NPGNonlinear Processes in GeophysicsNPGNonlin. Processes Geophys.1607-7946Copernicus GmbHGöttingen, Germany10.5194/npg-22-499-2015Earthquake sequencing: chimera states with Kuramoto model dynamics on directed graphsVasudevanK.vasudeva@ucalgary.caCaversM.WareA.Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, T2N 1N4, CanadaK. Vasudevan (vasudeva@ucalgary.ca)8September201522549951226January201520February201531May201511August2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://npg.copernicus.org/articles/22/499/2015/npg-22-499-2015.htmlThe full text article is available as a PDF file from https://npg.copernicus.org/articles/22/499/2015/npg-22-499-2015.pdf
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.
Introduction
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,
MW, provides an estimate for all medium to large earthquake
magnitudes. 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.
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.
Mathematical model of the earthquake sequencing
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, α, with non-local coupling effects terms added
explicitly:
θ˙i=ωi-1N∑j=1NGijsinθi-θj+α.
Here, θ˙i is the time derivative of the phase of the
ith oscillator. The angle α (0 ≤α≤π/2)
corresponds to the phase lag between oscillators i and j. Gij is the
non-local coupling function that depends on the shortest path length,
dij, between oscillators i and j in the complex network:
Gij=Ke-κdij.K is the global coupling strength and κ is the strength of the
non-local coupling. For convenience, we use a constant natural frequency for
all the oscillators, i.e. homogeneous case, and, thus, we could use
ωi= 0 for i= 1, …, N. Although we have not
investigated the influence of the global coupling strength on the
steady-state solution of the Kuramoto model, we treat this term as constant,
in particular K= 1, based on observations made by Zhu et al. (2014).
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, α, in Eq. (1) is also in order. Panaggio and
Abrams (2014) interpret the phase lag as an approximation for a time-delayed
coupling when the delay is small. The value of α used is
(π/2) - 0.10. In some ways, we treat the phase lag as a proxy for
time delay. As Panaggio and Abrams (2014) demonstrate in their paper, the
value of α determines a balance between order and disorder. We have
not done an exhaustive search on the α parameter for the cases
discussed in this paper.
Partitioning of the global seismicity map:
(a) 128 × 128 gridding of the latitude–longitude map.
(b) 1024 × 1024 gridding of the latitude–longitude map.
Earthquakes of magnitudes exceeding or equal to Mw= 5.5 and
location depth not exceeding 70 km for the time period from January 1970 to
September 2014 constitute the glacial seismicity map. Earthquake information
was downloaded from IRIS (Incorporated Research Institutions for Seismology).
The earthquake frequency used in the maps is plotted on a log(log) display
scale, with larger circles representing higher frequencies.
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 MW= 5.5 observed to a depth
of 70 km. We partition the general latitude–longitude map of the earthquake
events into a grid. We show two maps of such grid matrices (Fig. 1). A cell
in a smaller grid (128 × 128) could have higher multiplicity of
earthquake events than that in a large grid (1024 × 1024). We
consider the coordinates of the topological center of each cell to represent
the coordinates of the earthquake events that fall into that cell. Thus, we
explore the effect of hubs and community effects by looking at transition
probability matrices generated from grids of different orders such as 128,
192, 256, 512, and 1024 representing the seismicity map on a global
longitude–latitude grid (Table 1). We compute the transition probability matrix and
the shortest-path distance matrix for the directed graphs resulting from the
catalogue considered. To keep the Kuramoto model simple, we assume a constant
phase lag, α, in the phase of the ensemble of oscillators. The value
of α used is (π/2) - 0.10. We relax this condition in
subsequent simulations. The most difficult parameter to deal with here is the
period of quiescence after the energy release following a certain stress
threshold. We incorporate the build-up of the threshold effect indirectly by
positing the inclusion of earthquake recurrences in transition probability
matrices. Here, we use the spatio-temporal recurrences based on the
record-breaking model of Davidsen et al. (2008). In all our initial
simulations, we ignore the influence of amplitude effects on the stability of
the chimera states. We carry out simulations on the Kuramoto model for
200 000 time steps for the 128 × 128 oscillator grid matrices and
for the 1024 × 1024 oscillator grid matrices. We report here the
preliminary results of our simulations.
