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.

Earthquakes of differing magnitudes occur at different locations and depths
in many tectonically active regions of the earth. The magnitude is the most
widely used and theoretically studied earthquake parameter (Kanamori and
Anderson, 1975; Hanks and Kanamori, 1979). The moment magnitude scale,

The earthquake catalogues have recently become the basis for Markov chain models of earthquake sequencing to explore probabilistic forecasting from the point of view of seismic hazard analysis (Nava et al., 2005; Cavers and Vasudevan, 2015). Cavers and Vasudevan (2015) have incorporated the spatio-temporal complexity of the earthquake recurrences (Davidsen et al., 2008; Vasudevan et al., 2010) into their Markov chain model.

Intrinsic to earthquake sequencing studies is the observation made on scaling behavior and earthquake cycles (Turcotte, 1997; Rundle et al., 2002, 2003). In this regard, fractal and fractal-rate stochastic point processes were found to be useful (Thurner et al., 1997). Telesca et al. (2011) applied such models to earthquake sequencing. Vasudevan and Cavers (2013) have recently extended the application of this model to study time-correlative behavior in earthquake sequencing by carrying out Fano factor and Allan factor analysis of a time series of state-to-state transition frequencies of a Markov chain.

One aspect of earthquake sequencing that requires a close look is a model for the non-linear dynamics of earthquakes. In this paper, we investigate the synchronization behavior of weakly coupled “earthquake oscillations”. Such oscillations in the earth's crust and the epileptic brain show certain commonalities in that the distributions of energies and recurrence times exhibit similar power-law behavior (Herz and Hopfield, 1995; Rundle et al., 2003; Osorio et al., 2010; Chialvo, 2010). A growing interest in understanding the behavior of earthquakes and epileptic seizures with a view to exploring possible forecasting methods is one reason for the present study. In the case of epileptic seizures, the non-linear dynamics of pulse-coupled neuronal oscillations as an alternative to the Kuramoto (1975) model are under close scrutiny (Rothkegel and Lehnertz, 2014). To our knowledge, neither a simple Kuramoto model nor a modification of it has been worked out for earthquake sequencing studies. Mirollo and Strogatz (1990), Kuramoto (1991) and Rothkegel and Lehnertz (2014) considered the synchronization of pulse-coupled oscillators in which single oscillators release energy rapidly when they reach a trigger threshold and become quiescent for some time until they reach the trigger threshold again. Examples falling into this category are earthquakes and spiking neuronal activities (Herz and Hopfield, 1995; Beggs and Plenz, 2003; Rundle et al., 2002, 2003; Scholz, 2010; Karsai et al., 2012; Rothkegel and Lehnertz, 2014). Herz and Hopfield (1995) studied the collective oscillations with pulse-coupled threshold elements on a fault system to capture the earthquake processes. There are two timescales: the first is given by the fault dynamics defining the duration of the earthquake, and the second timescale is given by the recurrence time between “characteristic events”, the largest earthquakes on a fault. The known recurrence times on several fault systems are 6 to 8 orders of magnitude longer than the duration of single events. Rundle et al. (2002) examined the self-organization in “leaky” threshold systems such as networks of earthquake faults. In their paper, they argued that on the “macroscopic” scale of regional earthquake fault systems, self-organization leads to the appearance of phase dynamics and a state vector whose rotations would characterize the evolution of earthquake activity in the system. Scholz (2010) invoked the Kuramoto model to represent the fault interactions, although no numerical synchronization–simulation results were presented. He postulated that the common occurrence of triggering of a large earthquake by other earthquakes on nearby faults and the observation of space–time clustering of large earthquakes in the paleoseismic record were both indicators of synchronization occurring between faults. However, we need to bear in mind here that incorporating fault–fault interactions on a global scale involving all the networks of earthquake faults is formidable and nearly impossible. In this paper, we modify the simple non-linear mathematical model, the Kuramoto model with a phase lag, for the sequencing of global earthquake data. We show here that the solutions to the Kuramoto model with phase lag and with non-local coupling effects reveal the co-existence of synchronized and asynchronized states or chimera states for certain parameter values. We use this model as a precursor to our planned studies on other mathematical models such as integrate and fire models.

