Formal Tools for Exploring Complexity

Much of the discussion in this paper is concerned with clarifying the concepts of order, disorder, and complexity, and the formulation of their corresponding measures for a complex system. We begin with a review of the concept of entropy because this was the first measure of order in physical systems and understanding the evolution of this concept will also make clearer the evolution of our understanding of complexity. Thus, we start from thermodynamic entropy, along with the ancillary notion of equilibrium, and end with the dynamics of chaos and fractional dimensions. In particular, we use the normalized entropy explicitly to define parameters for order, disorder and complexity. In addition we find that the generalized fractional dimension of a complex dynamical process is intimately related to one of the modern concepts of entropy used as a measure of complexity. We conclude that the generalized fractional dimension and the approximate entropy are two measures of complexity that have demonstrated their value in providing quantitative measures of the complexity of dynamical systems.