Complexity and the Fractional Calculus

We study complex processes whose evolution in time rests on the occurrence of a large and random number of events. The mean time interval between two consecutive critical events is infinite, thereby violating the ergodic condition and activating at the same time a stochastic central limit theorem that supports the hypothesis that the Mittag-Leffler function is a universal property of nature. The time evolution of these complex systems is properly generated by means of fractional differential equations, thus leading to the interpretation of fractional trajectories as the average over many random trajectories each of which satisfies the stochastic central limit theorem and the condition for the Mittag-Leffler universality. 1. Introduction The fractional calculus has developed in a number of significant ways in the recent past. Sokolov et al. [1] maintain that this calculus was restricted to the field of mathematics until the last decade of the twentieth century, when it became very popular among physicists as a powerful way to describe the dynamics of a variety of complex physical phenomena. For example, anomalous diffusion was described using fractional diffusion equations [2, 3]; viscoelastic materials were modeled using fractional Langevin equations [4]; and complex dynamic systems could be governed using fractional control [5]. In the last decade the concept of fractional dynamics has gained further attention in the statistical and chemical physics communities [6]. Fractional differential equations have also been successfully applied to neural dynamics [7, 8] and ecology [9] as well as to traditional fields of engineering [10, 11] namely [12, 13]. Of particular interest to the authors is the growing literature on extending systems of nonlinear dynamic equations having strange attractor solutions to fractional nonlinear equations. Such extensions were typically made by replacing integer-valued derivatives by fractional derivatives; for example, in the Lorenz system [14–16], in the chaotic rigid body motion of gyros [17], in Hopfield-type neural networks [8], and in the immune model of HIV infection [18], to name a few. These replacements were made in attempts to incorporate dynamic mechanisms thought to be important that could not be captured by the traditional models, for example, complexity in the form of memory in time and non-locality in space. The results of extending these nonlinear models has been to apparently introduce dissipation into the dynamics such that the solution on the strange attractor collapses to that of a stable fixed point. The