Grid sizes and the number of oscillators corresponding to non-zero cells.
We report the Kuramoto model experimental results for oscillators resulting
from 128 × 128, 192 × 192, 256 × 256,
512 × 512, and 1024 × 1024 grids of the latitude–longitude
map of the earthquakes. We consider a total of 13 190 earthquakes. We
construct the transition probability and the shortest-path distance matrices
for the grids without (“non-recurrence” results) and with the consideration
of the spatio-temporally complex recurrences (“recurrence” results), as
shown in Figs. 2 and 3.
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 i is defined as
〈ωi〉=limτ→∞1T∫t0t0+Tθ˙idt.
Here, we take T= 1000 so that the effective angular velocities of the
oscillators are averaged over the last 1000 time steps. We take
t0= 199 001 for the 128 × 128 grid and for the
1024 × 1024 grid.
128 × 128 gridded map: (a) transition probability matrix without
recurrences. (b) Transition probability matrix with recurrences.
(c) Shortest-path distance matrix without recurrences. (d) Shortest-path distance
matrix with recurrences. In (a) and (b), the transition frequencies used in
the maps are plotted using a log(log) display scale, with larger circles
representing higher frequencies.
A 1024 × 1024 gridded map: (a) transition probability matrix
without recurrences. (b) Transition probability matrix with recurrences.
(c) Shortest-path distance matrix without recurrences. (d) Shortest-path distance
matrix with recurrences. In (a) and (b), the transition frequencies used in
the maps are plotted using a log(log) display scale, with larger circles
representing higher frequencies.
The fluctuation of the instantaneous angular velocity, σi, of an
oscillator i around its effective velocity is defined as
σi2=limT→∞1T∫t0t0+Tθ˙i-〈ωi〉2dt.
If σi= 0, then oscillator i rotates at a constant angular
velocity. We show the non-recurrence and recurrence results obtained from the
behavior of the last 1000 time steps of the simulations involving
200 000 time steps. We present the results for the 128 × 128 grid
without and with recurrences in Figs. 4 and 5 for the three attributes using
κ= 0.10. Figures 6 and 7 show these attributes for the
1024 × 1024 grid without and with recurrences, respectively, for
κ= 0.1.
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 × 128 grid for
κ= 0.10, suggesting that the Kuramoto model has reached the
steady state.
We investigate the influence of the non-local coupling coefficient,
κ, on the chimera index for each grid and summarize our results for
the non-recurrent 128 × 128 and 1024 × 1024 grids in Fig. 9.
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.
DiscussionBuilding the directed graph
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 Mw= 5.5
and the depths of occurrence not exceeding 70 km, recorded between
January 1970 and September 2014. One difference in the two figures lies in
the coarseness of the gridding, with the first one being coarser than the
second. A cursory glance at the figures immediately suggests the relevance of
plate tectonics in that most earthquakes seem to occur at and around plate
boundaries. A broad classification of these earthquakes could consist of the
following categories: strike-slip earthquakes, subduction-zone seismicity,
oceanic earthquakes, continental extensional regimes, intraplate earthquakes,
and slow earthquakes (Scholz, 2002). The interplay between these remains a
topic of research among seismologists. In general, fault systems play an
important role in understanding the cause and recurrence of earthquakes.
Scholz (2002) provides an excellent account of the mechanics of earthquakes
and faulting. Ben-Zion and Sammis (2003) examined the
continuum-Euclidean, granular, and fractal views of the geometrical,
mechanical, and mathematical nature of faults and concluded that many aspects
of the observed spatio-temporal complexity of earthquakes and faults might be
explained using the continuum-Euclidean model. They contended that a
continuum-based description would provide a long-term attractor for
structural evolution of fault zones at all scales. The underpinning point in
these works is the importance of the faulting in earthquake processes.
Earthquakes are known to occur at different depths. Excepting in instances
where there are surface ruptures as a result of earthquakes, fault zones at
seismogenic depths in kilometers cannot be directly observed (Ben-Zion and
Sammis, 2003). Continued geological mapping and high-resolution geophysical
measurements afford a mechanism to improve our understanding of the fault
zones.