As alluded to earlier, there is a quiescence period between earthquakes in an earthquake zone, also known as the recurrence times. Since the globally recorded earthquake data are only available for a short time period, incorporating the recurrence times into the earthquake catalogue is impossible. Here, we consider the model proposed by Davidsen et al. (2008) to include the spatio-temporal complexity of recurrences by identifying the earthquakes occurring in close proximity to any occurred event in the record-breaking sequence. In this paper, we also 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.

The Kuramoto (1975) model for a large number of weakly coupled oscillators
has become a standard template in non-linear dynamical studies, pertinent to
synchronization behavior, following the ground-breaking study of
Winfree (1967). To apply this model to earthquake sequencing studies, we need
to make a few justifiable assumptions and to incorporate certain essential
features of earthquakes that we have come to know. For example, plate motions
and, hence, plate tectonics (Stein, 1993; Kagan et al., 2010; DeMets et al.,
2010) suggest that most of the earthquakes occur in and around plate
boundaries because of the varying plate motions of the plates that uniquely
encompass the earth's crust. In particular, different plates move at
different rates and along different orientations, resulting in stress-field
changes at the plate boundaries. When stress-field accumulation reaches, at a
particular location or in a zone, a certain critical threshold, energy is
released in the form of an earthquake. The relaxed system goes through the
stress-build-up process again, a similar mechanism being operative in
neuronal communication dynamics. We assume that there is a uniform stress
increase during the quiescent period. Collective synchronization of
threshold-coupled or pulse-coupled oscillators would be a candidate for such
a study (Mirollo and Strogatz, 1990; Kuramoto, 1991; Rothkegel and Lehnertz,
2014). However, we defer the extension of their approach to earthquake
sequencing studies to a future date. Since the quiescence period is 6 to
8 orders of magnitude longer than the event time duration, it would be an
ideal platform on which to carry out this study. We surmise that the behavior
of earthquake cycles noted in earthquake sequencing does not lend support,
however, to a full synchronization or full asynchronization as a solution to
this non-dynamics problem. One proven modification is the inclusion of
non-local effects of the geometry of the system that has been shown to lead
to a co-existence of partially synchronized and partially asynchronized
states of oscillators as a steady-state solution. Such states, addressed as
chimera states, are the subject of recent theoretical and experimental
studies (Kuramoto and Battogtokh, 2002; Abrams and Strogatz, 2004; Abrams et
al., 2008; Ko and Ermentrout, 2008; Omel'chenko et al., 2008; Sethia et al.,
2008; Sheeba et al., 2009; Laing, 2009a, b; Laing et al., 2012; Martens et
al., 2013; Yao et al., 2013; Rothkegel and Lehnertz, 2014; Kapitaniak et al.,
2014; Pazó and Montbrió, 2014; Panaggio and Abrams, 2014; Zhu et al.,
2014; Gupta et al., 2014; Vasudevan and Cavers, 2014a, b). We focus our
present study on defining a Kuramoto model with a phase lag that would
accommodate the existence of chimera states. The Kuramoto model has been
extensively studied for a system made up of a large number of weakly coupled
oscillators, where most of the physical problems are finite and can be
described as non-linear dynamics on complex networks (Acebrón et al.,
2005; Arenas et al., 2008). In the realm of graph theory, complex
networks can be cast as either undirected or directed graphs. In our studies
on earthquake sequencing, we consider a directed graph as a representation of
an earthquake complex network. The occurrence of chimera states as solutions
to non-linear dynamics on both undirected and directed graphs has recently
been investigated (Zhu et al., 2014; Vasudevan and Cavers, 2014a). 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). For the
earthquake sequencing studies here, we use the following Kuramoto model with
a phase lag,

We would like to stress that the model in Eq. (1) is not a pulse-coupled or threshold-coupled oscillator model. Although it would be appropriate to consider a variation of the Kuramoto model such as the Shinomoto–Kuramoto model (Shinomoto and Kuramoto, 1986; Sakaguchi et al., 1988; Lindner et al., 2004), we limit ourselves to a simpler model that does not include the excitable behavior of the model. We intend to use this model as a precursor to our planned studies on other mathematical models such as integrate and fire models.

