Distribution Of Recurrence Times Research Articles

Effect of noise-induced crisis of attractor on characteristics of Poincaré recurrence

The effect of noise on the Poincare recurrence time has been studied in a one-dimensional map with quasi-attractor. The validity of the Kac theorem in application to noise systems is substantiated. Results of calculations of the distribution of recurrence times in a noisy system are presented.

Characteristic slip for five great earthquakes along the Fuyun fault in China

The seismic hazard associated with individual faults can be assessed from the distribution of slip and the recurrence time of earthquakes. Analysis of the offsets of stream channels and terraces caused during five historical earthquakes on the Fuyun fault, China, reveals characteristic fault slip of about six metres for each event. The seismic hazard associated with an individual fault can be assessed from the distributions of slip and recurrence times of earthquakes. However, seismic cycle models1 that aim to predict rupture lengths and fault displacements of successive earthquakes on one fault remain poorly validated. It is therefore unknown whether individual fault segments rupture independently, producing earthquakes with a diverse range of magnitudes and recurrence times, or slip by characteristic amounts, with characteristic magnitudes. Here we use high-resolution satellite data to document the horizontal offsets of stream channels and terraces created by strike-slip motion on the Fuyun fault, Xinjiang, China, during five historical earthquakes. We find that the Ms 7.9 11 August 1931 earthquake produced a surface rupture with a length of 160 km, dispersed over three different fault segments. The 290 measured stream channel and terrace offsets record an average slip of 6.3 m. We use the degree of preservation of geomorphological markers to assign relative ages to individual fault offsets and identify at least four distinct older earthquakes. We find that these older earthquakes also produced fault offsets with a similar distribution to the 1931 earthquake. As the slip distributions during five successive earthquakes were so similar, we conclude that ruptures on the Fuyun fault obey a characteristic slip model.

Extreme Events on Complex Networks

A wide spectrum of extreme events ranging from traffic jams to floods take place on networks. Motivated by these, we employ a random walk model for transport and obtain analytical and numerical results for the extreme events on networks. They reveal an unforeseen, and yet a robust, feature: small degree nodes of a network are more likely to encounter extreme events than the hubs. Further, we also study the recurrence time distribution and scaling of the probabilities for extreme events. These results suggest a revision of design principles and can be used as an input for designing the nodes of a network so as to smoothly handle extreme events.

Earthquake correlations and networks: A comparative study

We quantify the correlation between earthquakes and use the same to extract causally connected earthquake pairs. Our correlation metric is a variation on the one introduced by Baiesi and Paczuski [M. Baiesi and M. Paczuski, Phys. Rev. E 69, 066106 (2004)]. A network of earthquakes is then constructed from the time-ordered catalog and with links between the more correlated ones. A list of recurrences to each of the earthquakes is identified employing correlation thresholds to demarcate the most meaningful ones in each cluster. Data pertaining to three different seismic regions (viz., California, Japan, and the Himalayas) are comparatively analyzed using such a network model. The distribution of recurrence lengths and recurrence times are two of the key features analyzed to draw conclusions about the universal aspects of such a network model. We find that the unimodal feature of recurrence length distribution, which helps to associate typical rupture lengths with different magnitude earthquakes, is robust across the different seismic regions. The out-degree of the networks shows a hub structure rooted on the large magnitude earthquakes. In-degree distribution is seen to be dependent on the density of events in the neighborhood. Power laws, with two regimes having different exponents, are obtained with recurrence time distribution. The first regime confirms the Omori law for aftershocks while the second regime, with a faster falloff for the larger recurrence times, establishes that pure spatial recurrences also follow a power-law distribution. The crossover to the second power-law regime can be taken to be signaling the end of the aftershock regime in an objective fashion.

