Abstract
Abstract. Earthquake sequencing studies allow us to investigate empirical relationships among spatio-temporal parameters describing the complexity of earthquake properties. We have recently studied the relevance of Markov chain models to draw information from global earthquake catalogues. In these studies, we considered directed graphs as graph theoretic representations of the Markov chain model and analyzed their properties. Here, we look at earthquake sequencing itself as a directed graph. In general, earthquakes are occurrences resulting from significant stress interactions among faults. As a result, stress-field fluctuations evolve continuously. We propose that they are akin to the dynamics of the collective behavior of weakly coupled non-linear oscillators. Since mapping of global stress-field fluctuations in real time at all scales is an impossible task, we consider an earthquake zone as a proxy for a collection of weakly coupled oscillators, the dynamics of which would be appropriate for the ubiquitous Kuramoto model. In the present work, we apply the Kuramoto model with phase lag to the non-linear dynamics on a directed graph of a sequence of earthquakes. For directed graphs with certain properties, the Kuramoto model yields synchronization, and inclusion of non-local effects evokes the occurrence of chimera states or the co-existence of synchronous and asynchronous behavior of oscillators. In this paper, we show how we build the directed graphs derived from global seismicity data. Then, we present conditions under which chimera states could occur and, subsequently, point out the role of the Kuramoto model in understanding the evolution of synchronous and asynchronous regions. We surmise that one implication of the emergence of chimera states will lead to investigation of the present and other mathematical models in detail to generate global chimera-state maps similar to global seismicity maps for earthquake forecasting studies.
Highlights
Earthquakes of differing magnitudes occur at different locations and depths in many tectonically active regions of the earth
We investigate the Kuramoto model with a phase lag for the sequencing of global earthquake data influenced by the recurrences to point out the emergence of chimera states under certain conditions
As a precursor to studying earthquake sequencing with real data from the earthquake catalogues, we investigated the Kuramoto model on synthetic networks that mimic Erdös–Rényi random networks, small-world networks, and scale-free networks and directed graphs adapted from them, and examined chimera-state solutions (Vasudevan and Cavers, 2014a)
Summary
Earthquakes of differing magnitudes occur at different locations and depths in many tectonically active regions of the earth. Continuous recording and analysis of earthquakes that occur in different regions of the earth have led to earthquake catalogues. These catalogues carry information about the epicenter and the estimated hypocenter, the time and the magnitude of the earthquakes, leading to a set of empirical rules for different earthquake regions and the global seismicity (Omori, 1895; Gutenberg and Richter, 1954; Bath, 1965; Bufe and Varnes, 1993; Utsu et al, 1995; Ogata, 2011). The empirical rules allow us to understand and expand on the inter-relationships between the earthquake magnitude and the frequency of occurrence of events, and the main shocks and their aftershocks in space and in time
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