ABSTRACT
From a given time series X(t), an m-dimensional vector V(m, T) in phase space can be constructed by the m-history with time delay T: V(m, T) = {X(t), X(t + T), . . ., X[t + (m – 1)T]}, where m is the embedding dimension of phase space (Takens 1981). This is a powerful tool in developing numerical algorithms of nonlinear dynamics, since it is much easier to observe only one variable than to analyze a complex multi-dimensional system. The phase portrait in two-dimensional phase space X(t + T) versus X(t) gives clear picture of the underlying dynamics of a time series. Figure 5.2 displays the phase portrait of the three models using the sample data of Figure 5.1. The nearly uniform cloud of points in Figure 5.2a closely resembles the phase portrait of random noise (with infinite degree of freedom). The curved image in Figure 5.2b is characteristic of the one-dimensional unimodal discrete chaos LCD with a fractal dimension less than one. The spiral pattern in Figure 5.2c is typical of a RCC strange attractor whose dimensionality is a fractal number between two and three. Its spiral orbit differs from periodic cycles.