Open mushrooms: stickiness revisited

We investigate mushroom billiards, a class of dynamical systems with sharply divided phase space. For typical values of the control parameter of the system ρ, an infinite number of marginally unstable periodic orbits (MUPOs) exist making the system sticky in the sense that unstable orbits approach regular regions in phase space and thus exhibit quasi-regular behaviour for long periods of time. The problem of finding these MUPOs is expressed as the well-known problem of finding optimal rational approximations of a real number, subject to some system-specific constraints. By introducing a generalized mushroom and using properties of continued fractions, we describe for the first time a zero measure set of control parameter values ρ ∊ (0, 1) for which all MUPOs are destroyed and therefore the system is less sticky, leading to a different power law exponent for the Poincaré recurrence time distribution statistics. The open mushroom (billiard with a hole) is then considered in order to quantify the stickiness exhibited due to MUPOs and exact leading order expressions for the algebraic decay of the survival probability function are calculated for mushrooms with triangular and rectangular stems. Numerical simulations are also performed which confirm our predictions for both sticky and less sticky mushrooms.

Network of earthquakes and recurrences therein

We quantify the correlation between earthquakes and use the same to distinguish between relevant causally connected earthquakes. Our correlation metric is a variation on the one introduced by Baiesi and Paczuski (Phys Rev E 69:066,106, 2004). A network of earthquakes is constructed, which is time-ordered and with links between the more correlated ones. Data pertaining to the California region has been used in the study. Recurrences to earthquakes are identified employing correlation thresholds to demarcate the most meaningful ones in each cluster. The distribution of recurrence lengths and recurrence times are analyzed subsequently to extract information about the complex dynamics. We find that the unimodal feature of recurrence lengths helps to associate typical rupture lengths with different magnitude earthquakes. The out-degree of the network shows a hub structure rooted on the large magnitude earthquakes. In-degree distribution is seen to be dependent on the density of events in the neighborhood. Power laws are also obtained with recurrence time distribution agreeing with the Omori law.

Earthquake Recurrence in Simulated Fault Systems

We employ a computationally efficient fault system earthquake simulator, RSQSim, to explore effects of earthquake nucleation and fault system geometry on earthquake occurrence. The simulations incorporate rate- and state-dependent friction, high-resolution representations of fault systems, and quasi-dynamic rupture propagation. Faults are represented as continuous planar surfaces, surfaces with a random fractal roughness, and discontinuous fractally segmented faults. Simulated earthquake catalogs have up to 106 earthquakes that span a magnitude range from ∼M4.5 to M8. The seismicity has strong temporal and spatial clustering in the form of foreshocks and aftershocks and occasional large-earthquake pairs. Fault system geometry plays the primary role in establishing the characteristics of stress evolution that control earthquake recurrence statistics. Empirical density distributions of earthquake recurrence times at a specific point on a fault depend strongly on magnitude and take a variety of complex forms that change with position within the fault system. Because fault system geometry is an observable that greatly impacts recurrence statistics, we propose using fault system earthquake simulators to define the empirical probability density distributions for use in regional assessments of earthquake probabilities.

Recurrence of Large Earthquakes: Bayesian Inference from Catalogs in the Presence of Magnitude Uncertainties

We present a Bayesian method that allows continuous updating the aperiodicity of the recurrence time distribution of large earthquakes based on a catalog with magnitudes above a completeness threshold. The approach uses a recently proposed renewal model for seismicity and allows the inclusion of magnitude uncertainties in a straightforward manner. Errors accounting for grouped magnitudes and random errors are studied and discussed. The results indicate that a stable and realistic value of the aperiodicity can be predicted in an early state of seismicity evolution, even though only a small number of large earthquakes has occurred to date. Furthermore, we demonstrate that magnitude uncertainties can drastically influence the results and can therefore not be neglected. We show how to correct for the bias caused by magnitude errors. For the region of Parkfield we find that the aperiodicity, or the coefficient of variation, is clearly higher than in studies which are solely based on the large earthquakes.

Bayesian nonparametric inference for earthquake recurrence time distributions in different tectonic regimes

The objective of this paper is to estimate the probability distribution of the time between consecutive earthquakes by employing two equally powerful tools: for the geology, a database of individual seismogenic sources, and for the statistics, assuming that the unknown distribution is a random measure, an estimation method based on the stochastic simulation of Markov chains. The resort to sophisticated instruments is motivated by the particular situation of Italy, where a complex tectonic model is combined with infrequent, medium‐size earthquakes. The quality and the length of the parametric catalogue of Italian earthquakes provide for a generous data set but bring with them problems of incompleteness and uncertainty regarding the parametrization of the events. The pointwise estimate of the interevent time density functions makes it possible to calculate the occurrence probability depending on the date of the last event at different forecasting horizons.

