Distribution Of Recurrence Times Research Articles

Статистика времен возвращения при анализе хаоса в динамических системах

We propose a method of delimiting the existence domains of periodic, quasi-periodic and chaotic solutions of dynamical systems in the parameter space. The method is based on an analysis of the sampling of the times of passing a phase point from the previous intersection of the Poincaré plane to the next one (or from one local maximum to the next). An algorithm for generating a sample of recurrence times with subsequent analysis of the histogram of the obtained sample is constructed. A simple measure of histogram filling allows us to separate periodic and chaotic modes, as well as to estimate the degree of chaoticity of intermediate modes. On simple model signals it is shown that the distribution of recurrence times gives information not contained in the spectral densities of the signal. Then, on the example of the classical Lorentz system, it is shown how a simple measure of filling the histogram of recurrence times allows us to obtain a visual map of modes. The paper presents the results of a comparative analysis of power spectral density and histograms of recurrence times for different modes realized in the Lorentz system at different values of the control parameter (Rayleigh number).

Improved bounds for the solutions of renewal equations

Sequences of non-decreasing (non-increasing) lower (upper) bounds for the renewal-type equation as well as for the renewal function which are improvements of the famous corresponding bounds of Marshal [(1973). Linear bounds on the renewal function. SIAM Journal on Applied Mathematics 24(2): 245–250] are given. Also, sequences such bounds converging to the ordinary renewal function are obtained for several reliability classes of the lifetime distributions of the inter-arrival times, which are refinements of all of the existing known corresponding bounds. For the first time, a lower bound for the renewal function with DMRL lifetimes is given. Finally, sequences of such improved bounds are given for the ordinary renewal density as well as for the right-tail of the distribution of the forward recurrence time.

Sheared granular matter and the empirical relations of seismicity.

The frictional instability associated with earthquake initiation and earthquake dynamics is believed to be mainly controlled by the dynamics of fragmented rocks within the fault gauge. Principal features of the emerging seismicity (e.g., intermittent dynamics and broad time and/or energy scales) have been replicated by simple experimental setups, which involve a slowly driven slider on top of granular matter, for example. Yet these setups are often physically limited and might not allow one to determine the underlying nature of specific features and, hence, the universality and generality of the experimental observations. Here, we address this challenge by a numerical study of a spring-slider experiment based on two-dimensional discrete element method simulations, which allows us to control the properties of the granular matter and of the surface of the slider, for example. Upon quasistatic loading, stick-slip-type behavior emerges which is contrasted by a stable sliding regime at finite driving rates, in agreement with experimental observations. Across large parameter ranges for damping, interparticle friction, particle polydispersity, etc., the earthquake-like dynamics associated with the former regime results in several robust scale-free statistical features also observed in experiments. At first sight, these closely resemble the main empirical relations of tectonic seismicity at geological scales. This includes the Gutenberg-Richter distribution of event sizes, the Omori-Utsu-type decay of aftershock rates, as well as the aftershock productivity relation and broad recurrence time distributions. Yet, we show that the correlations associated with tectonic aftershocks are absent such that the origin of the Omori-Utsu relation, the aftershock productivity relation, and Båth's relation in the simulations is fundamentally different from the case of tectonic seismicity. This, we believe, is mainly due to a lack of macroscale relaxation processes that are closely tied to the generation of real aftershocks. We argue that the same is true for previous laboratory experiments.

Potential Well in Poincaré Recurrence

From a physical/dynamical system perspective, the potential well represents the proportional mass of points that escape the neighbourhood of a given point. In the last 20 years, several works have shown the importance of this quantity to obtain precise approximations for several recurrence time distributions in mixing stochastic processes and dynamical systems. Besides providing a review of the different scaling factors used in the literature in recurrence times, the present work contributes two new results: (1) For -mixing and -mixing processes, we give a new exponential approximation for hitting and return times using the potential well as the scaling parameter. The error terms are explicit and sharp. (2) We analyse the uniform positivity of the potential well. Our results apply to processes on countable alphabets and do not assume a complete grammar.

Stickiness in generic low-dimensional Hamiltonian systems: A recurrence-time statistics approach.

