Potential of Non-Linear Dynamics Methods for Analysis of Geophysical Series and Seismicity

   Recent studies show that earthquakes display the signs of dynamically
   complex systems. Recently the set of traditional (linear) research methods
   has been greatly expanded by non-linear methods obtained from the theory
   of non-linear dynamics and chaos; many researches are aimed at evaluation
   of non-linear characteristics and properties of natural and artificial
   systems. To evaluate some characteristics it is necessary to restore the
   system's phase dynamics, while for other characteristics it is not
   necessary. To evaluate the deterministic component of stochastic process
   in test examples and in real seismological data it is a good practice to
   use the methods of IFS dumpiness test and RP- recurrence plots. The
   system's dynamics has also been reconstructed. It has been shown that
   during experimental actions of current pulses there appears a
   deterministic component in the seismic mode of Northern Tien Shan. The
   Gutenberg-Richter law is traditionally used for analysis of seismicity of
   any given region. To describe these systems it was proposed to use the
   generalization of power-series distribution, the so-called Tsallis
   statistics. This generalization includes the limit cases and the
   Gutenberg-Richter law, as well as the classic statistics of Boltzmann for
   the system of noninteracting elements. In particular, the catalogue-based
   distributions are well approximated by the Tsallis distribution function
   which may indicate the regularity of the weakest events flow. From the
   viewpoint of the Tsallis non-extensive statistics, deviation from the
   Gutenberg-Richter law for weak events does not necessarily indicate the
   omission of some events by seismic network.