provides an interesting set of strange attractors. © 1997 Elsevier Science Ltd 253 The visualization of time histories by themselves may not always provide definite evidence of chaotic behaviour or classify the nature of chaos (e.g. transient chaos). As a result qualitative methods to study the behaviour of the system further include the visualization of phase plane and Poincare maps. Phase portraits by themselves may provide a clue to chaos, but a modified technique called Poincare maps, which is basically plots of stroboscopic sampling of the phase plane will indicate if the structure of the plot is fractal. More details of fractals can be found in work by Barnsley [3], Farmer [4] and Mandelbrot [5]. A map which looks fractal is a strong indicator of chaos. Enlargements of the plot will indicate if a strange attractor exists. A strange attractor is basically a chaotic attractor, which visually looks as if complex stretching and folding have taken place. Parametric studies can indicate if more than one attractor exists, and plots of basins of attraction between the attractors can be obtained for a set of initial conditions [6, 7]. However, problems become apparent if the system has noisy input, or if there is large dissipation. In problems where the dimension is greater than 3, a visual examination of the Poincare map alone is not sufficient. It has been generally accepted that a positive Lyapunov exponent and a non-integer fractal dimension is a good indicator of chaos. Reliably obtaining the Lyapunov's exponent or fractal dimension with a limited amount of data needs careful attention and the generation of the fractal dimension i n particular needs intensive computation. For many numerically produced time series, where noise is not much of a problem, the computation of the largest Lyapunov exponent can be definitive. There are, however, the difficult issues of choosing a proper time Chaos & Graphics