Colloquium: Fractional calculus view of complexity: A tutorial

The fractional calculus has been part of the mathematics and science literature for 310 years. However, it is only in the past decade or so that it has drawn the attention of mainstream science as a way to describe the dynamics of complex phenomena with long-term memory, spatial heterogeneity, along with nonstationary and nonergodic statistics. The most recent application encompasses complex networks, which require new ways of thinking about the world. Part of the new cognition is provided by the fractional calculus description of temporal and topological complexity. Consequently, this Colloquium is not so much a tutorial on the mathematics of the fractional calculus as it is an exploration of how complex phenomena in the physical, social, and life sciences that have eluded traditional mathematical modeling become less mysterious when certain historical assumptions such as differentiability are discarded and the ordinary calculus is replaced with the fractional calculus. Exemplars considered include the fractional differential equations describing the dynamics of viscoelastic materials, turbulence, foraging, and phase transitions in complex social networks.

Figure 1

In this classic view of the tea party, when the Mad Hatter, White Rabbit, and Door Mouse discuss a fractal dimensional Seripinski gasket, Alice sees a Euclidean cube.

Figure 2

Left: Probability of a given number of neurons involved in a neuronal avalanche. From [67]. Right: The time interval between neuronal spikes. From [55]. Both are inverse power laws. Note that such “laws” are valid over a limited domain and the mechanism by which they are truncated is often of interest.

Figure 3

The curve is a graph of the GWF using the series Eq. (1). The function depicted by the curve obeys a scaling relation (2) characterized by a fractional dimension.

Figure 4

The smoke plume from chimneys is seen to expand with distance from the source. [86] speculated that this could be described by a Weierstrass function since the turbulent velocity field of the wind was not differentiable.

Figure 5

Cartoon of a two-dimensional random walk or drunkard’s walk. From [38].

Figure 6

Data from stress relaxation experiments using polyisobutylene in units of $\text{dynes}/{\mathrm{mm}}^2$ at constant strain for two different initial conditions are indicated by the discrete data points. Upper: The solid curve is the MLF with index $\alpha =0.60$. The short-dashed curve is the inverse power law, and the long-dashed curve is the stretched exponential. Lower: The dashed curve is the MLF and the boxes are data. The two fits use the MLF with different parameter values. Adapted from [35].

Figure 7

The dashed curves are the exponentials for the average opinion of an isolated individual. The dotted curves are the fits of the MLF to the erratic network dynamics calculated using the DMM for a network of ${10}^4$ elements. Left: Subcritical; middle: critical; right: supercritical. From [115].

Figure 8

Measured probability density of changes of the wind speed over 4 s $P(\delta u_{\tau }/\sigma )$, obtained from a wind measurement at the German North Sea coastline. The solid curve corresponds to a Gaussian distribution having the same mean and variance as the observational data. The measurements of the velocity deviations from the mean $\delta u_{\tau }$ are normalized to a standard deviation of $\sigma =0.8\mathrm{m}/\mathrm{s}$ and the data are plotted on a log-linear graph. From [16].

Figure 9

[45] pointed out the scale-invariant nature of the GPS foraging track of the albatross with the increased resolution of the track from left to right.