No abstract

SUMMARY In analyses of phenomena leading to fractal distributions there prevail opinions that the causes of such distributions are critical phenomena in phase transitions or SOC models. Power distributions are also associated with deterministic chaos generated by low‐dimensional non‐linear systems. Non‐linearity is considered to be a necessary condition for scale invariance and fractal statistics. However, an explanation of what kinds of non‐linearities lead to fractal distributions and how they do it is lacking. In this contribution, the influence of the non‐linearity of a model on the output behaviour is investigated. In many cases the relation between the ‘input’x and the ‘output’y of a system can be expressed by a non‐linear transformation y=g(x). The kind of non‐linearity of a model that transforms an input random variable with exponential distribution into a variable with a long‐tail distribution over a wide range of scales is analysed. A very wide class of non‐linearly increasing functions g leads to power‐like output distributions. Non‐linear relations of the type y=g(x) can be solutions of random differential equations or of a system of random differential equations that describe some physical phenomena. Various kinds of random differential equations are analysed and discussed. The contribution shows in an elementary way the causes of universality of fractal distributions in many branches of science. A simple geophysical example of crack tip propagation is included.

Earthquakes are the result of fracture of the earth's material due to tectonic stress.The fracture of solids is preceded by the appearance of numerous microcracks which can propagate and coalesce. The state of the crack system is described by the size distribution function which satisfies an integro-differential kinetic equation. Under two different assumptions concerning the fusion cross-section, exponential and inverse-power solutions are deduced. The exponential form of the crack size distribution is observed in many damage experiments for metals but the inverse-power law is typical for more brittle materials like rocks. A second way of analysis of the evolution of crack populations is based on the division of all cracks into n groups, then a set of n nonlinear differential equations is obtained. This model is more handy for numerical calculations and was used for long-term evolution of the crack system.

We introduce a new method of characterizing the seismic complex systems using a procedure of transformation from complex networks into time series. The undirected complex network is constructed from seismic hypocenters data. Network nodes are marked by their connectivity. The walk on the graph following the time of succeeding seismic events generates the connectivity time series which contains, both the space and time, features of seismic processes. This procedure was applied to four seismic data sets registered in Chile. It was shown that multifractality of constructed connectivity time series changes due to the particular geophysics characteristics of the seismic zones-it decreases with the occurrence of large earthquakes-and shows the spatiotemporal organization of these seismic systems.