We analyze the structure and stickiness in the chaotic components of generic Hamiltonian systems with divided phase space. Following the method proposed recently in Lozej and Robnik [Phys. Rev. E 98, 022220 (2018)2470-004510.1103/PhysRevE.98.022220], the sticky regions are identified using the statistics of recurrence times of a single chaotic orbit into cells dividing the phase space into a grid. We perform extensive numerical studies of three example systems: the Chirikov standard map, the family of Robnik billiards, and the family of lemon billiards. The filling of the cells is compared to the random model of chaotic diffusion, introduced in Robnik et al. [J. Phys. A: Math. Gen. 30, L803 (1997)JPHAC50305-447010.1088/0305-4470/30/23/003] for the description of transport in the phase spaces of ergodic systems. The model is based on the assumption of completely uncorrelated cell visits because of the strongly chaotic dynamics of the orbit and the distribution of recurrence times is exponential. In generic systems the stickiness induces correlations in the cell visits. The distribution of recurrence times exhibits a separation of timescales because of the dynamical trapping. We model the recurrence time distributions to cells inside sticky areas as a mixture of exponential distributions with different decay times. We introduce the variable S, which is the ratio between the standard deviation and the mean of the recurrence times as a measure of stickiness. We use S to globally assess the distributions of recurrence times. We find that in the bulk of the chaotic sea S=1, while S>1 in areas of stickiness. We present the results in the form of animated grayscale plots of the variable S in the largest chaotic component for the three example systems, included as supplemental material to this paper.

Ergodicity, mixing, and recurrence in the three rotor problem.

In the classical three rotor problem, three equal point masses move on a circle subject to attractive cosine potentials of strength g. In the center of mass frame, energy E is the only known conserved quantity. In earlier works [Krishnaswami and Senapati, Indian Acad. Sci. Conf. Ser. 2(1), 139 (2019), and Chaos 29(12), 123121 (2019)], an order-chaos-order transition was discovered in this system along with a band of global chaos for 5.33g≤E≤5.6g. Here, we provide numerical evidence for ergodicity and mixing in this band. The distributions of relative angles and angular momenta along generic trajectories are shown to approach the corresponding distributions over constant energy hypersurfaces (weighted by the Liouville measure) as a power-law in time. Moreover, trajectories emanating from a small volume are shown to become uniformly distributed over constant energy hypersurfaces, indicating that the dynamics is mixing. Outside this band, ergodicity and mixing fail, though the distributions of angular momenta over constant energy hypersurfaces show interesting phase transitions from Wignerian to bimodal with increasing energy. Finally, in the band of global chaos, the distribution of recurrence times to finite size cells is found to follow an exponential law with the mean recurrence time satisfying a scaling law involving an exponent consistent with global chaos and ergodicity.

An Advanced Statistical Method for Point Process Modelling With Missing Event Histories

Researchers are always challenged in developing history-dependent point process models for recurrence events such as system failures when the early event history is missing. The raison d'être in most cases is the estimation of model parameters. Even for simple renewal point processes, the model estimation is difficult since the distribution of backward recurrence time is not explicitly given, except for the limiting distribution case. To fill the gap, this article establishes an effective statistical method for parameter estimation for history-dependent point process models with partial missing history. The proposed method addresses the missing history issue through a data augmentation (DA) technique integrated with a Markov Chain Monte Carlo (MCMC) simulation technique. The key novelty is the creation of an acceptance/rejection criterion to assure the validity of the augmented missing history. Next, the augmented and observed histories are combined to form a `complete' history which is in turn used for model estimation. The estimated parameters are then used to generate new missing history. An iterative procedure is implemented to warrant stationary distributions for the estimates after burn-in. The validity and efficiency of the proposed method are demonstrated using two simulation studies and one real-life case study of modelling municipal water pipe failures from different model estimation perspectives.