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 × 128 gridded map without recurrences using
κ= 0.10. (a) Stationary phase angle. (b) Effective angular
velocity. (c) Fluctuations in instantaneous angular velocity.
Three attributes of a chimera state of the 1693
oscillators for a 128 × 128 gridded map with recurrences using
κ= 0.10. (a) Stationary phase angle. (b) Effective angular
velocity. (c) Fluctuations in instantaneous angular velocity.
Three attributes of a chimera state of the
7697 oscillators for a 1024 × 1024 gridded map without recurrences using
κ= 0.10. (a) Stationary phase angle. (b) Effective angular
velocity. (c) Fluctuations in instantaneous angular velocity.
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 × 128 and
1024 × 1024 grids. The oscillator index is determined by the grid
partition with non-zero cells labelled in row-by-row order. A log(log)
display scale is used to highlight the “clustering”. The level of
clustering along the first leading off-diagonal elements of the transition
matrix is highlighted and indicates the partitioning and the relative
significance of the seismicity zones in the globe. However, this does not
invoke any causality argument. Since the multiplicity of the earthquakes in
the cells of the grids used varies from “zero” to a large number, for the reason mentioned
concerning the Euclidean geometry mentioned earlier, inter-cell and
intra-cell transitions populate the transition matrices. These transition
matrices are not symmetric. The non-linear dynamics of weakly coupled
oscillators on such matrices has not been fully understood.
Three attributes of a chimera state of the
7697 oscillators for a 1024 × 1024 gridded map with recurrences using
κ= 0.10. (a) Stationary phase angle. (b) Effective angular
velocity. (c) Fluctuations in instantaneous angular velocity.
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.
Synchronization
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:
θ˙i=ωi+Ki∑j=1Nsinθj(t)-θi(t)(i=1,…,N),
where θi is the phase of the ith oscillator,
θ˙i(t) is the first derivative of the phase
of the ith oscillator with time, ωi is the natural
frequency of the oscillator, Ki is the strength of coupling of the
ith oscillator to other oscillators and N is the size of the population of the
oscillators. The frequencies ωi are chosen from a uniform
distribution. Kuramoto (1975) demonstrated that synchronization was
accomplished in the case of mean-field coupling with
Ki=KN>0(i=1,…,N)
in the above equation. We can describe the Kuramoto model in a simpler form
by introducing the complex-valued order parameter r(t):
Z=r(t)eiψ(t)=1N∑j=1Neiθj(t),
where Ψ(t) is the average phase and r(t) honours
0 ≤r(t)≤ 1. The expression of the Kuramoto model becomes
θ˙i(t)=ωi+Krsinψ-θi(t)(i=1,…,N).
The collective behavior of all the oscillators is monitored by examining the
time evolution of the order parameter, r (Kuramoto, 1975; Strogatz, 2000;
Pikovsky et al., 2003; Strogatz, 2003). The order parameter
can assume values in the range 0 to 1 including the limits. From this, it is
obvious that each oscillator is connected to the common average phase with
the coupling strength is given by Kr. A value of “0” for r corresponds
to total incoherence, i.e. no phase locking of the phases of the oscillators;
a value of “1” for r corresponds to full coherence, i.e. phase locking
of all the phases of the oscillators. The time evolution of the Kuramoto
model can be monitored either by looking at the polar plots of the phases on
a unit circle (Kuramoto, 1975) or by following the plot of the order
parameter, r, as a function of the coupling strength, K. Acebrón et
al. (2005) have provided a comprehensive review of the Kuramoto model.
