Global Aspects of the Dissipative Dynamical Systems. I: -- Statistical Identification and Fractal Properties of the Lorenz Chaos --

Several empirical concepts which become practically necessary to characterize the ergodic motion in the dissipative systems are explained concretely by carrying out with the Lorenz model. Chaotic motions are characterized not only by the geometrical features in the phase space but also by the statistical aspects in the time series of the dynamical variables. These global aspects are understood in terms of the fractal properties appearing in the dimension spectrum of the strange attractor and the Hausdorff dimension of the time series. Furthermore, the internal stiffness and markovian order of the chaotic orbit are discussed in relation to the time series analysis based on the auto-regressive model. The numerical methods proposed here will be applied to the problem of the chaotic response under the periodic perturbation (in Part II).