A comment on the phase-lag parameter,

Partitioning of the global seismicity map:

Here, we construct a directed graph of earthquake events from the
Incorporated Institutions for Seismology (IRIS) earthquake catalogue for the
time period between 1970 and 2014. We consider earthquake events with
magnitudes exceeding or equal to

Grid sizes and the number of oscillators corresponding to non-zero cells.

We report the Kuramoto model experimental results for oscillators resulting
from 128

To represent the results, we use snapshots of three attributes (Zhu et
al., 2014): (i) the phase profile, (ii) the effective angular
velocities of oscillators and (iii) the fluctuation of the instantaneous
angular velocity of oscillators. The effective angular velocity of
oscillator

128

A 1024

The fluctuation of the instantaneous angular velocity,

Whether or not the Kuramoto model reaches the steady state, we examine the
ratio of the number of coherent or synchronous oscillators to the total
number of oscillators or “chimera index” as a function of the number of
time steps. Here, we carry out 200 000 time steps. After every 20 000 time
steps, we look at the chimera index for the last 1000 time steps. As an
example, in Fig. 8, we find the asymptotic behavior of the scatter of the
chimera index for ten such intervals for the 128

We investigate the influence of the non-local coupling coefficient,

Most of the initial computations reported in this work were on a HP C7000 chassis cluster system with dual-core 2.4 GHz AMD Opetron processors at the high-performance computing facility at the University of Calgary. We carried out a series of runs for 200 000 time steps on a Mac Pro Six-Core Intel Xeon E5 3.5 GHz, 16 GB RAM desktop work station and on a Dell PowerEdge R910 with Intel Xeon E7-4870 2.40 GHz 256 GB RAM processors, and we used the Matlab ODE113 solver to solve the Kuramoto model.

Earthquake sequencing is a well-studied problem in earthquake seismological
communities around the globe, and yet it hides a suite of phenomenological
mysteries that stand in the way of successful earthquake forecasting. One of
the first steps in carrying out any investigative work on earthquake
sequencing is to look at the global seismicity map such as the one posted by
IRIS on a regular basis, with continuous updating of the associated
catalogue. In Fig. 1a and b, we summarize the cumulative results of the
catalogue for magnitudes of earthquakes exceeding

Rundle et al. (2003) took a statistical physics approach in emphasizing the significance of faults and fault systems as high-dimensional non-linear dynamical systems characterized by a wide range of scales in both space and time, from centimeters to thousands of kilometers, and from seconds to many thousands of years. The signature of the residual behavior in these systems is chaotic and complex. Understanding the coupling between different space scales and timescales to comprehend the non-linear dynamics of the fault systems is not an easy problem. In this regard, any attempt to explore the possibilities that accrue from non-linear dynamics studies is welcome.

In earlier studies on model and theoretical seismicity (Burridge and Knopoff, 1967; Vieira, 1999), special attention was paid to finding out whether chaos was present in the symmetric non-linear two-block Burridge–Knopff model for earthquakes. Vieira (1999) demonstrated with a three-block system the appearance of synchronized chaos. A consequence of this study was the speculation that earthquake faults, which are generally coupled through the elastic media in the earth's crust, could in principle synchronize even when they have an irregular chaotic dynamics (Vieira, 1999). Going one step further would be to suggest that the occurrence of earthquakes and the space–timescale patterns they leave behind is a sound proxy for modeling and theoretical studies of the fault systems. It is this point that is pursued in this work.

