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.

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.

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 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.