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 to analyze a complex 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. With the fixed point solution (the so-called zero-dimensional attractor), the dynamical system is represented by only one point in the phase portrait. For periodic solution (the onedimensional attractor), its portrait is a closed loop. Figure 4.2 displays the phase portrait of the three models. The nearly uniform cloud of points in Figure 4.2a closely resembles the phase portrait of random noise (with infinite degree of freedom). The curved image in Figure 4.2b is characteristic of the onedimensional unimodal chaos in discrete time. The spiral pattern in Figure 4.2c is typical of a strange attractor whose dimensionality is not an integer. Its wandering orbit differs from periodic cycles.