Accumulation of unstable periodic orbits and the stickiness in the two-dimensional piecewise linear map

We investigate the stickiness of the two-dimensional piecewise linear map with a family of marginal unstable periodic orbits (FMUPOs), and show that a series of unstable periodic orbits accumulating to FMUPOs plays a significant role to give rise to the power law correlation of trajectories. We can explicitly specify the sticky zone in which unstable periodic orbits whose stability increases algebraically exist, and find that there exists a hierarchy in accumulating periodic orbits. In particular, the periodic orbits with linearly increasing stability play the role of fundamental cycles as in the hyperbolic systems, which allows us to apply the method of cycle expansion. We also study the recurrence time distribution, especially discussing the position and size of the recurrence region. Following the definition adopted in one-dimensional maps, we show that the recurrence time distribution has an exponential part in the short time regime and an asymptotic power law part. The analysis on the crossover time T(c)(*) between these two regimes implies T(c)(*) approximately -log[micro(R)] where micro(R) denotes the area of the recurrence region.

Beyond Word Frequency: Bursts, Lulls, and Scaling in the Temporal Distributions of Words

BackgroundZipf's discovery that word frequency distributions obey a power law established parallels between biological and physical processes, and language, laying the groundwork for a complex systems perspective on human communication. More recent research has also identified scaling regularities in the dynamics underlying the successive occurrences of events, suggesting the possibility of similar findings for language as well.Methodology/Principal FindingsBy considering frequent words in USENET discussion groups and in disparate databases where the language has different levels of formality, here we show that the distributions of distances between successive occurrences of the same word display bursty deviations from a Poisson process and are well characterized by a stretched exponential (Weibull) scaling. The extent of this deviation depends strongly on semantic type – a measure of the logicality of each word – and less strongly on frequency. We develop a generative model of this behavior that fully determines the dynamics of word usage.Conclusions/SignificanceRecurrence patterns of words are well described by a stretched exponential distribution of recurrence times, an empirical scaling that cannot be anticipated from Zipf's law. Because the use of words provides a uniquely precise and powerful lens on human thought and activity, our findings also have implications for other overt manifestations of collective human dynamics.

A note on testing regime switching assumption based on recurrence times

By perceiving a regime switching model as an example of a nonlinear dynamical system, we employ recurrence time distribution of a chosen event to derive a test statistic for testing the null hypothesis of one regime versus the alternative of having two or more regimes involved in a time series. This simple chi-square type of statistic is compared with existing likelihood ratio based ones that are all, in general, very complex in construction. The power of our proposed test is rather satisfactory and the computing load required is significantly more economical.

Statistical prediction of the next great earthquake around Tehran, Iran

The analysis of seismic activity variations with space and time is a complex problem. Several statistical methods have been adopted to study these variations. One of the tasks that has attracted the attention of the seismological and statistical community is to explain seismicity patterns by statistical models and apply the results for earthquake prediction. Here the probability distribution of recurrence times as described by Exponential, Gamma, Lognormal, Pareto, Rayleigh and Weibull probability distributions and the idea of conditional probability has been applied to predict the next great (Ms ≥ 6.0 and Ms ≥ 6.5) earthquake around Tehran ( r ≤ 200 km). Conditional probability specifies the likelihood that a given earthquake will happen within a specified time. This likelihood is based on the information about past earthquake occurrences in the given region and the basic assumption that future seismic activity will follow the pattern of past activity. The rapid growth of Tehran to approximately 12 million inhabitants has resulted in a much more rapid increase in its vulnerability to natural disasters, especially earthquakes. Several earthquakes affected this region in the past, mostly on the Mosha, Taleqan, Eyvankey and Garmsar faults. The estimated recurrence times for Exponential, Gamma, Lognormal, Pareto, Rayleigh and Weibull distributions has been computed to be 66.64, 14.79, 26.88, 2.37, 67.58 and 80.47, respectively. Accordingly, one may expect that a large damaging earthquake may occur around Tehran approximately every 10 years.