Three attributes of a chimera state of the 1693
oscillators for a 128

Three attributes of a chimera state of the 1693
oscillators for a 128

Three attributes of a chimera state of the
7697 oscillators for a 1024

In this study, we focus on the non-linear dynamics of weakly coupled
oscillators. Each oscillator (corresponding to the occurrence of an
earthquake) is a proxy for a fault system or network with known information
on its location, the time when the earthquake event occurred, and magnitude.
This defines an element in the earthquake sequence. A continued sequence of
events is represented as a directed graph (Vasudevan et al., 2010; Cavers and
Vasudevan, 2014; Vasudevan and Cavers, 2014b) with the vertices representing
the earthquakes (and their attributes) and the arcs the connecting links
between neighbors in a sequence. Figures 2a and 3a show the transition
matrices for the directed graphs of the two grids, 128

Three attributes of a chimera state of the
7697 oscillators for a 1024

As mentioned earlier, the quiescence period between earthquakes in an earthquake zone is what we interpret here as a recurrence period. Studies on plate-boundary motions (Bird, 2003; DeMets et al., 2010; Stein, 1993) will provide an insight into the recurrence period for earthquakes in certain major fault zones. Even in instances where knowledge of the recurrence periods is known, it is usually punctuated by random fluctuations, the statistics of which are not unknown. The quiescence period is analogous to the process in human brains that precedes epileptic seizures (Berg et al., 2006; Rothkegel and Lehnertz, 2014), the structure of which has been modeled using pulse-coupled phase oscillators. Such pulse coupling or threshold coupling remains to be quantified for earthquakes. We defer this aspect of the work to future studies. Furthermore, the historical seismicity data set is short and, therefore, any information to be drawn from global records will be insufficient. However, the recurrence model introduced by Davidsen et al. (2008) offers a simple remedy to the problem identifying the earthquakes occurring in close proximity to any occurred event in the record-breaking sequence. Incorporating this feature into the transition matrices results in modified transition matrices, as shown in Figs. 2b and 3b. We propose that accounting for the quiescence period in this manner opens additional options such as feedback effects on the non-linear dynamics of weakly coupled oscillators.

Scholz (2010) argued for the role of synchronization in fault interactions
and earthquake clustering and for the usefulness of the Kuramoto model.
Kuramoto (1975) proposed a mathematical model of phase oscillators
interacting at arbitrary intrinsic frequencies and coupled through a sine of
their phase differences. He suggested the following equations for each
oscillator in the system:

For the stability of the solution from the Kuramoto model, use of a large population of oscillators for calculability in the thermodynamic limit is a pre-requisite. Over the last decade, efforts have gone into considering a finite number of oscillators satisfying the original conditions of the Kuramoto model. Easing the restrictions on the interaction model can be cast as an investigation of synchronization on complex networks. This would allow one to relate the complex topology and the heterogeneity of the network to the synchronization behavior.

We rewrite the original Kuramoto model for a complex network corresponding
to undirected and directed graphs as

While the synchronization and asynchronization studies on earthquake
sequencing are important in terms of the Kuramoto model given in Eq. (9),
very little attention has been paid to the co-existence of synchronized and
asynchronized states or the chimera states. Kuramoto and Battogtokh (2002)
and Abrams and Strogatz (2004) paved the way for such a study by including
the non-local effects in the Kuramoto model, as expressed in Eqs. (1) and
(2). Non-local effects mean simply the inclusion of geometry effects. For the
global seismicity map considered in this study, we generated the
shortest-path distance matrices with and without the inclusion of recurrences
(Figs. 2c, 3c, 2d, and 3d). The shortest-path algorithm encapsulates both the
cascading effects of earthquakes and the negation of long-range distance
effects. In this study, we kept the global coupling strength constant and
allowed the non-local coupling strength,