Comparing the sandpile model with targeted triggering and the Olami-Feder-Christensen model as models of seismicity using recurrence network analysis

Slowly driven sandpile models has found applications in modelling earthquakes due to the observed power law statistics in its magnitude distributions, like the behaviour of earthquakes. Adding a probability to target the most susceptible site in the grid, the sandpile model recovers even the spatio-temporal statistics of earthquake events. In this work, we compare the sandpile model with targeted triggering to the Olami-Feder-Christensen (OFC) model: a standard earthquake model that also exhibits self-organized criticality. The sandpile model captures the magnitude distributions of earthquake events at a value of targeted triggering probability p = [0.004,0.007]. The triggering probability value p = 1.0, showing that the most susceptible site is always triggered, follows the magnitude distribution of the OFC model. A comparison was done by constructing a record-breaking recurrence network for the events. Spatial and magnitude criteria set the temporally directed links between events across the entire record. Both the models recover power-law exponents comparable to those previously obtained for earthquake data, which is 1.0 for recurrence distance and recurrence time distributions, and 2.1 for the in-degree distributions for the farthest recurrence criteria. The sandpile model with targeted triggering exhibits a behaviour in between a slowly driven sandpile and the OFC model.

Locking on a megathrust as a cause of distributed faulting and fault-jumping earthquakes

Seismicity on large faults is often characterized in terms of an independent recurrence time and magnitude distribution, which forms the basis for calculating future earthquake probabilities. The underlying assumption is that the driving mechanism for earthquakes on any particular fault is uniquely linked to that fault, determined by the rate of long-term creep on its deeper extension. However, our modelling of nearly 20 years of Global Positioning System data along the obliquely converging plate boundary in New Zealand shows that interseismic stress accumulation can be independent of the properties of the numerous crustal faults and controlled by locking on the megathrust. In this way, the interseismic driving mechanism for large crustal earthquakes is not linked directly to the individual major faults that rupture. This scenario predicts large-magnitude earthquakes with complex multifault ruptures in broad zones that ‘jump’ from fault to fault, following the contours of stress/strain loading. This can explain the November 2016 Mw7.8 Kaikoura earthquake that shattered the plate boundary in central New Zealand. Repeated episodes of this would create the observed complex array of active faults with the appearance of coherent slip. Our analysis opens up the possibility to use long-term Global Positioning System data to identify this type of earthquake behaviour. Earthquakes that jump from fault to fault in subduction zones can be explained by locking on the plate interface, according to GPS data from New Zealand where the 2016 Kaikoura earthquake produced a complex array of crustal ruptures.

Have we seen the largest earthquakes in eastern North America?

The assumed magnitude of the largest future earthquakes, Mmax, is crucial in assessing seismic hazard, especially for critical facilities like nuclear power plants. Estimates are made using various methods and often prove too low, as for the 2011 Tohoku, Japan, earthquake. Estimating Mmax is particularly challenging within tectonic plates, where large earthquakes are infrequent, vary in location and time, and often occur on previously unrecognized faults. For example, it is unclear whether the short historical record includes the largest possible earthquakes along the eastern continental margin of North America. We explore this issue by generating synthetic earthquake histories and sampling them over a few hundred years. Due to the short histories, the maximum magnitudes appearing most often in a sub-catalog, Mmaxa, are often smaller than the maximum magnitude in the parent catalog, Mmaxp, that can occur. Future earthquakes along the continental margin may thus be significantly larger than those observed to date. More generally, these simulations demonstrate that the largest earthquake in a catalog likely reflects a combination of catalog length, a region’s earthquake productivity, and relative proportion of small to large events. For regions with low seismicity, small variations in b value, the ratio of large to small events, due to sampling has a significant impact on the expected recurrence times of large magnitude earthquakes. Although the precise likelihood of observing Mmaxp depends on the distribution of recurrence times, a catalog shorter than an earthquake’s mean recurrence time will likely not contain an event of that size. As a result, Mmax cannot always be reliably estimated from earthquake catalogs.

Extreme events in systems with discontinuous boundaries

We describe a class of extreme events emerging in systems with discontinuous boundaries that exhibit stick-slip dynamics. This kind of systems is capable of generating extreme events when the system trajectory sticks to a sliding set and slides for a relatively large distance along the discontinuous boundary to one of the system's subspaces. We present two examples of such systems, a microelectromechanical cantilever and a driven class-B laser, where this type of extreme events appear. We show that their forecasting is possible by monitoring the sliding trajectory along the discontinuous boundary. In both cases, the probability distribution of recurrence times exhibits a power-law behavior, typical for extreme events. The results of this study can be of interest for engineering applications, for example, to predict extremely large-amplitude oscillations in cantilevers, systems with friction, lasers, and other systems with discontinuous boundaries.