Transport and diffusion in the embedding map

We study the transport properties of passive inertial particles in two-dimensional (2D) incompressible flows. Here, the particle dynamics is represented by the four-dimensional dissipative embedding map of the 2D area-preserving standard map which models the incompressible flow. The system is a model for impurity dynamics in a fluid and is characterized by two parameters, the inertia parameter alpha and the dissipation parameter gamma . The aerosol regime, where the particles are denser than the fluid, and the bubble regime, where they are less dense than the fluid, correspond to the parameter regimes alpha>1 and alpha<1 , respectively. Earlier studies of this system show a rich phase diagram with dynamical regimes corresponding to periodic orbits, chaotic structures, and mixed regimes. We obtain the statistical characterizers of transport for this system in these dynamical regimes. These are the recurrence time statistics, the diffusion exponent, and the distribution of jump lengths. The recurrence time distribution shows a power-law tail in the dynamical regimes, where there is preferential concentration of particles in sticky regions of the phase space, and an exponential decay in mixing regimes. The diffusion exponent shows behavior of three types-normal, subdiffusive, and superdiffusive, depending on the parameter regimes. Phase diagrams of the system are constructed to differentiate different types of diffusion behavior, as well as the behavior of the absolute drift. We correlate the dynamical regimes seen for the system at different parameter values with the transport properties observed at these regimes and in the behavior of the transients. This system also shows the existence of a crisis and unstable dimension variability at certain parameter values. The signature of the unstable dimension variability is seen in the statistical characterizers of transport. We discuss the implications of our results for realistic systems.

Finite-size effects on the statistics of extreme events in the BTW model

The BTW Abelian sandpile model is a prominent example of systems showing self-organised criticality (SOC) in the infinite size limit. We study finite-size effects with special focus on the statistics of extreme events, i.e., of particularly large avalanches. Not only the avalanche size probability distribution, but also the mutual independence of large avalanches in the critical state is affected by finite-size effects. Instead of a Poissonian recurrencetime distribution, in the finite system we find a repulsion of extreme events that depends on the avalanche size and not on the respective probability. The dependence of these effects on the system size is investigated and some data collapse is found. Our results imply that SOC is an unsuitable mechanism for the explanation of extreme events which occur in clusters.

Recurrent Large Earthquakes in a Fault Region: What Can Be Inferred from Small and Intermediate Events?

We present a renewal model for the recurrence of large earthquakes in a fault zone consisting of a major fault and surrounding smaller faults with Gutenberg–Richter-type seismicity represented by seismic moment release drawn from a truncated power-law distribution. The recurrence times of characteristic earthquakes for the major fault are explored. It is continuously loaded (plate motion) and undergoes positive and negative fluctuations due to adjacent smaller faults, with a large number N eq of such changes between two major earthquakes. Because the distribution has a finite variance, in the limit N eq→∞ the central limit theorem implies that the recurrence times follow a Brownian passage-time (BPT) distribution. This allows us to calculate individual recurrence-time distributions for specific fault zones without tuning free parameters: the mean recurrence time can be estimated from geological or paleoseismic data, and the standard deviation is determined from the frequency-size distribution, namely, the Richter b -value, of an earthquake catalog. The approach is demonstrated for the Parkfield segment of the San Andreas fault in California as well as for a long simulation of a numerical fault model. Assuming power-law distributed earthquake magnitudes up to the size of the recurrent Parkfield event ( M 6), we find a coefficient of variation that is higher than the value obtained by a direct fit of the BPT distribution to seven large earthquakes. We argue that the BPT distribution is a reasonable choice for seismic hazard assessment because it governs not only Brownian motion with drift but also models with power-law statistics for the recurrence of large earthquakes in an asymptotic limit. Finally, we find that the condition b =1 separates two regimes: in the first ( b 1) it is greater.

