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We investigate the emergent dynamics when the nonlinear Dieterich-Ruina rate and state friction law is attached to a Burridge-Knopoff spring-block model. We derive both the discrete equations and the continuum formulation governing the system in this framework. The discrete system (ODEs) exhibits both periodic and chaotic motion, where the system’s transition to chaos is size-dependent, i.e. how many blocks are considered. From the discrete model we derive the nonlinear elastic wave equation by taking the continuum limit. This results in a nonlinear partial differential equation (PDE) and we find that both temporal and spatial chaos ensues when the same parameter is increased. This critical parameter value needed for the onset of chaos in the continuous model is much smaller than the value needed in the case of a single block and we discuss the implications this has on dynamic modeling of earthquake rupture with this specific friction law. Most importantly, these results suggest that the friction law is scale-dependent, thus caution should be taken when attaching a friction law derived at laboratory scales to full-scale earthquake rupture models. Furthermore, we find solutions where the initial slip pulse propagates like a traveling wave, or remains localized in space, suggesting the presence of soliton and breather solutions. We discuss the significance of these pulse-like solutions and how they can be understood as a proxy for the propagation of the rupture front across the fault surface during an earthquake. We compute analytically the conditions for soliton solutions and by exploring the resulting parameter space, we introduce a possible method for determining a range of suitable parameter values to be used in future dynamic earthquake modeling.

Conditions under which a single oscillator model coupled with Dieterich-Ruina's rate and state dependent friction exhibits chaotic dynamics is studied. Properties of spring-block models are discussed. The parameter values of the system are explored and the corresponding numerical solutions presented. Bifurcation analysis is performed to determine the bifurcations and stability of stationary solutions and we find that the system undergoes a Hopf bifurcation to a periodic orbit. This periodic orbit then undergoes a period doubling cascade into a strange attractor, recognized as broadband noise in the power spectrum. The implications for earthquakes are discussed.

We show that both the slip distribution and the peak ground acceleration (PGA) for the 1999 Chi Chi earthquake can be described by Levy laws. Furthermore, we found that the tails of the probability density functions (PDF) characterizing the slip and the PGA are governed by a parameter, the Levy index, with almost the same values as predicted by the Central Limit theorem. The PDF tail controls the frequency at which extreme large events can occur. These events are the large stress drops-asperities distributed over the fault surface and the large PGA observed in the ground motion. Our results suggest that the frequency of these events is coupled; the PDF of the PGA is a direct consequence of the PDF of the asperities. The physical rationale responsible for the coupling is the principle of superposition of wave signals characterized by random properties that are governed by the Central Limit theorem.

Climatic variability can be characterized by invariant quantities arising from the analysis of scaling properties of paleoclimatic records. In this paper we discuss a stochastic model that reproduces the variability and the long-range correlation observed in dendrochronological time series. We have found that non-Gaussian distributions are better suited to describe the climatic variability embedded in these data. Our results indicate that Gaussian distribution fails to capture the large fluctuation -extreme events-that characterized climatic variability in these time series. This might have applications on the study of extreme weather events on future climate scenarios.

Based on the results obtained by analysing a large number of dynamic strike slip and dipping fault ruptures we construct a kinematic rupture model generator that incorporates some of the key source parameters extracted from the dynamic rupture models. In this kinematic rupture model, the slip rate function includes the final slip, the rise time, the local rupture velocity and the peak time (a measure of the impulsive part of the slip rate function). A four-dimensional correlation matrix is used to describe the spatial interdependency between the four source parameters—each of which is modelled as correlated random field. Each source parameter has a different marginal distribution determined from the analysis of the dynamic models. The marginal distributions are allowed to change as a function of distance from the hypocentre. The autocorrelation of each parameter is modelled by a power spectrum that follows a power law. The value of the spectral decay of the power law for the different source parameters is based on the results obtained from the dynamic rupture models. Finally, the values of rise time and peak time are adjusted such that the moment rate function fits a Brune spectrum for a specified corner frequency. We validate the rupture model generator using observed strong motion near field recording for the 1994 Northridge earthquake and the 1989 Loma Prieta earthquake. We perform comparisons with the rupture model generator of Liu et al. We find that only considering the response spectral bias of a best model is not sufficient to validate a rupture model generator. Thus we test the predictive power of the two rupture model generators if multiple ruptures are computed. Overall, the new rupture model generator yields a better prediction, compared to Liu et al., for the two validation events, especially in predicting observed PGV values and pseudospectral velocity at low frequencies (<1 Hz). [ABSTRACT FROM AUTHOR]

In this study we present evidence that nonlinearity can be directly observed in acceleration time histories such as those recorded at the Wildlife Refuge and Kushiro Port downhole arrays from the 1987 Superstition Hills, California, and the 1993 Kushiro-Oki, Japan, earthquakes, respectively. These accelerograms and others compiled in this study present a characteristic waveform composed of intermittent high-frequency peaks riding on a low-frequency carrier. In addition, soil amplification of the surface records is strongly observed compared to their downhole counterpart; this is contrary to the expected amplification reduction produced by the nonlinear soil behavior. Laboratory studies show that the physical mechanism that produces such phenomena is the dilatant nature of cohesionless soils, which introduces the partial recovery of the shear strength under cyclic loads. This recovery translates into the ability to produce large deformations followed by large and spiky shear stresses. The spikes observed in the acceleration records are directly related to these periods of dilatancy and generation of pore pressure. These results are significant in strong-motion seismology because these spikes produce large if not the largest acceleration. They are site related, not source related. Using the in situ observations from the Kushiro Port downhole array, we have modeled the 1993 Kushiro-Oki earthquake. The synthetic accelerograms show the development of intermittent behavior--high frequency peaks--as observed in the recorded acceleration time histories. Shear modulus degradation due to pore pressure produces large strains in the soil with large amplification in the low-frequency band of the ground motion. We also modeled data from the 1987 Superstition Hills earthquake recorded at the Wildlife Refuge station. The results show the importance of better soil characterization when pore pressure may develop and the effects of dilatancy in the understanding of nonlinear site response. [ABSTRACT FROM AUTHOR]