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
θ˙i(t)=ωi+∑j=1NKijaijsinθj(t)-θi(t)(i=1,…,N),
where Kij is the coupling strength between pairs of connected
oscillators and aij refers to the elements of the adjacency or
connectivity matrix. Much effort has gone into understanding the role of the
coupling strength (Hong et al., 2002; Arenas et al., 2008; Dörfler et
al., 2013) in the synchronization behavior of small-world and scale-free
graphs. Here, we leave the coupling strength term a constant, unlike in the
model under the thermodynamic limit where the size of the population, N,
enters explicitly in the coupling strength term as a divisor. The structure
of the adjacency matrix decides essentially the nature of the interaction
term made up of the sine coupling of the phases. Vasudevan and Cavers (2014a)
have investigated the synchronization behavior of the random graphs under
different rewiring probabilities and the scale-free graphs from a spectral
graph theory point of view. These studies did not include a study on the
effect of clustering on the synchronization. In this regard, the work of
McGraw and Menzinger (2005) is quite appealing. They conclude that for random
networks and scale-free networks, increased clustering promotes the
synchronization of the most connected nodes (hubs) even though it inhibits
global synchronization. We see the role of the effect of clustering on the
nature of synchronization behavior in earthquake sequencing studies and will
conduct a separate study. Whether or not we reach similar
conclusions for directed graphs, we have recently investigated synthetic
networks that mimic real data structures (Vasudevan and Cavers, 2014a). In
this regard, it is worth mentioning that synchronization of Kuramoto
oscillators in directed networks has been subjected to a detailed study
(Restrepo et al., 2006).
Chimera states
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, κ, to vary from one
simulation to the next one, similar to what was done in the recent work of
Zhu et al. (2014).
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.
Simulation results and analysis
The Kuramoto model simulation with non-local coupling effects
(κ= 0.10) with a phase lag, as expressed in Eq. (1) for a
128 × 128 grid transition probability, and the corresponding
shortest-path distance matrices, lead to snapshots of three attributes:
(i) the phase profile, (ii) the effective angular velocities of oscillators,
and (iii) the fluctuation of the instantaneous angular velocity of
oscillators. We did not sort the results according to an increase in the
values of the attributes. Figure 4a–c show that, for a case of no
recurrences, there exists a chimera state. The ensemble averages from the
last 1000 time steps of the 200 000 time steps in the numerical simulations
reveal the co-existence of synchronous and asynchronous oscillators. This
means that some of the cells in the grid show a
synchronous behavior and some others do not. In this particular case of no
recurrences (Fig. 4), the number of synchronous oscillators to the number of
asynchronous oscillators is large. In the case of recurrences, as shown in
Fig. 5, this ratio is much larger. Also, the chimera pattern of the
synchronized and asynchronized components of the oscillators is similar to
what was observed by Abrams and Strogatz (2004). Figures 4a and 5a are the
first evidence of the possible existence of a chimera state in earthquake
sequencing. Figures 4b, c, 5b, and c confirm this.
Chimera index as a function of time steps for the 128 × 128
grid without recurrences for κ= 0.10.
Going from the 128 × 128 grid to the 1024 × 1024 grid,
there are more non-zero cells with multiplicity of earthquakes at least 1.
The number of oscillators is substantially larger, 1693 vs. 7697. Figures 6
and 7 reveal the chimera state as the steady-state solution to the non-linear
dynamics on weakly coupled oscillators without and with recurrences for the
1024 × 1024 grid with the non-local coupling coefficient,
κ= 0.1. Figure 8 shows the behavior of the chimera index as a
function of the number of time steps for the non-recurrent 128 × 128
grid with the non-local coupling coefficient, κ, set at 0.10. The
purpose of this figure is to demonstrate that the asymptotic behavior of the
chimera index with an increase in the number of time steps could be used to
look at the steady-state solution of the Kuramoto model.
We looked at the influence of the non-local coupling coefficient, κ,
on the ratio of the number of coherent oscillators to the total number of
oscillators for both the 128 × 128 and 1024 × 1024 grids
without recurrences in Fig. 9. A similar observation is made for the case of
recurrences. As the non-local coupling coefficient, κ, increases
from 0.01 to 1.0, the ratio decreases. For values of κ approaching 0,
the non-local Kuramoto model acts as a simple Kuramoto model in that there is
full synchronization for the global coupling parameter, A (or used as K
in the literature), of 1.0. What is surprising to begin with is that, as
κ approaches 1, the steady-state solution becomes more asynchronized.
Investigations on the effect of the non-local coupling effect parameter,
κ, on the steady-state solution of the phase angle distribution in the
chimera state (Figs. 10a, b, 11a and b) suggest that for both the
128 × 128 grid and the 1024 × 1024 grid, for larger
κ values, the number of asynchronous oscillators is larger, and for
smaller κ values, the presence of synchronous oscillators becomes
dominant. For in-between values, i.e. between 1.0 and 0.03, the nature of the
chimera states changes.