Return interval distribution of extreme events and long-term memory

The distribution of recurrence times or return intervals between extreme events is important to characterize and understand the behavior of physical systems and phenomena in many disciplines. It is well known that many physical processes in nature and society display long-range correlations. Hence, in the last few years, considerable research effort has been directed towards studying the distribution of return intervals for long-range correlated time series. Based on numerical simulations, it was shown that the return interval distributions are of stretched exponential type. In this paper, we obtain an analytical expression for the distribution of return intervals in long-range correlated time series which holds good when the average return intervals are large. We show that the distribution is actually a product of power law and a stretched exponential form. We also discuss the regimes of validity and perform detailed studies on how the return interval distribution depends on the threshold used to define extreme events.

Non-Poissonian earthquake occurrence in coupled stress release models and its effect on seismic hazard

SUMMARY Most seismic hazard estimations are based on the assumption of a Poisson process for earthquake occurrence, even though both observations and models indicate a departure of real seismic sequences from this simplistic assumption. Instrumental earthquake catalogues show earthquake clustering on regional scales while the elastic rebound theory predicts a periodic recurrence of characteristic earthquakes on longer timescales for individual events. Recent implementations of time-dependent hazard calculations in California and Japan are based on quasi-periodic recurrences of fault ruptures according to renewal models such as the Brownian Passage Time model. However, these renewal models neglect earthquake interactions and the dependence on the stressing history which might destroy any regularity of earthquake recurrences in reality. To explore this, we investigate the (coupled) stress release model, a stochastic version of the elastic rebound hypothesis. In particular, we are interested in the time-variability of the occurrence of large earthquakes and its sensitivity to the occurrence of Gutenberg‐Richter type earthquake activity and fault interactions. Our results show that in general large earthquakes occur quasi-periodically in the model: the occurrence probability of large earthquakes is strongly decreased shortly after a strong event and becomes constant on longer timescales. Although possible stress-interaction between adjacent fault zones does not affect the recurrence time distributions in each zone significantly, it leads to a temporal clustering of events on larger regional scales. The non-random characteristics, especially the quasi-periodic behaviour of large earthquakes, are even more pronounced if stress changes due to small earthquakes are less important. The recurrence-time distribution for the largest events is characterized by a coefficient of variation from 0.6 to 0.84 depending on the relative importance of small earthquakes.

Sensitivity analysis of the parameters of earthquake recurrence time power law scaling

The stability of the power law scaling of earthquake recurrence time distribution in a given space–time window is investigated, taking into account the magnitude of completeness and the effective starting time of aftershock sequences in earthquake catalogs from Southern California and Japan. A new method is introduced for sampling at different distances from a network of target events. This method allows the recurrence times to be sampled many times on the same area. Two power laws with unknown exponents are assumed to govern short- and long-recurrence-time ranges. This assumption is developed analytically and shown to imply simple correlation between these power laws. In practice, the results show that this correlation structure is not satisfied for short magnitude cutoffs (m c = 2.5, 3.5, 4.5), and hence the recurrence time distribution departs from the power law scaling. The scaling parameters obtained from the stack of the distributions corresponding to different magnitude thresholds are quite different for different regions of study. It is also found that significantly different scaling parameters adjust the distribution for different magnitude thresholds. In particular, the power law exponents decrease when the magnitude cutoff increases, resulting in a slower decrease of the recurrence time distribution, especially for short time ranges. For example, in the case of Japan, the exponent p2 of the power law scaling at large recurrence times follows roughly the relation: \(p_2 \left( {m_c } \right)=-0.07m_c +2.7;m_c \ge 3.5\), where m c is the magnitude cutoff. In case of Southern California, it is shown that Weibull distribution provides a better alternative fit to the data for moderate and large time scales.

Poincaré Recurrences from the Perspective of Transient Chaos

We obtain a description of the Poincaré recurrences of chaotic systems in terms of the ergodic theory of transient chaos. It is based on the equivalence between the recurrence time distribution and an escape time distribution obtained by leaking the system and taking a special initial ensemble. This ensemble is atypical in terms of the natural measure of the leaked system, the conditionally invariant measure. Accordingly, for general initial ensembles, the average recurrence and escape times are different. However, we show that the decay rate of these distributions is always the same. Our results remain valid for Hamiltonian systems with mixed phase space and validate a split of the chaotic saddle in hyperbolic and nonhyperbolic components.