| A parameter-free and easy-to-implement method is designed for determining the global embedding dimension in nonlinear time series analysis using the theory of deterministic chaos. An effective and reliable method for determining the dynamical dimension is also developed. The robustness and stability of these two methods are demonstrated by tests using a variety of model dynamical systems. They are shown to overcome some of the difficulties of existing leading methods. Their applicability to real-world data is demonstrated through the analysis of sunspot numbers and Canadian lynx data.; A systematic study of Lyapunov exponents, fractal dimensions and predictabilities for the surface weather over the eastern United States is performed by analyzing 15 years of hourly sea-level pressures at 88 meteorological stations. The global and dynamical dimensions are determined by the two methods developed in this work. It is found that the Lyapunov exponent spectra of the local weather systems all have the same sign pattern {dollar}{lcub}+++0---{rcub},{dollar} indicating strong hyperchaos. This study reveals that the Lyapunov exponent spectrum, fractal dimension, and predictability of the local weather systems all have a well-defined geographical variability, gradually varying with latitude. The error doubling times, as estimated from the Kolmogorov entropy, indicate that the surface weather in the South Atlantic and South Central regions of the United States can be predicted, on average, for double the lead times than those in the Northeast and Midwest regions. The low fractal dimension estimates made in this study provide strong evidence for the existence of low-dimensional attractors for the surface weather. The systematic geographical variation of fractal dimensions and predictabilities, and the existence of well-defined local attractors support the hypothesis that the atmosphere is a set of loosely coupled subsystems and that the fractal dimensions are those of the local attractors. |