Symmetry-breaking phenomena like chimera states have also been observed for two-cluster networks of oscillators with a Lorentizian frequency distribution (Montbrió et al., 2004) for all values of time delay. A crucial result by Laing (2009a, 2009b) extends the previous observation to oscillators with heterogeneous frequencies. Also interesting to observe in this regard is that these heterogeneities can lead to new bifurcations, allowing for alternating synchrony between the distinct populations over time. Ko and Ermentrout (2008) demonstrated the presence of chimera-like states when the coupling strengths were heterogeneous. The last study used coupled Morris–Lecar oscillators. Although there is overwhelming evidence for the existence of chimera states in the presence of time delay or phase lag, all of our initial Kuramoto model simulations on the directed graph transition matrices and the associated shortest-path distance matrices included a constant phase lag only.

A postulation for the existence of evolving chimera states in data from earthquake catalogues has certain implications. For instance, it would pave the way to understanding the evolving alterations in stress-field fluctuations in fault zones frequented by earthquakes. Also, it would suggest a need to consider steps to quantify partially or fully the ratio of the number of synchronized oscillators to the total number of oscillators. The steps would involve extensive testing of the dependence of the parameters and additional mathematical models. We interpret the zones with synchronized oscillators as the ones being susceptible to earthquakes and the zones with asynchronized oscillators as the ones going through a quiescence period. The hope is that confirmation of chimera states in earthquake sequencing would signal a possible use for earthquake forecasting studies.

The Kuramoto model simulation with non-local coupling effects
(

Chimera index as a function of time steps for the 128

Going from the 128

We looked at the influence of the non-local coupling coefficient,

The outcome of each one of the simulations described for both the
non-recurrence and recurrence cases contains synchronous and asynchronous
vectors. Mapping these vectors on the respective grids (128

Influence of the non-local coupling coefficient parameter,

Chimera-state map of the synchronous and asynchronous oscillators as
a steady-state solution for a

Earthquake sequencing is an intriguing research topic. The dynamics involved
in the evolution of earthquake sequencing are complex. Very much has been
understood, and yet the evolving picture is incomplete. In this regard, the
work of Scholz (2010) acted as a catalyst in us investigating the
synchronization aspect of earthquakes using the Kuramoto model. To name a
few, the works of Vieira (1999), Rundle et al. (2002, 2003), Kuramoto and
Battogtokh (2002), Abrams and Strogatz (2004), and Laing (2009a, b) have
helped us take this step forward with this work. We summarize below the main
points of this paper and also point out the direction in which we are going:

Earthquake sequencing from the IRIS earthquake catalogue browser can be
expressed as a transition matrix of a directed graph. Partitioning of the
latitude–longitude grid of the globe into grids of finite dimensions such as
128

Inclusion of the non-local effects in the Kuramoto model of the directed
graphs is tested for different values of the non-local coupling coefficient,

For a non-local coupling strength,

As the non-local coupling strength,

As the non-local coupling strength is increased from 0.1 to 1.0, there is a steady increase in the asynchronous behavior.

The recurrence results support the presence of chimera states for both
128

There is still a nagging question about which non-local coupling coefficient would be an ideal candidate for understanding the global stress-field fluctuations.

In general, the hypothesis that all networks of earthquake faults around the globe go through full synchronization still needs to be strongly tested. On the other hand, the prevalence of chimera states or multi-chimera states is an attractive option to understand the earthquake sequencing.

We believe that there is, now, a mechanism available to us to explore and seek an answer to the non-linear dynamics of earthquake oscillations.

We would like to thank two anonymous referees for their constructive criticisms and helpful suggestions that helped us improve the initial manuscript. We would like to express deep gratitude to the Department of Mathematics and Statistics for support and computing time and to the generosity and hospitality shown at the Max Planck Institute for the Physics of Complex Systems at Dresden, Germany, during K. Vasudevan's short visit to the institute last summer. We thank Incorporated Institutions for Seismology (IRIS) for the information on the global earthquake catalogue and the high-performance computing at the University of Calgary. Edited by: J. Kurths Reviewed by: two anonymous referees