Structure, size, and statistical properties of chaotic components in a mixed-type Hamiltonian system.

We perform a detailed study of the chaotic component in mixed-type Hamiltonian systems on the example of a family of billiards [introduced by Robnik in J. Phys. A: Math. Gen. 16, 3971 (1983)JPHAC50305-447010.1088/0305-4470/16/17/014]. The phase space is divided into a grid of cells and a chaotic orbit is iterated a large number of times. The structure of the chaotic component is discerned from the cells visited by the chaotic orbit. The fractal dimension of the border of the chaotic component for various values of the billiard shape parameter is determined with the box-counting method. The cell-filling dynamics is compared to a model of uncorrelated motion, the so-called random model [Robnik et al. J.Phys. A: Math. Gen. 30, L803 (1997)JPHAC50305-447010.1088/0305-4470/30/23/003], and deviations attributed to sticky objects in the phase space are found. The statistics of the number of orbit visits to the cells is analyzed and found to be in agreement with the random model in the long run. The stickiness of the various structures in the phase space is quantified in terms of the cell recurrence times. The recurrence time distributions in a few selected cells as well as the mean and standard deviation of recurrence times for all cells are analyzed. The standard deviation of cell recurrence time is found to be a good quantifier of stickiness on a global scale. Three methods for determining the measure of the chaotic component are compared and the measure is calculated for various values of the billiard shape parameter. Lastly, the decay of correlations and the diffusion of momenta is analyzed.

The Masses and Accretion Rates of White Dwarfs in Classical and Recurrent Novae

Abstract Models have long predicted that the frequency-averaged masses of white dwarfs (WDs) in Galactic classical novae are twice as large as those of field WDs. Only a handful of dynamically well-determined nova WDs masses have been published, leaving the theoretical predictions poorly tested. The recurrence time distributions and mass accretion rate distributions of novae are even more poorly known. To address these deficiencies, we have combined our extensive simulations of nova eruptions with the Strope et al. and Schaefer databases of outburst characteristics of Galactic classical and recurrent novae (RNe) to determine the masses of 92 WDs in novae. We find that the mean mass (frequency-averaged mean mass) of 82 Galactic classical novae is 1.06 (1.13) M ⊙, while the mean mass of 10 RNe is 1.31 M ⊙. These masses, and the observed nova outburst amplitude and decline time distributions allow us to determine the long-term mass accretion rate distribution of classical novae. Remarkably, that value is just 1.3 × 10−10 M ⊙ yr−1, which is an order of magnitude smaller than that of cataclysmic binaries in the decades before and after classical nova eruptions. This predicts that old novae become low-mass transfer rate systems, and hence dwarf novae, for most of the time between nova eruptions. We determine the mass accretion rates of each of the 10 known Galactic recurrent nova, finding them to be in the range of 10−7–10−8 M ⊙ yr−1. We are able to predict the recurrence time distribution of novae and compare it with the predictions of population synthesis models.

Testing extreme dependence in financial time series

Financial interdependence indicates a process through which transmission of shock originating in the financial market of one economy spreads to others. This paper provides a new idea of Residual and Recurrence Times of high or low values for bivariate time series to detect extreme dependence or contagion. In presence of financial extreme dependence, the distributions of residual and recurrence times are not the same. We examine the equality of two distributions using the permutation test. In comparison to other methods in multivariate extreme value theory, our proposed method does not need the i.i.d. assumption. Our method can handle the situation where the extremes for different components do not occur at the same time. We justify our methods in two ways: first using thorough simulation studies and then applying the proposed method to real data on weekly stock indices from sixteen markets.