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 × 128 or 1024 × 1024
grids) should reveal the “non-readiness or readiness” cells or zones for
earthquakes. One such map for a 128 × 128 grid without recurrences for
κ= 0.10 is shown in Fig. 12. This qualitative description of the
evolutionary dynamics of the earthquake sequencing is highly instructive.
Influence of the non-local coupling coefficient parameter,
κ, on the ratio of the number of synchronized
oscillators to the total number of oscillators for both the 128 × 128 and the
1024 × 1024 grids without recurrences.
(a) Effect of the non-local coupling coefficient parameter,
κ, on the evolution and disappearance of the chimera states for the
128 × 128 grid without recurrence. Stationary phase angle
as a function of the oscillator index: kappa, κ= 1.0 (top left
panel); kappa, κ= 0.3 (top right panel); kappa,
κ= 0.1 (bottom left panel); kappa, κ= 0.03 (bottom
right panel). (b) Effect of the non-local coupling coefficient
parameter, κ, on evolution and disappearance of the chimera states for
the 128 × 128 grid with recurrence. Stationary phase angle
as a function of the oscillator index: kappa, κ= 1.0 (top left
panel); kappa, κ= 0.3 (top right panel); kappa,
κ= 0.1 (bottom left panel); kappa, κ= 0.03 (bottom
right panel).
(a) Effect of the non-local coupling coefficient parameter,
κ, on the evolution and disappearance of the chimera states for the
1024 × 1024 grid without recurrence. Stationary phase angle
as a function of the oscillator index: kappa, κ= 1.0 (top left
panel); kappa, κ= 0.3 (top right panel); kappa,
κ= 0.1 (bottom left panel); kappa, κ= 0.03 (bottom
right panel). (b) Effect of the non-local coupling coefficient
parameter, κ, on evolution and disappearance of the chimera states
for the 1024 × 1024 grid with recurrence. Stationary phase
angle as a function of the oscillator index: kappa, κ= 1.0 (top
left panel); kappa, κ= 0.3 (top right panel); kappa,
κ= 0.1 (bottom left panel); kappa, κ= 0.03 (bottom
right panel).
Chimera-state map of the synchronous and asynchronous oscillators as
a steady-state solution for a non-recurrence case. The non-local
coupling coefficient parameter, κ, is 0.1. Blue dots refer to the
asynchronous oscillators and red dots to the synchronous oscillators.
Conclusions and future work
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 × 128, 192 × 192, 256 × 256,
512 × 512, and 1024 × 1024 grids result in differing
dimensions of transition matrices of oscillators in increasing order.
Short-path distance matrices for the latter are generated concurrently to
study the non-local effects used in the Kuramoto model.
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, κ.
For a non-local coupling strength, κ, of 0.10, the Kuramoto model
yields chimera states as a steady-state solution, i.e. co-existence of
synchronized and asynchronized states. This is true for all the grid sizes
considered. Differences exist in the ratio of the number of coherent oscillators
to the number of incoherent oscillators.
As the non-local coupling strength, κ, is lowered from 1.0 to 0.01,
there is a general tendency towards an increase in synchronization, as is
expected. While this general trend is observed for directed graphs generated
from grids of orders 128, 192, 256, and 512, the graph from the
1024 × 1024 grid reveals the presence of the chimera state.
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 × 128 and 1024 × 1024 grids. However, it is quite intriguing to find out that
the asynchronous oscillators come from a sub-set of the oscillators in both cases.
There is still a nagging question about which non-local coupling coefficient
would be an ideal candidate for understanding the global stress-field fluctuations.
Figure 12 illustrates an example of how a chimera state could be displayed
on the map grid. Imposing geophysical and geodetic constraints on the
earthquake zones in terms of heterogeneity of the natural frequencies would
provide a quantitative answer to the above question.
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.
Needless to say, the role of the parameters such as the heterogeneity of the
oscillators as expressed in the natural frequency of the oscillators, the
variability of the time-delay corrections instead of a constant time delay,
and the heterogeneity of the non-local coupling strength and the global
coupling strength in the present Kuramoto model, remain to be investigated.
Work is currently in progress.
Acknowledgements
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
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