Abstract
Abstract This chapter explores generalizations of fractal dimensions and correlation dimensions (generalized dimensions) that allow us to characterize more complex dynamics. These concepts are somewhat more sophisticated than those explored in the previous chapter but their exploration is worthwhile because they are closely related to the theory of statistical mechanics and thermodynamics with many analogies that link these subjects to dynamics. Embedding schemes are used to generate state spaces from time series recorded from numerical work or from experiments. In favourable circumstances it is sufficient to build the embedding space from the time series of a single dynamical variable. The calculation of correlation dimensions is used to illustrate these procedures. The Kaplan–Yorke conjecture links the (average) Lyanpunov exponents of a system to the dimension of its state space attractor. Sets of unstable periodic orbits can be used to characterize chaotic behaviour. Connections with partition functions in statistical mechanics, wavelet analysis and q-calculus are introduced.
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