Maximal Repetition and Zero Entropy Rate

Maximal repetition of a string is the maximal length of a repeated substring. This paper investigates maximal repetition of strings drawn from stochastic processes. Strengthening previous results, two new bounds for the almost sure growth rate of maximal repetition are identified: an upper bound in terms of conditional R\'enyi entropy of order $\gamma>1$ given a sufficiently long past and a lower bound in terms of unconditional Shannon entropy ($\gamma=1$). Both the upper and the lower bound can be proved using an inequality for the distribution of recurrence time. We also supply an alternative proof of the lower bound which makes use of an inequality for the expectation of subword complexity. In particular, it is shown that a power-law logarithmic growth of maximal repetition with respect to the string length, recently observed for texts in natural language, may hold only if the conditional R\'enyi entropy rate given a sufficiently long past equals zero. According to this observation, natural language cannot be faithfully modeled by a typical hidden Markov process, which is a class of basic language models used in computational linguistics.

Regularity of a renewal process estimated from binary data.

Assessment of the regularity of a sequence of events over time is important for clinical decision-making as well as informing public health policy. Our motivating example involves determining the effect of an intervention on the regularity of HIV self-testing behavior among high-risk individuals when exact self-testing times are not recorded. Assuming that these unobserved testing times follow a renewal process, the goals of this work are to develop suitable methods for estimating its distributional parameters when only the presence or absence of at least one event per subject in each of several observation windows is recorded. We propose two approaches to estimation and inference: a likelihood-based discrete survival model using only time to first event; and a potentially more efficient quasi-likelihood approach based on the forward recurrence time distribution using all available data. Regularity is quantified and estimated by the coefficient of variation (CV) of the interevent time distribution. Focusing on the gamma renewal process, where the shape parameter of the corresponding interevent time distribution has a monotone relationship with its CV, we conduct simulation studies to evaluate the performance of the proposed methods. We then apply them to our motivating example, concluding that the use of text message reminders significantly improves the regularity of self-testing, but not its frequency. A discussion on interesting directions for further research is provided.

Transport, diffusion, and energy studies in the Arnold-Beltrami-Childress map

We study the transport and diffusion properties of passive inertial particles described by a six-dimensional dissipative bailout embedding map. The base map chosen for the study is the three-dimensional incompressible Arnold-Beltrami-Childress (ABC) map chosen as a representation of volume preserving flows. There are two distinct cases: the two-action and the one-action cases, depending on whether two or one of the parameters (A,B,C) exceed 1. The embedded map dynamics is governed by two parameters (α,γ), which quantify the mass density ratio and dissipation, respectively. There are important differences between the aerosol (α<1) and the bubble (α>1) regimes. We have studied the diffusive behavior of the system and constructed the phase diagram in the parameter space by computing the diffusion exponents η. Three classes have been broadly classified-subdiffusive transport (η<1), normal diffusion (η≈1), and superdiffusion (η>1) with η≈2 referred to as the ballistic regime. Correlating the diffusive phase diagram with the phase diagram for dynamical regimes seen earlier, we find that the hyperchaotic bubble regime is largely correlated with normal and superdiffusive behavior. In contrast, in the aerosol regime, ballistic superdiffusion is seen in regions that largely show periodic dynamical behaviors, whereas subdiffusive behavior is seen in both periodic and chaotic regimes. The probability distributions of the diffusion exponents show power-law scaling for both aerosol and bubbles in the superdiffusive regimes. We further study the Poincáre recurrence times statistics of the system. Here, we find that recurrence time distributions show power law regimes due to the existence of partial barriers to transport in the phase space. Moreover, the plot of average particle kinetic energies versus the mass density ratio for the two-action case exhibits a devil's staircase-like structure for higher dissipation values. We explain these results and discuss their implications for realistic systems.

