Fractional dynamics and MDS visualization of earthquake phenomena

Abstract

This paper analyses earthquake data in the perspective of dynamical systems and frac-tional calculus (FC). This new standpoint uses Multidimensional Scaling (MDS) as a power-ful clustering and visualization tool. FC extends the concepts of integrals and derivatives to non-integer and complex orders. MDS is a technique that produces spatial or geometric representations of complex objects, such that those objects that are perceived to be similar in some sense are placed on the MDS maps forming clusters. In this study, over three million seismic occurrences, covering the period from January 1, 1904 up to March 14, 2012 are analysed. The events are characterized by their magnitude and spatiotemporal distri-butions and are divided into fifty groups, according to the Flinn-Engdahl (F-E) seismic regions of Earth. Several correlation indices are proposed to quantify the similarities among regions. MDS maps are proven as an intuitive and useful visual representation of the com-plex relationships that are present among seismic events, which may not be perceived on traditional geographic maps. Therefore, MDS constitutes a valid alternative to classic visu-alization tools for understanding the global behaviour of earthquakes.

Analysis of global terrorism dynamics by means of entropy and state space portrait

“…FC models are able to capture long-range phenomena, often overlooked by standard differential equations, being an interesting tool for analyzing phenomena that occur in many dynamical systems with complex behavior [6,25,31,32,37,38,42,43,60].…”

Approximate Analytical Solutions of Time Fractional Whitham–Broer–Kaup Equations by a Residual Power Series Method

“…In the past few decades, fractional calculus theory has played an important role in the fields of fluid mechanics, physics, entropy and engineering [2][3][4][5]. By using fractional calculus, some physical models and engineering processes can be described more reasonably and applicably.…”

Integer and fractional-order entropy analysis of earthquake data series

“…Such behavior increases stress, while strain energy accumulates around the fault surfaces. Whenever the stress is high enough to break through the asperities, the accumulated energy is released and the plates move suddenly causing an earthquake [31].…”

“…A common property to CS is the absence of a characteristic length-scale, meaning that CS reveals frequency-size power law (PL) behavior [7,31,34,39]. PL distributions have been associated with systems with memory, as is the case of fractional-order systems [4,22].…”

“…The GR law can also be stated in terms of energy released by earthquakes. In this case the number of earthquakes with energy greater than ε, N (ε), behaves as a PL, with critical exponent τ ∈ [0.8; 1.05] [31,47], while:…”