Testing stress shadowing effects at the South American subduction zone

The seismic gap hypothesis assumes that a characteristic earthquake is followed by a long period with a reduced occurrence probability for the next large event on the same fault segment, as a consequence of the induced stress shadow. The gap model is commonly accepted by geologists and is often used for time-dependent seismic hazard estimations. However, systematic and rigorous tests to verify the seismic gap model have often failed so far, which might be partially related to limited data and too tight model assumptions. In this study, we relax the assumption of a characteristic size and location of repeating earthquakes and analyse one of the best available data sets, namely the historical record of major earthquakes along a 3000 km long linear segment of the South American subduction zone. To test whether a stress shadow effect is observable, we compiled a comprehensive catalogue of mega-thrust earthquakes along this plate boundary from 1520 to 2015 containing 174 earthquakes with Mw > 6.5. In our new testing approach, we analyse the time span between an earthquake and the last event that ruptured the epicentre location, where we consider the impact of the uncertainties of epicentres and rupture extensions. Assuming uniform boundary conditions along the trench, we compare the distribution of these recurrence times with simple recurrence models. We find that the distribution is in all cases almost exponentially distributed corresponding to a random (Poissonian) process; despite some tendencies for clustering of the Mw ≥ 7 events and a weak quasi-periodicity of the Mw ≥ 8 earthquakes, respectively. To verify whether the absence of a clear stress shadow signal is related to physical assumptions or data uncertainties, we perform simulations of a physics-based stochastic earthquake model considering rate and state-dependent earthquake nucleation, which are adapted to the observations with regard to the number of events, spatial extend, size distribution and involved uncertainties. Our simulations show that the catalogue uncertainties lead to a significant blurring of the theoretically peaked distribution, but the distribution would be still distinguishable from the observed one for Mw ≥ 7 events. However, considering the stress transfer to adjacent fault segments and heterogeneous instead of constant stress drop within the rupture zone can explain the observed recurrence time distribution. We conclude that simplified recurrence models, ignoring the complexity of the underlying physical process, cannot be applied for forecasting the Mw ≥ 7 earthquake occurrence at this plate boundary.

Statistical properties of the one-dimensional Burridge-Knopoff model of earthquakes obeying the rate- and state-dependent friction law.

Statistical properties of the one-dimensional spring-block (Burridge-Knopoff) model of earthquakes obeying the rate- and state-dependent friction law are studied by extensive computer simulations. The quantities computed include the magnitude distribution, the rupture-length distribution, the main shock recurrence-time distribution, the seismic-time correlations before and after the main shock, the mean slip amount, and the mean stress drop at the main shock, etc. Events of the model can be classified into two distinct categories. One tends to be unilateral with its epicenter located at the rim of the rupture zone of the preceding event, while the other tends to be bilateral with enhanced "characteristic" features resembling the so-called "asperity." For both types of events, the distribution of the rupture length L_{r} exhibits an exponential behavior at larger sizes, ≈exp[-L_{r}/L_{0}] with a characteristic "seismic correlation length" L_{0}. The mean slip as well as the mean stress drop tends to be rupture-length independent for larger events. The continuum limit of the model is examined, where the model is found to exhibit pronounced characteristic features. In the continuum limit, the characteristic rupture length L_{0} is estimated to be ∼100 [km]. This means that, even in a hypothetical homogenous infinite fault, events cannot be indefinitely large in the exponential sense, the upper limit being of order ∼10^{3} kilometers. Implications to real seismicity are discussed.

Recurrence Time Distributions of Large Earthquakes in Eastern Iran

The eastern part of the Iranian plateau is characterized by large and infrequent earthquakes with recurrence intervals of more than several hundred years. Given that previous observations and paleoseismological studies are insufficient for forecasting large earthquakes, we have developed a physics‐based synthetic seismicity model for the fault system in eastern Iran using the Virtual Quake simulator. The model is independent of the seismic catalogs, however, it is tuned to match available earthquakes records. We show that the modeled seismicity rates and empirical frequency–magnitude distributions are consistent within the uncertainty of the empirical relations. Furthermore, our synthetic catalog agrees with previous paleoseismological investigations. From the resulting catalog, we can obtain the statistical distributions of recurrence times and waiting times for large earthquakes in the region as a whole and for the individual faults. In agreement with previous earthquake simulator studies, we find that the Poisson (time‐independent) distribution best describes the recurrence times of large earthquakes in the region as a whole. Large earthquakes on the individual faults show quasi‐periodic behavior, and for most faults can be well represented by the Weibull distribution. We present the corresponding time‐dependent conditional probabilities for large earthquakes throughout the region and for a few selected individual faults. Long‐term simulations indicate that waiting times for large earthquakes on specific faults are strongly dependent on the fault system configuration. Faults confined by other active faults, or consisting of several segments, host large earthquakes with more distributed recurrence times. Moreover, the obtained long‐term synthetic seismic catalog shows that fault interactions clearly have a major effect on occurrence of large earthquakes, and hence should be taken into account for seismic‐hazard assessment.